# Bernoulli numbers

Bernoulli numbers
You are encouraged to solve this task according to the task description, using any language you may know.

Bernoulli numbers are used in some series expansions of several functions   (trigonometric, hyperbolic, gamma, etc.),   and are extremely important in number theory and analysis.

Note that there are two definitions of Bernoulli numbers;   this task will be using the modern usage   (as per   The National Institute of Standards and Technology convention).

The   nth   Bernoulli number is expressed as   Bn.

•   show the Bernoulli numbers   B0   through   B60.
•   suppress the output of values which are equal to zero.   (Other than   B1 , all odd Bernoulli numbers have a value of zero.)
•   express the Bernoulli numbers as fractions  (most are improper fractions).
•   the fractions should be reduced.
•   index each number in some way so that it can be discerned which Bernoulli number is being displayed.
•   align the solidi   (/)   if used (extra credit).

An algorithm

The Akiyama–Tanigawa algorithm for the "second Bernoulli numbers" as taken from wikipedia is as follows:

 for m from 0 by 1 to n do
A[m] ← 1/(m+1)
for j from m by -1 to 1 do
A[j-1] ← j×(A[j-1] - A[j])
return A[0] (which is Bn)


## ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

Uses Algol 68G's LONG LONG INT which has allows for large numbers of digits.

BEGIN    # Show the non-zero Bernoulli numbers B0 to B60                  #    # as rational numbers                                            #    # Uses code from the Arithmetic/Rational task modified to use    #    # LONG LONG INT to allow for the large number of digits requried #     PR precision 100 PR # sets the precision of LONG LONG INT        #     # Code from the Arithmetic/Rational task                         #    # ============================================================== #    MODE FRAC = STRUCT( LONG LONG INT num #erator#,  den #ominator#);    PROC gcd = (LONG LONG INT a, b) LONG LONG INT: # greatest common divisor #       (a = 0 | b |: b = 0 | a |: ABS a > ABS b  | gcd(b, a MOD b) | gcd(a, b MOD a));     PROC lcm = (LONG LONG INT a, b)LONG LONG INT: # least common multiple #       a OVER gcd(a, b) * b;     PRIO // = 9; # higher then the ** operator #     OP // = (LONG LONG INT num, den)FRAC: ( # initialise and normalise #       LONG LONG INT common = gcd(num, den);       IF den < 0 THEN         ( -num OVER common, -den OVER common)       ELSE         ( num OVER common, den OVER common)       FI     );    OP + = (FRAC a, b)FRAC: (       LONG LONG INT common = lcm(den OF a, den OF b);       FRAC result := ( common OVER den OF a * num OF a + common OVER den OF b * num OF b, common );       num OF result//den OF result    );    OP - = (FRAC a, b)FRAC: a + -b,       * = (FRAC a, b)FRAC: (           LONG LONG INT num = num OF a * num OF b,           den = den OF a * den OF b;           LONG LONG INT common = gcd(num, den);           (num OVER common) // (den OVER common)         );    OP - = (FRAC frac)FRAC: (-num OF frac, den OF frac);    # ============================================================== #    # end code from the Arithmetic/Rational task                     #     # Additional FRACrelated operators                               #    OP *  = ( INT a, FRAC b )FRAC: ( num OF b * a ) // den OF b;    OP // = ( INT a, INT  b )FRAC: LONG LONG INT( a ) // LONG LONG INT( b );     # returns the nth Bernoulli number, n must be >= 0               #    # Uses the algorithm suggested by the task, so B(1) is +1/2      #    PROC bernoulli = ( INT n )FRAC:         IF n < 0         THEN # n is out of range # 0 // 1         ELSE # n is valid        #            [ 0 : n ]FRAC a;            FOR i FROM LWB a TO UPB a DO a[ i ] := 0 // 1 OD;            FOR m FROM 0 TO n DO                 a[ m ] := 1 // ( m + 1 );                FOR j FROM m BY -1 TO 1 DO                    a[ j - 1 ] := j * ( a[ j - 1 ] - a[ j ] )                OD            OD;            a[ 0 ]         FI # bernoulli # ;     FOR n FROM 0 TO 60 DO        FRAC bn := bernoulli( n );        IF num OF bn /= 0 THEN            # have a non-0 Bn #            print( ( "B(", whole( n, -2 ), ") ", whole( num OF bn, -50 ), " / ", whole( den OF bn, 0 ), newline ) )        FI    ODEND
Output:
B( 0)                                                  1 / 1
B( 1)                                                  1 / 2
B( 2)                                                  1 / 6
B( 4)                                                 -1 / 30
B( 6)                                                  1 / 42
B( 8)                                                 -1 / 30
B(10)                                                  5 / 66
B(12)                                               -691 / 2730
B(14)                                                  7 / 6
B(16)                                              -3617 / 510
B(18)                                              43867 / 798
B(20)                                            -174611 / 330
B(22)                                             854513 / 138
B(24)                                         -236364091 / 2730
B(26)                                            8553103 / 6
B(28)                                       -23749461029 / 870
B(30)                                      8615841276005 / 14322
B(32)                                     -7709321041217 / 510
B(34)                                      2577687858367 / 6
B(36)                              -26315271553053477373 / 1919190
B(38)                                   2929993913841559 / 6
B(40)                             -261082718496449122051 / 13530
B(42)                             1520097643918070802691 / 1806
B(44)                           -27833269579301024235023 / 690
B(46)                           596451111593912163277961 / 282
B(48)                      -5609403368997817686249127547 / 46410
B(50)                        495057205241079648212477525 / 66
B(52)                    -801165718135489957347924991853 / 1590
B(54)                   29149963634884862421418123812691 / 798
B(56)                -2479392929313226753685415739663229 / 870
B(58)                84483613348880041862046775994036021 / 354
B(60)       -1215233140483755572040304994079820246041491 / 56786730


## C

Library: GMP
 #include <stdlib.h>#include <gmp.h> #define mpq_for(buf, op, n)\    do {\        size_t i;\        for (i = 0; i < (n); ++i)\            mpq_##op(buf[i]);\    } while (0) void bernoulli(mpq_t rop, unsigned int n){    unsigned int m, j;    mpq_t *a = malloc(sizeof(mpq_t) * (n + 1));    mpq_for(a, init, n + 1);     for (m = 0; m <= n; ++m) {        mpq_set_ui(a[m], 1, m + 1);        for (j = m; j > 0; --j) {            mpq_sub(a[j-1], a[j], a[j-1]);            mpq_set_ui(rop, j, 1);            mpq_mul(a[j-1], a[j-1], rop);        }    }     mpq_set(rop, a[0]);    mpq_for(a, clear, n + 1);    free(a);} int main(void){    mpq_t rop;    mpz_t n, d;    mpq_init(rop);    mpz_inits(n, d, NULL);     unsigned int i;    for (i = 0; i <= 60; ++i) {        bernoulli(rop, i);        if (mpq_cmp_ui(rop, 0, 1)) {            mpq_get_num(n, rop);            mpq_get_den(d, rop);            gmp_printf("B(%-2u) = %44Zd / %Zd\n", i, n, d);        }    }     mpz_clears(n, d, NULL);    mpq_clear(rop);    return 0;}
Output:
B(0 ) =                                            1 / 1
B(1 ) =                                           -1 / 2
B(2 ) =                                            1 / 6
B(4 ) =                                           -1 / 30
B(6 ) =                                            1 / 42
B(8 ) =                                           -1 / 30
B(10) =                                            5 / 66
B(12) =                                         -691 / 2730
B(14) =                                            7 / 6
B(16) =                                        -3617 / 510
B(18) =                                        43867 / 798
B(20) =                                      -174611 / 330
B(22) =                                       854513 / 138
B(24) =                                   -236364091 / 2730
B(26) =                                      8553103 / 6
B(28) =                                 -23749461029 / 870
B(30) =                                8615841276005 / 14322
B(32) =                               -7709321041217 / 510
B(34) =                                2577687858367 / 6
B(36) =                        -26315271553053477373 / 1919190
B(38) =                             2929993913841559 / 6
B(40) =                       -261082718496449122051 / 13530
B(42) =                       1520097643918070802691 / 1806
B(44) =                     -27833269579301024235023 / 690
B(46) =                     596451111593912163277961 / 282
B(48) =                -5609403368997817686249127547 / 46410
B(50) =                  495057205241079648212477525 / 66
B(52) =              -801165718135489957347924991853 / 1590
B(54) =             29149963634884862421418123812691 / 798
B(56) =          -2479392929313226753685415739663229 / 870
B(58) =          84483613348880041862046775994036021 / 354
B(60) = -1215233140483755572040304994079820246041491 / 56786730


## C++

### Using Boost | C++11

Library: boost
  /** * Configured with: --prefix=/Library/Developer/CommandLineTools/usr --with-gxx-include-dir=/usr/include/c++/4.2.1 * Apple LLVM version 9.1.0 (clang-902.0.39.1) * Target: x86_64-apple-darwin17.5.0 * Thread model: posix*/ #include <iostream> //std::cout#include <iostream> //formatting#include <vector> //Container#include <boost/rational.hpp> // Rationals#include <boost/multiprecision/cpp_int.hpp> //1024bit precision  typedef boost::rational<boost::multiprecision::int1024_t> rational; // reduce boilerplate rational bernulli(size_t n){      auto out = std::vector<rational>();      for(size_t m=0;m<=n;m++){         out.emplace_back(1,(m+1)); // automatically constructs object         for (size_t j = m;j>=1;j--){             out[j-1] = rational(j) * (out[j-1]-out[j]);         }     }     return out[0]; } int main() {    for(size_t n = 0; n <= 60;n+=n>=2?2:1){        auto b = bernulli(n);        std::cout << "B("<<std::right<<std::setw(2)<<n<<") = ";        std::cout << std::right<<std::setw(44)<<b.numerator();        std::cout << " / " << b.denominator() <<std::endl;    }     return 0;}
Output:
B( 0) =                                            1 / 1
B( 1) =                                            1 / 2
B( 2) =                                            1 / 6
B( 4) =                                           -1 / 30
B( 6) =                                            1 / 42
B( 8) =                                           -1 / 30
B(10) =                                            5 / 66
B(12) =                                         -691 / 2730
B(14) =                                            7 / 6
B(16) =                                        -3617 / 510
B(18) =                                        43867 / 798
B(20) =                                      -174611 / 330
B(22) =                                       854513 / 138
B(24) =                                   -236364091 / 2730
B(26) =                                      8553103 / 6
B(28) =                                 -23749461029 / 870
B(30) =                                8615841276005 / 14322
B(32) =                               -7709321041217 / 510
B(34) =                                2577687858367 / 6
B(36) =                        -26315271553053477373 / 1919190
B(38) =                             2929993913841559 / 6
B(40) =                       -261082718496449122051 / 13530
B(42) =                       1520097643918070802691 / 1806
B(44) =                     -27833269579301024235023 / 690
B(46) =                     596451111593912163277961 / 282
B(48) =                -5609403368997817686249127547 / 46410
B(50) =                  495057205241079648212477525 / 66
B(52) =              -801165718135489957347924991853 / 1590
B(54) =             29149963634884862421418123812691 / 798
B(56) =          -2479392929313226753685415739663229 / 870
B(58) =          84483613348880041862046775994036021 / 354
B(60) = -1215233140483755572040304994079820246041491 / 56786730


## C#

### Using Mpir.NET

Library: Mpir.NET

Translation of the C implementation

 using Mpir.NET;using System; namespace Bernoulli{    class Program    {        private static void bernoulli(mpq_t rop, uint n)        {            mpq_t[] a = new mpq_t[n + 1];             for (uint i = 0; i < n + 1; i++)            {                a[i] = new mpq_t();            }             for (uint m = 0; m <= n; ++m)            {                mpir.mpq_set_ui(a[m], 1, m + 1);                 for (uint j = m; j > 0; --j)                {                    mpir.mpq_sub(a[j - 1], a[j], a[j - 1]);                    mpir.mpq_set_ui(rop, j, 1);                    mpir.mpq_mul(a[j - 1], a[j - 1], rop);                }                 mpir.mpq_set(rop, a[0]);            }        }         static void Main(string[] args)        {            mpq_t rop = new mpq_t();            mpz_t n = new mpz_t();            mpz_t d = new mpz_t();             for (uint  i = 0; i <= 60; ++i)             {                bernoulli(rop, i);                 if (mpir.mpq_cmp_ui(rop, 0, 1) != 0)                 {                    mpir.mpq_get_num(n, rop);                    mpir.mpq_get_den(d, rop);                    Console.WriteLine(string.Format("B({0, 2}) = {1, 44} / {2}", i, n, d));                }            }             Console.ReadKey();        }    }}
Output:
B(0 ) =                                            1 / 1
B(1 ) =                                           -1 / 2
B(2 ) =                                            1 / 6
B(4 ) =                                           -1 / 30
B(6 ) =                                            1 / 42
B(8 ) =                                           -1 / 30
B(10) =                                            5 / 66
B(12) =                                         -691 / 2730
B(14) =                                            7 / 6
B(16) =                                        -3617 / 510
B(18) =                                        43867 / 798
B(20) =                                      -174611 / 330
B(22) =                                       854513 / 138
B(24) =                                   -236364091 / 2730
B(26) =                                      8553103 / 6
B(28) =                                 -23749461029 / 870
B(30) =                                8615841276005 / 14322
B(32) =                               -7709321041217 / 510
B(34) =                                2577687858367 / 6
B(36) =                        -26315271553053477373 / 1919190
B(38) =                             2929993913841559 / 6
B(40) =                       -261082718496449122051 / 13530
B(42) =                       1520097643918070802691 / 1806
B(44) =                     -27833269579301024235023 / 690
B(46) =                     596451111593912163277961 / 282
B(48) =                -5609403368997817686249127547 / 46410
B(50) =                  495057205241079648212477525 / 66
B(52) =              -801165718135489957347924991853 / 1590
B(54) =             29149963634884862421418123812691 / 798
B(56) =          -2479392929313226753685415739663229 / 870
B(58) =          84483613348880041862046775994036021 / 354
B(60) = -1215233140483755572040304994079820246041491 / 56786730


### Using Math.NET

 using System;using System.Console;using System.Linq;using MathNet.Numerics; namespace Rosettacode.Rational.CS{    class Program    {        private static readonly Func<int, BigRational> ℚ = BigRational.FromInt;         private static BigRational CalculateBernoulli(int n)        {            var a = InitializeArray(n);             foreach(var m in Enumerable.Range(1,n))            {                a[m] = ℚ(1) / (ℚ(m) + ℚ(1));                 for (var j = m; j >= 1; j--)                {                    a[j-1] = ℚ(j) * (a[j-1] - a[j]);                }            }             return a[0];        }         private static BigRational[] InitializeArray(int n)        {            var a = new BigRational[n + 1];             for (var x = 0; x < a.Length; x++)            {                a[x] = ℚ(x + 1);            }             return a;        }         static void Main()        {            Enumerable.Range(0, 61) // the second parameter is the number of range elements, and is not the final item of the range.                .Select(n => new {N = n, BernoulliNumber = CalculateBernoulli(n)})                .Where(b => !b.BernoulliNumber.Numerator.IsZero)                .Select(b => string.Format("B({0, 2}) = {1, 44} / {2}", b.N, b.BernoulliNumber.Numerator, b.BernoulliNumber.Denominator))                .ToList()                .ForEach(WriteLine);        }    }}
Output:
B( 0) =                                            1 / 1
B( 1) =                                            1 / 2
B( 2) =                                            1 / 6
B( 4) =                                           -1 / 30
B( 6) =                                            1 / 42
B( 8) =                                           -1 / 30
B(10) =                                            5 / 66
B(12) =                                         -691 / 2730
B(14) =                                            7 / 6
B(16) =                                        -3617 / 510
B(18) =                                        43867 / 798
B(20) =                                      -174611 / 330
B(22) =                                       854513 / 138
B(24) =                                   -236364091 / 2730
B(26) =                                      8553103 / 6
B(28) =                                 -23749461029 / 870
B(30) =                                8615841276005 / 14322
B(32) =                               -7709321041217 / 510
B(34) =                                2577687858367 / 6
B(36) =                        -26315271553053477373 / 1919190
B(38) =                             2929993913841559 / 6
B(40) =                       -261082718496449122051 / 13530
B(42) =                       1520097643918070802691 / 1806
B(44) =                     -27833269579301024235023 / 690
B(46) =                     596451111593912163277961 / 282
B(48) =                -5609403368997817686249127547 / 46410
B(50) =                  495057205241079648212477525 / 66
B(52) =              -801165718135489957347924991853 / 1590
B(54) =             29149963634884862421418123812691 / 798
B(56) =          -2479392929313226753685415739663229 / 870
B(58) =          84483613348880041862046775994036021 / 354
B(60) = -1215233140483755572040304994079820246041491 / 56786730


### Using System.Numerics

Algo based on the example provided in the header of this RC page (the one from Wikipedia).
Extra feature - one can override the default of 60 by supplying a suitable number on the command line. The column widths are not hard-coded, but will adapt to the widths of the items listed.

using System;using System.Numerics;using System.Collections.Generic; namespace bern{    class Program    {        struct BerNum { public int index; public BigInteger Numer, Denomin; };        static int w1 = 1, w2 = 1; // widths for formatting output        static int max = 60; // default maximum, can override on command line         // returns nth Bernoulli number        static BerNum CalcBernoulli(int n)        {            BerNum res;            BigInteger f;            BigInteger[] nu = new BigInteger[n + 1],                         de = new BigInteger[n + 1];            for (int m = 0; m <= n; m++)            {                nu[m] = 1; de[m] = m + 1;                for (int j = m; j > 0; j--)                    if ((f = BigInteger.GreatestCommonDivisor(                        nu[j - 1] = j * (de[j] * nu[j - 1] - de[j - 1] * nu[j]),                        de[j - 1] *= de[j])) != BigInteger.One)                    { nu[j - 1] /= f; de[j - 1] /= f; }            }            res.index = n; res.Numer = nu[0]; res.Denomin = de[0];            w1 = Math.Max(n.ToString().Length, w1);             // ratchet up widths            w2 = Math.Max(res.Numer.ToString().Length, w2);            if (max > 50) Console.Write("."); // progress dots appear for larger values            return res;        }         static void Main(string[] args)        {            List<BerNum> BNumbList = new List<BerNum>();            // defaults to 60 when no (or invalid) command line parameter is present            if (args.Length > 0) {                int.TryParse(args[0], out max);                if (max < 1 || max > Int16.MaxValue) max = 60;                if (args[0] == "0") max = 0;            }            for (int i = 0; i <= max; i++) // fill list with values            {                BerNum BNumb = CalcBernoulli(i);                if (BNumb.Numer != BigInteger.Zero) BNumbList.Add(BNumb);            }            if (max > 50) Console.WriteLine();            string strFmt = "B({0, " + w1.ToString() + "}) = {1, " + w2.ToString() + "} / {2}";            // display formatted list            foreach (BerNum bn in BNumbList)                Console.WriteLine(strFmt , bn.index, bn.Numer, bn.Denomin);            if (System.Diagnostics.Debugger.IsAttached) Console.Read();        }    }}
Output:

Default (nothing entered on command line):

.............................................................
B( 0) =                                            1 / 1
B( 1) =                                            1 / 2
B( 2) =                                            1 / 6
B( 4) =                                           -1 / 30
B( 6) =                                            1 / 42
B( 8) =                                           -1 / 30
B(10) =                                            5 / 66
B(12) =                                         -691 / 2730
B(14) =                                            7 / 6
B(16) =                                        -3617 / 510
B(18) =                                        43867 / 798
B(20) =                                      -174611 / 330
B(22) =                                       854513 / 138
B(24) =                                   -236364091 / 2730
B(26) =                                      8553103 / 6
B(28) =                                 -23749461029 / 870
B(30) =                                8615841276005 / 14322
B(32) =                               -7709321041217 / 510
B(34) =                                2577687858367 / 6
B(36) =                        -26315271553053477373 / 1919190
B(38) =                             2929993913841559 / 6
B(40) =                       -261082718496449122051 / 13530
B(42) =                       1520097643918070802691 / 1806
B(44) =                     -27833269579301024235023 / 690
B(46) =                     596451111593912163277961 / 282
B(48) =                -5609403368997817686249127547 / 46410
B(50) =                  495057205241079648212477525 / 66
B(52) =              -801165718135489957347924991853 / 1590
B(54) =             29149963634884862421418123812691 / 798
B(56) =          -2479392929313226753685415739663229 / 870
B(58) =          84483613348880041862046775994036021 / 354
B(60) = -1215233140483755572040304994079820246041491 / 56786730

Output with "8" entered on command line:

B(0) =  1 / 1
B(1) =  1 / 2
B(2) =  1 / 6
B(4) = -1 / 30
B(6) =  1 / 42
B(8) = -1 / 30

Output with "126" entered on the command line:

...............................................................................................................................
B(  0) =                                                                                                                      1 / 1
B(  1) =                                                                                                                      1 / 2
B(  2) =                                                                                                                      1 / 6
B(  4) =                                                                                                                     -1 / 30
B(  6) =                                                                                                                      1 / 42
B(  8) =                                                                                                                     -1 / 30
B( 10) =                                                                                                                      5 / 66
B( 12) =                                                                                                                   -691 / 2730
B( 14) =                                                                                                                      7 / 6
B( 16) =                                                                                                                  -3617 / 510
B( 18) =                                                                                                                  43867 / 798
B( 20) =                                                                                                                -174611 / 330
B( 22) =                                                                                                                 854513 / 138
B( 24) =                                                                                                             -236364091 / 2730
B( 26) =                                                                                                                8553103 / 6
B( 28) =                                                                                                           -23749461029 / 870
B( 30) =                                                                                                          8615841276005 / 14322
B( 32) =                                                                                                         -7709321041217 / 510
B( 34) =                                                                                                          2577687858367 / 6
B( 36) =                                                                                                  -26315271553053477373 / 1919190
B( 38) =                                                                                                       2929993913841559 / 6
B( 40) =                                                                                                 -261082718496449122051 / 13530
B( 42) =                                                                                                 1520097643918070802691 / 1806
B( 44) =                                                                                               -27833269579301024235023 / 690
B( 46) =                                                                                               596451111593912163277961 / 282
B( 48) =                                                                                          -5609403368997817686249127547 / 46410
B( 50) =                                                                                            495057205241079648212477525 / 66
B( 52) =                                                                                        -801165718135489957347924991853 / 1590
B( 54) =                                                                                       29149963634884862421418123812691 / 798
B( 56) =                                                                                    -2479392929313226753685415739663229 / 870
B( 58) =                                                                                    84483613348880041862046775994036021 / 354
B( 60) =                                                                           -1215233140483755572040304994079820246041491 / 56786730
B( 62) =                                                                                 12300585434086858541953039857403386151 / 6
B( 64) =                                                                            -106783830147866529886385444979142647942017 / 510
B( 66) =                                                                         1472600022126335654051619428551932342241899101 / 64722
B( 68) =                                                                          -78773130858718728141909149208474606244347001 / 30
B( 70) =                                                                      1505381347333367003803076567377857208511438160235 / 4686
B( 72) =                                                               -5827954961669944110438277244641067365282488301844260429 / 140100870
B( 74) =                                                                     34152417289221168014330073731472635186688307783087 / 6
B( 76) =                                                                 -24655088825935372707687196040585199904365267828865801 / 30
B( 78) =                                                              414846365575400828295179035549542073492199375372400483487 / 3318
B( 80) =                                                         -4603784299479457646935574969019046849794257872751288919656867 / 230010
B( 82) =                                                          1677014149185145836823154509786269900207736027570253414881613 / 498
B( 84) =                                                   -2024576195935290360231131160111731009989917391198090877281083932477 / 3404310
B( 86) =                                                        660714619417678653573847847426261496277830686653388931761996983 / 6
B( 88) =                                                -1311426488674017507995511424019311843345750275572028644296919890574047 / 61410
B( 90) =                                              1179057279021082799884123351249215083775254949669647116231545215727922535 / 272118
B( 92) =                                             -1295585948207537527989427828538576749659341483719435143023316326829946247 / 1410
B( 94) =                                              1220813806579744469607301679413201203958508415202696621436215105284649447 / 6
B( 96) =                                     -211600449597266513097597728109824233673043954389060234150638733420050668349987259 / 4501770
B( 98) =                                          67908260672905495624051117546403605607342195728504487509073961249992947058239 / 6
B(100) =                                   -94598037819122125295227433069493721872702841533066936133385696204311395415197247711 / 33330
B(102) =                                  3204019410860907078243020782116241775491817197152717450679002501086861530836678158791 / 4326
B(104) =                               -319533631363830011287103352796174274671189606078272738327103470162849568365549721224053 / 1590
B(106) =                              36373903172617414408151820151593427169231298640581690038930816378281879873386202346572901 / 642
B(108) =                     -3469342247847828789552088659323852541399766785760491146870005891371501266319724897592306597338057 / 209191710
B(110) =                         7645992940484742892248134246724347500528752413412307906683593870759797606269585779977930217515 / 1518
B(112) =                  -2650879602155099713352597214685162014443151499192509896451788427680966756514875515366781203552600109 / 1671270
B(114) =                     21737832319369163333310761086652991475721156679090831360806110114933605484234593650904188618562649 / 42
B(116) =                -309553916571842976912513458033841416869004128064329844245504045721008957524571968271388199595754752259 / 1770
B(118) =                 366963119969713111534947151585585006684606361080699204301059440676414485045806461889371776354517095799 / 6
B(120) =     -51507486535079109061843996857849983274095170353262675213092869167199297474922985358811329367077682677803282070131 / 2328255930
B(122) =            49633666079262581912532637475990757438722790311060139770309311793150683214100431329033113678098037968564431 / 6
B(124) =        -95876775334247128750774903107542444620578830013297336819553512729358593354435944413631943610268472689094609001 / 30
B(126) = 5556330281949274850616324408918951380525567307126747246796782304333594286400508981287241419934529638692081513802696639 / 4357878


## Crystal

Translation of Ruby.

require "big" class Bernoulli  include Iterator(Tuple(Int32, BigRational))   def initialize    @a = [] of BigRational    @m = 0  end   def next    @a << BigRational.new(1, @m+1)    @m.downto(1) { |j| @a[j-1] = j*(@a[j-1] - @a[j]) }    v = @m.odd? && @m != 1 ? BigRational.new(0, 1) : @a.first    return {@m, v}  ensure    @m += 1  endend b = Bernoulli.newbn = b.first(61).to_a max_width = bn.map { |_, v| v.numerator.to_s.size }.maxbn.reject { |i, v| v.zero? }.each do |i, v|  puts "B(%2i) = %*i/%i" % [i, max_width, v.numerator, v.denominator]end
Output:
B( 0) =                                            1/1
B( 1) =                                            1/2
B( 2) =                                            1/6
B( 4) =                                           -1/30
B( 6) =                                            1/42
B( 8) =                                           -1/30
B(10) =                                            5/66
B(12) =                                         -691/2730
B(14) =                                            7/6
B(16) =                                        -3617/510
B(18) =                                        43867/798
B(20) =                                      -174611/330
B(22) =                                       854513/138
B(24) =                                   -236364091/2730
B(26) =                                      8553103/6
B(28) =                                 -23749461029/870
B(30) =                                8615841276005/14322
B(32) =                               -7709321041217/510
B(34) =                                2577687858367/6
B(36) =                        -26315271553053477373/1919190
B(38) =                             2929993913841559/6
B(40) =                       -261082718496449122051/13530
B(42) =                       1520097643918070802691/1806
B(44) =                     -27833269579301024235023/690
B(46) =                     596451111593912163277961/282
B(48) =                -5609403368997817686249127547/46410
B(50) =                  495057205241079648212477525/66
B(52) =              -801165718135489957347924991853/1590
B(54) =             29149963634884862421418123812691/798
B(56) =          -2479392929313226753685415739663229/870
B(58) =          84483613348880041862046775994036021/354
B(60) = -1215233140483755572040304994079820246041491/56786730


## Clojure

  ns test-project-intellij.core  (:gen-class)) (defn a-t [n]  " Used Akiyama-Tanigawa algorithm with a single loop rather than double nested loop "  " Clojure does fractional arithmetic automatically so that part is easy "  (loop [m 0         j m         A (vec (map #(/ 1 %) (range 1 (+ n 2))))] ; Prefil A(m) with 1/(m+1), for m = 1 to n    (cond                                          ; Three way conditional allows single loop      (>= j 1) (recur m (dec j) (assoc A (dec j) (* j (- (nth A (dec j)) (nth A j))))) ; A[j-1] ← j×(A[j-1] - A[j]) ;      (< m n) (recur (inc m) (inc m) A)                                                 ; increment m, reset j = m      :else (nth A 0)))) (defn format-ans [ans]  " Formats answer so that '/' is aligned for all answers "  (if (= ans 1)  (format "%50d / %8d" 1 1)  (format "%50d / %8d" (numerator ans) (denominator ans)))) ;; Generate a set of results for [0 1 2 4 ... 60](doseq [q (flatten [0 1 (range 2 62 2)])        :let [ans (a-t q)]]  (println q ":" (format-ans ans)))
Output:
0 :                                                  1 /        1
1 :                                                  1 /        2
2 :                                                  1 /        6
4 :                                                 -1 /       30
6 :                                                  1 /       42
8 :                                                 -1 /       30
10 :                                                  5 /       66
12 :                                               -691 /     2730
14 :                                                  7 /        6
16 :                                              -3617 /      510
18 :                                              43867 /      798
20 :                                            -174611 /      330
22 :                                             854513 /      138
24 :                                         -236364091 /     2730
26 :                                            8553103 /        6
28 :                                       -23749461029 /      870
30 :                                      8615841276005 /    14322
32 :                                     -7709321041217 /      510
34 :                                      2577687858367 /        6
36 :                              -26315271553053477373 /  1919190
38 :                                   2929993913841559 /        6
40 :                             -261082718496449122051 /    13530
42 :                             1520097643918070802691 /     1806
44 :                           -27833269579301024235023 /      690
46 :                           596451111593912163277961 /      282
48 :                      -5609403368997817686249127547 /    46410
50 :                        495057205241079648212477525 /       66
52 :                    -801165718135489957347924991853 /     1590
54 :                   29149963634884862421418123812691 /      798
56 :                -2479392929313226753685415739663229 /      870
58 :                84483613348880041862046775994036021 /      354
60 :       -1215233140483755572040304994079820246041491 / 56786730


## Common Lisp

An implementation of the simple algorithm.

Be advised that the pseudocode algorithm specifies (j * (a[j-1] - a[j])) in the inner loop; implementing that as-is gives the wrong value (1/2) where n = 1, whereas subtracting a[j]-a[j-1] yields the correct value (B[1]=-1/2). See the numerator list.

(defun bernouilli (n)  (loop with a = (make-array (list (1+ n)))     for m from 0 to n do       (setf (aref a m) (/ 1 (+ m 1)))       (loop for j from m downto 1 do            (setf (aref a (- j 1))                  (* j (- (aref a j) (aref a (- j 1))))))     finally (return (aref a 0)))) ;;Print outputs to stdout: (loop for n from 0 to 60 do     (let ((b (bernouilli n)))       (when (not (zerop b))         (format t "~a: ~a~%" n b))))  ;;For the "extra credit" challenge, we need to align the slashes. (let (results)  ;;collect the results  (loop for n from 0 to 60 do       (let ((b (bernouilli n)))         (when (not (zerop b)) (push (cons b n) results))))  ;;parse the numerators into strings; save the greatest length in max-length  (let ((max-length (apply #'max (mapcar (lambda (r)                                           (length (format nil "~a" (numerator r))))                                         (mapcar #'car results)))))    ;;Print the numbers with using the fixed-width formatter: ~Nd, where N is    ;;the number of leading spaces. We can't just pass in the width variable    ;;but we can splice together a formatting string that includes it.     ;;We also can't use the fixed-width formatter on a ratio, so we have to split    ;;the ratio and splice it back together like idiots.    (loop for n in (mapcar #'cdr (reverse results))          for r in (mapcar #'car (reverse results)) do         (format t (concatenate 'string                                "B(~2d): ~"                                (format nil "~a" max-length)                                "d/~a~%")                 n                 (numerator r)                 (denominator r)))))
Output:
B( 0):                                            1/1
B( 1):                                           -1/2
B( 2):                                            1/6
B( 4):                                           -1/30
B( 6):                                            1/42
B( 8):                                           -1/30
B(10):                                            5/66
B(12):                                         -691/2730
B(14):                                            7/6
B(16):                                        -3617/510
B(18):                                        43867/798
B(20):                                      -174611/330
B(22):                                       854513/138
B(24):                                   -236364091/2730
B(26):                                      8553103/6
B(28):                                 -23749461029/870
B(30):                                8615841276005/14322
B(32):                               -7709321041217/510
B(34):                                2577687858367/6
B(36):                        -26315271553053477373/1919190
B(38):                             2929993913841559/6
B(40):                       -261082718496449122051/13530
B(42):                       1520097643918070802691/1806
B(44):                     -27833269579301024235023/690
B(46):                     596451111593912163277961/282
B(48):                -5609403368997817686249127547/46410
B(50):                  495057205241079648212477525/66
B(52):              -801165718135489957347924991853/1590
B(54):             29149963634884862421418123812691/798
B(56):          -2479392929313226753685415739663229/870
B(58):          84483613348880041862046775994036021/354
B(60): -1215233140483755572040304994079820246041491/56786730


## D

This uses the D module from the Arithmetic/Rational task.

Translation of: Python
import std.stdio, std.range, std.algorithm, std.conv, arithmetic_rational; auto bernoulli(in uint n) pure nothrow /*@safe*/ {    auto A = new Rational[n + 1];    foreach (immutable m; 0 .. n + 1) {        A[m] = Rational(1, m + 1);        foreach_reverse (immutable j; 1 .. m + 1)            A[j - 1] = j * (A[j - 1] - A[j]);    }    return A[0];} void main() {    immutable berns = 61.iota.map!bernoulli.enumerate.filter!(t => t[1]).array;    immutable width = berns.map!(b => b[1].numerator.text.length).reduce!max;    foreach (immutable b; berns)        writefln("B(%2d) = %*d/%d", b[0], width, b[1].tupleof);}

The output is exactly the same as the Python entry.

## EchoLisp

 This example is in need of improvement: Try to show B1   within the output proper as   -1/2.

Only 'small' rationals are supported in EchoLisp, i.e numerator and demominator < 2^31. So, we create a class of 'large' rationals, supported by the bigint library, and then apply the magic formula.

 (lib 'bigint) ;; lerge numbers(lib 'gloops) ;; classes (define-class Rational null ((a :initform #0) (b :initform #1)))(define-method tostring  (Rational) (lambda (r) (format "%50d / %d" r.a r.b)))(define-method normalize (Rational) (lambda (r) ;; divide a and b by gcd		 (let ((g (gcd r.a r.b)))		 (set! r.a (/ r.a g)) (set! r.b (/ r.b g)) 		 (when (< r.b 0) (set! r.a ( - r.a)) (set! r.b (- r.b))) ;; denominator > 0 		r))) (define-method initialize (Rational) (lambda (r) (normalize r)))(define-method add (Rational) (lambda (r n)  ;; + Rational any number			(normalize (Rational (+ (* (+ #0 n) r.b) r.a) r.b))))(define-method add (Rational Rational) (lambda (r q) ;;; + Rational Rational			(normalize (Rational (+ (* r.a q.b) (* r.b q.a)) (* r.b q.b)))))(define-method sub (Rational Rational) (lambda (r q) 			(normalize (Rational (- (* r.a q.b) (* r.b q.a)) (* r.b q.b)))))(define-method mul (Rational Rational) (lambda (r q) 			(normalize (Rational  (* r.a q.a)  (* r.b q.b)))))(define-method mul (Rational) (lambda (r n) 			(normalize (Rational  (* r.a (+ #0 n))  r.b ))))(define-method div (Rational Rational) (lambda (r q) 			(normalize (Rational  (* r.a q.b)  (* r.b q.a)))))
Output:
 ;; Bernoulli numbers;; http://rosettacode.org/wiki/Bernoulli_numbers(define A (make-vector 100 0)) (define (B n)(for ((m (1+ n))) ;; #1 creates a large integer	(vector-set! A m (Rational #1 (+ #1 m)))	(for ((j (in-range m 0 -1)))	  (vector-set! A (1- j) 	  	(mul (sub (vector-ref A (1- j)) (vector-ref A j)) j))))	  (vector-ref A 0))     (for ((b (in-range 0 62 2))) (writeln b (B b)))  →  0                                                      1 / 1    2                                                      1 / 6    4                                                     -1 / 30    6                                                      1 / 42    8                                                     -1 / 30    10                                                      5 / 66    12                                                   -691 / 2730    14                                                      7 / 6    16                                                  -3617 / 510    18                                                  43867 / 798    20                                                -174611 / 330    22                                                 854513 / 138    24                                             -236364091 / 2730    26                                                8553103 / 6    28                                           -23749461029 / 870    30                                          8615841276005 / 14322    32                                         -7709321041217 / 510    34                                          2577687858367 / 6    36                                  -26315271553053477373 / 1919190    38                                       2929993913841559 / 6    40                                 -261082718496449122051 / 13530    42                                 1520097643918070802691 / 1806    44                               -27833269579301024235023 / 690    46                               596451111593912163277961 / 282    48                          -5609403368997817686249127547 / 46410    50                            495057205241079648212477525 / 66    52                        -801165718135489957347924991853 / 1590    54                       29149963634884862421418123812691 / 798    56                    -2479392929313226753685415739663229 / 870    58                    84483613348880041862046775994036021 / 354    60           -1215233140483755572040304994079820246041491 / 56786730       (B 1) → 1 / 2

## Elixir

defmodule Bernoulli do  defmodule Rational do    import Kernel, except: [div: 2]     defstruct numerator: 0, denominator: 1     def new(numerator, denominator\\1) do      sign = if numerator * denominator < 0, do: -1, else: 1      {numerator, denominator} = {abs(numerator), abs(denominator)}      gcd = gcd(numerator, denominator)      %Rational{numerator: sign * Kernel.div(numerator, gcd),                denominator: Kernel.div(denominator, gcd)}    end     def sub(a, b) do      new(a.numerator * b.denominator - b.numerator * a.denominator,          a.denominator * b.denominator)    end     def mul(a, b) when is_integer(a) do      new(a * b.numerator, b.denominator)    end     defp gcd(a,0), do: a    defp gcd(a,b), do: gcd(b, rem(a,b))  end   def numbers(n) do    Stream.transform(0..n, {}, fn m,acc ->      acc = Tuple.append(acc, Rational.new(1,m+1))      if m>0 do        new =           Enum.reduce(m..1, acc, fn j,ar ->            put_elem(ar, j-1, Rational.mul(j, Rational.sub(elem(ar,j-1), elem(ar,j))))          end)        {[elem(new,0)], new}      else        {[elem(acc,0)], acc}      end    end) |> Enum.to_list  end   def task(n \\ 61) do    b_nums = numbers(n)    width  = Enum.map(b_nums, fn b -> b.numerator |> to_string |> String.length end)             |> Enum.max    format = 'B(~2w) = ~#{width}w / ~w~n'    Enum.with_index(b_nums)    |> Enum.each(fn {b,i} ->         if b.numerator != 0, do: :io.fwrite format, [i, b.numerator, b.denominator]       end)  endend Bernoulli.task
Output:
B( 0) =                                            1 / 1
B( 1) =                                            1 / 2
B( 2) =                                            1 / 6
B( 4) =                                           -1 / 30
B( 6) =                                            1 / 42
B( 8) =                                           -1 / 30
B(10) =                                            5 / 66
B(12) =                                         -691 / 2730
B(14) =                                            7 / 6
B(16) =                                        -3617 / 510
B(18) =                                        43867 / 798
B(20) =                                      -174611 / 330
B(22) =                                       854513 / 138
B(24) =                                   -236364091 / 2730
B(26) =                                      8553103 / 6
B(28) =                                 -23749461029 / 870
B(30) =                                8615841276005 / 14322
B(32) =                               -7709321041217 / 510
B(34) =                                2577687858367 / 6
B(36) =                        -26315271553053477373 / 1919190
B(38) =                             2929993913841559 / 6
B(40) =                       -261082718496449122051 / 13530
B(42) =                       1520097643918070802691 / 1806
B(44) =                     -27833269579301024235023 / 690
B(46) =                     596451111593912163277961 / 282
B(48) =                -5609403368997817686249127547 / 46410
B(50) =                  495057205241079648212477525 / 66
B(52) =              -801165718135489957347924991853 / 1590
B(54) =             29149963634884862421418123812691 / 798
B(56) =          -2479392929313226753685415739663229 / 870
B(58) =          84483613348880041862046775994036021 / 354
B(60) = -1215233140483755572040304994079820246041491 / 56786730


## F#

 open MathNet.Numericsopen Systemopen System.Collections.Generic let calculateBernoulli n =    let ℚ(x) = BigRational.FromInt x    let A = Array.init<BigRational> (n+1) (fun x -> ℚ(x+1))     for m in [1..n] do        A.[m] <- ℚ(1) / (ℚ(m) + ℚ(1))        for j in [m..(-1)..1] do            A.[j-1] <- ℚ(j) * (A.[j-1] - A.[j])    A.[0] [<EntryPoint>]let main argv =     for n in [0..60] do        let bernoulliNumber = calculateBernoulli n        match bernoulliNumber.Numerator.IsZero with         | false ->             let formatedString = String.Format("B({0, 2}) = {1, 44} / {2}", n, bernoulliNumber.Numerator, bernoulliNumber.Denominator)            printfn "%s" formatedString        | true ->             printf ""    0
Output:
B( 0) =                                            1 / 1
B( 1) =                                            1 / 2
B( 2) =                                            1 / 6
B( 4) =                                           -1 / 30
B( 6) =                                            1 / 42
B( 8) =                                           -1 / 30
B(10) =                                            5 / 66
B(12) =                                         -691 / 2730
B(14) =                                            7 / 6
B(16) =                                        -3617 / 510
B(18) =                                        43867 / 798
B(20) =                                      -174611 / 330
B(22) =                                       854513 / 138
B(24) =                                   -236364091 / 2730
B(26) =                                      8553103 / 6
B(28) =                                 -23749461029 / 870
B(30) =                                8615841276005 / 14322
B(32) =                               -7709321041217 / 510
B(34) =                                2577687858367 / 6
B(36) =                        -26315271553053477373 / 1919190
B(38) =                             2929993913841559 / 6
B(40) =                       -261082718496449122051 / 13530
B(42) =                       1520097643918070802691 / 1806
B(44) =                     -27833269579301024235023 / 690
B(46) =                     596451111593912163277961 / 282
B(48) =                -5609403368997817686249127547 / 46410
B(50) =                  495057205241079648212477525 / 66
B(52) =              -801165718135489957347924991853 / 1590
B(54) =             29149963634884862421418123812691 / 798
B(56) =          -2479392929313226753685415739663229 / 870
B(58) =          84483613348880041862046775994036021 / 354
B(60) = -1215233140483755572040304994079820246041491 / 56786730


## Factor

One could use the "bernoulli" word from the math.extras vocabulary as follows:

IN: scratchpad    [      0  1 1 "%2d : %d / %d\n" printf      1 -1 2 "%2d : %d / %d\n" printf      30 iota [        1 + 2 * dup bernoulli [ numerator ] [ denominator ] bi        "%2d : %d / %d\n" printf      ] each    ] time 0 : 1 / 1 1 : -1 / 2 2 : 1 / 6 4 : -1 / 30 6 : 1 / 42 8 : -1 / 3010 : 5 / 6612 : -691 / 273014 : 7 / 616 : -3617 / 51018 : 43867 / 79820 : -174611 / 33022 : 854513 / 13824 : -236364091 / 273026 : 8553103 / 628 : -23749461029 / 87030 : 8615841276005 / 1432232 : -7709321041217 / 51034 : 2577687858367 / 636 : -26315271553053477373 / 191919038 : 2929993913841559 / 640 : -261082718496449122051 / 1353042 : 1520097643918070802691 / 180644 : -27833269579301024235023 / 69046 : 596451111593912163277961 / 28248 : -5609403368997817686249127547 / 4641050 : 495057205241079648212477525 / 6652 : -801165718135489957347924991853 / 159054 : 29149963634884862421418123812691 / 79856 : -2479392929313226753685415739663229 / 87058 : 84483613348880041862046775994036021 / 35460 : -1215233140483755572040304994079820246041491 / 56786730Running time: 0.00489444 seconds

Alternatively a method described by Brent and Harvey (2011) in "Fast computation of Bernoulli, Tangent and Secant numbers" https://arxiv.org/pdf/1108.0286.pdf is shown.

:: bernoulli-numbers ( n -- )  n 1 + 0 <array> :> tab  1 1 tab set-nth  2 n [a,b] [| k |    k 1 - dup    tab nth *             k tab set-nth      ] each  2 n [a,b] [| k |       k n [a,b] [| j |         j tab nth            j k - 2 + *          j 1 - tab nth        j k - * +            j tab set-nth    ] each  ] each  1 :> s!  1 n [a,b] [| k |    k 2 * dup             2^ dup 1 - *             k tab nth             swap / *              s * k tab set-nth     s -1 * s!  ] each   0  1 1 "%2d : %d / %d\n" printf  1 -1 2 "%2d : %d / %d\n" printf  1 n [a,b] [| k |    k 2 * k tab nth    [ numerator ] [ denominator ] bi    "%2d : %d / %d\n" printf  ] each;

It gives the same result as the native implementation, but is slightly faster.

[ 30 bernoulli-numbers ] time...Running time: 0.004331652 seconds

## FreeBASIC

Library: GMP
' version 08-10-2016' compile with: fbc -s console' uses gmp #Include Once "gmp.bi" #Define max 60 Dim As Long nDim As ZString Ptr gmp_str :gmp_str = Allocate(1000) ' 1000 charDim Shared As Mpq_ptr tmp, big_jtmp = Allocate(Len(__mpq_struct)) :Mpq_init(tmp)big_j = Allocate(Len(__mpq_struct)) :Mpq_init(big_j) Dim Shared As Mpq_ptr a(max), b(max)For n = 0 To max  A(n) = Allocate(Len(__mpq_struct)) :Mpq_init(A(n))  B(n) = Allocate(Len(__mpq_struct)) :Mpq_init(B(n))Next Function Bernoulli(n As Integer) As Mpq_ptr   Dim As Long m, j   For m = 0 To n    Mpq_set_ui(A(m), 1, m + 1)    For j = m To 1 Step - 1      Mpq_sub(tmp, A(j - 1), A(j))      Mpq_set_ui(big_j, j, 1)                 'big_j = j      Mpq_mul(A(j - 1), big_j, tmp)    Next  Next   Return A(0)End Function ' ------=< MAIN >=------ For n = 0 To max  Mpq_set(B(n), Bernoulli(n))  Mpq_get_str(gmp_str, 10, B(n))  If *gmp_str <> "0" Then    If *gmp_str = "1" Then *gmp_str = "1/1"    Print Using "B(##) = "; n;    Print Space(45 - InStr(*gmp_str, "/")); *gmp_str  End IfNext  ' empty keyboard bufferWhile Inkey <> "" :WendPrint :Print "hit any key to end program"SleepEnd
Output:
B( 0) =                                            1/1
B( 1) =                                            1/2
B( 2) =                                            1/6
B( 4) =                                           -1/30
B( 6) =                                            1/42
B( 8) =                                           -1/30
B(10) =                                            5/66
B(12) =                                         -691/2730
B(14) =                                            7/6
B(16) =                                        -3617/510
B(18) =                                        43867/798
B(20) =                                      -174611/330
B(22) =                                       854513/138
B(24) =                                   -236364091/2730
B(26) =                                      8553103/6
B(28) =                                 -23749461029/870
B(30) =                                8615841276005/14322
B(32) =                               -7709321041217/510
B(34) =                                2577687858367/6
B(36) =                        -26315271553053477373/1919190
B(38) =                             2929993913841559/6
B(40) =                       -261082718496449122051/13530
B(42) =                       1520097643918070802691/1806
B(44) =                     -27833269579301024235023/690
B(46) =                     596451111593912163277961/282
B(48) =                -5609403368997817686249127547/46410
B(50) =                  495057205241079648212477525/66
B(52) =              -801165718135489957347924991853/1590
B(54) =             29149963634884862421418123812691/798
B(56) =          -2479392929313226753685415739663229/870
B(58) =          84483613348880041862046775994036021/354
B(60) = -1215233140483755572040304994079820246041491/56786730

## FunL

FunL has pre-defined function B in module integers, which is defined as:

import integers.choose def B( n ) = sum( 1/(k + 1)*sum((if 2|r then 1 else -1)*choose(k, r)*(r^n) | r <- 0..k) | k <- 0..n ) for i <- 0..60 if i == 1 or 2|i  printf( "B(%2d) = %s\n", i, B(i) )
Output:
B( 0) = 1
B( 1) = -1/2
B( 2) = 1/6
B( 4) = -1/30
B( 6) = 1/42
B( 8) = -1/30
B(10) = 5/66
B(12) = -691/2730
B(14) = 7/6
B(16) = -3617/510
B(18) = 43867/798
B(20) = -174611/330
B(22) = 854513/138
B(24) = -236364091/2730
B(26) = 8553103/6
B(28) = -23749461029/870
B(30) = 8615841276005/14322
B(32) = -7709321041217/510
B(34) = 2577687858367/6
B(36) = -26315271553053477373/1919190
B(38) = 2929993913841559/6
B(40) = -261082718496449122051/13530
B(42) = 1520097643918070802691/1806
B(44) = -27833269579301024235023/690
B(46) = 596451111593912163277961/282
B(48) = -5609403368997817686249127547/46410
B(50) = 495057205241079648212477525/66
B(52) = -801165718135489957347924991853/1590
B(54) = 29149963634884862421418123812691/798
B(56) = -2479392929313226753685415739663229/870
B(58) = 84483613348880041862046775994036021/354
B(60) = -1215233140483755572040304994079820246041491/56786730


## GAP

 This example is in need of improvement: B0 isn't shown.
for a in Filtered(List([1 .. 60], n -> [n, Bernoulli(n)]), x -> x[2] <> 0) do    Print(a, "\n");od; [ 1, -1/2 ][ 2, 1/6 ][ 4, -1/30 ][ 6, 1/42 ][ 8, -1/30 ][ 10, 5/66 ][ 12, -691/2730 ][ 14, 7/6 ][ 16, -3617/510 ][ 18, 43867/798 ][ 20, -174611/330 ][ 22, 854513/138 ][ 24, -236364091/2730 ][ 26, 8553103/6 ][ 28, -23749461029/870 ][ 30, 8615841276005/14322 ][ 32, -7709321041217/510 ][ 34, 2577687858367/6 ][ 36, -26315271553053477373/1919190 ][ 38, 2929993913841559/6 ][ 40, -261082718496449122051/13530 ][ 42, 1520097643918070802691/1806 ][ 44, -27833269579301024235023/690 ][ 46, 596451111593912163277961/282 ][ 48, -5609403368997817686249127547/46410 ][ 50, 495057205241079648212477525/66 ][ 52, -801165718135489957347924991853/1590 ][ 54, 29149963634884862421418123812691/798 ][ 56, -2479392929313226753685415739663229/870 ][ 58, 84483613348880041862046775994036021/354 ][ 60, -1215233140483755572040304994079820246041491/56786730 ]

## Go

package main import (	"fmt"	"math/big") func b(n int) *big.Rat {	var f big.Rat	a := make([]big.Rat, n+1)	for m := range a {		a[m].SetFrac64(1, int64(m+1))		for j := m; j >= 1; j-- {			d := &a[j-1]			d.Mul(f.SetInt64(int64(j)), d.Sub(d, &a[j]))		}	}	return f.Set(&a[0])} func main() {	for n := 0; n <= 60; n++ {		if b := b(n); b.Num().BitLen() > 0 {			fmt.Printf("B(%2d) =%45s/%s\n", n, b.Num(), b.Denom())		}	}}
Output:
B( 0) =                                            1/1
B( 1) =                                            1/2
B( 2) =                                            1/6
B( 4) =                                           -1/30
B( 6) =                                            1/42
B( 8) =                                           -1/30
B(10) =                                            5/66
B(12) =                                         -691/2730
B(14) =                                            7/6
B(16) =                                        -3617/510
B(18) =                                        43867/798
B(20) =                                      -174611/330
B(22) =                                       854513/138
B(24) =                                   -236364091/2730
B(26) =                                      8553103/6
B(28) =                                 -23749461029/870
B(30) =                                8615841276005/14322
B(32) =                               -7709321041217/510
B(34) =                                2577687858367/6
B(36) =                        -26315271553053477373/1919190
B(38) =                             2929993913841559/6
B(40) =                       -261082718496449122051/13530
B(42) =                       1520097643918070802691/1806
B(44) =                     -27833269579301024235023/690
B(46) =                     596451111593912163277961/282
B(48) =                -5609403368997817686249127547/46410
B(50) =                  495057205241079648212477525/66
B(52) =              -801165718135489957347924991853/1590
B(54) =             29149963634884862421418123812691/798
B(56) =          -2479392929313226753685415739663229/870
B(58) =          84483613348880041862046775994036021/354
B(60) = -1215233140483755572040304994079820246041491/56786730


This program works as a command line utility, that reads from stdin the number of elements to compute (default 60) and prints them in stdout. The implementation of the algorithm is in the function bernoullis. The rest is for printing the results.

import Data.Ratioimport System.Environment main = getArgs >>= printM . defaultArg  where    defaultArg as =      if null as        then 60        else read (head as) printM m =  mapM_ (putStrLn . printP) .  takeWhile ((<= m) . fst) . filter (\(_, b) -> b /= 0 % 1) . zip [0 ..] $bernoullis printP (i, r) = "B(" ++ show i ++ ") = " ++ show (numerator r) ++ "/" ++ show (denominator r) bernoullis = map head . iterate (ulli 1) . map berno$ enumFrom 0  where    berno i = 1 % (i + 1)    ulli _ [_] = []    ulli i (x:y:xs) = (i % 1) * (x - y) : ulli (i + 1) (y : xs)
Output:
B(0) = 1/1
B(1) = 1/2
B(2) = 1/6
B(4) = -1/30
B(6) = 1/42
B(8) = -1/30
B(10) = 5/66
B(12) = -691/2730
B(14) = 7/6
B(16) = -3617/510
B(18) = 43867/798
B(20) = -174611/330
B(22) = 854513/138
B(24) = -236364091/2730
B(26) = 8553103/6
B(28) = -23749461029/870
B(30) = 8615841276005/14322
B(32) = -7709321041217/510
B(34) = 2577687858367/6
B(36) = -26315271553053477373/1919190
B(38) = 2929993913841559/6
B(40) = -261082718496449122051/13530
B(42) = 1520097643918070802691/1806
B(44) = -27833269579301024235023/690
B(46) = 596451111593912163277961/282
B(48) = -5609403368997817686249127547/46410
B(50) = 495057205241079648212477525/66
B(52) = -801165718135489957347924991853/1590
B(54) = 29149963634884862421418123812691/798
B(56) = -2479392929313226753685415739663229/870
B(58) = 84483613348880041862046775994036021/354
B(60) = -1215233140483755572040304994079820246041491/56786730

import Data.Ratio (numerator, denominator, (%)) bernouillis :: Integer -> [Rational]bernouillis =  let faulhaber rs n = (:) =<< (-) 1 . sum zipWith ((*) . (n %)) [2 ..] rs in fmap head . tail . scanl faulhaber [] . enumFromTo 0 bernouilliTable :: Integer -> StringbernouilliTable = let row i x = [ concat ["B(", show i, ") = ", show n, "/", show (denominator x)] | let n = numerator x , n /= 0 ] in unlines . concat . zipWith row [0 ..] . bernouillis main :: IO ()main = putStrLn (bernouilliTable 60) Output: B(0) = 1/1 B(1) = 1/2 B(2) = 1/6 B(4) = -1/30 B(6) = 1/42 B(8) = -1/30 B(10) = 5/66 B(12) = -691/2730 B(14) = 7/6 B(16) = -3617/510 B(18) = 43867/798 B(20) = -174611/330 B(22) = 854513/138 B(24) = -236364091/2730 B(26) = 8553103/6 B(28) = -23749461029/870 B(30) = 8615841276005/14322 B(32) = -7709321041217/510 B(34) = 2577687858367/6 B(36) = -26315271553053477373/1919190 B(38) = 2929993913841559/6 B(40) = -261082718496449122051/13530 B(42) = 1520097643918070802691/1806 B(44) = -27833269579301024235023/690 B(46) = 596451111593912163277961/282 B(48) = -5609403368997817686249127547/46410 B(50) = 495057205241079648212477525/66 B(52) = -801165718135489957347924991853/1590 B(54) = 29149963634884862421418123812691/798 B(56) = -2479392929313226753685415739663229/870 B(58) = 84483613348880041862046775994036021/354 B(60) = -1215233140483755572040304994079820246041491/56786730 ## Icon and Unicon The following works in both languages: link "rational" procedure main(args) limit := integer(!args) | 60 every b := bernoulli(i := 0 to limit) do if b.numer > 0 then write(right(i,3),": ",align(rat2str(b),60))end procedure bernoulli(n) (A := table(0))[0] := rational(1,1,1) every m := 1 to n do { A[m] := rational(1,m+1,1) every j := m to 1 by -1 do A[j-1] := mpyrat(rational(j,1,1), subrat(A[j-1],A[j])) } return A[0]end procedure align(r,n) return repl(" ",n-find("/",r))||rend Sample run: ->bernoulli 60 0: (1/1) 1: (1/2) 2: (1/6) 4: (-1/30) 6: (1/42) 8: (-1/30) 10: (5/66) 12: (-691/2730) 14: (7/6) 16: (-3617/510) 18: (43867/798) 20: (-174611/330) 22: (854513/138) 24: (-236364091/2730) 26: (8553103/6) 28: (-23749461029/870) 30: (8615841276005/14322) 32: (-7709321041217/510) 34: (2577687858367/6) 36: (-26315271553053477373/1919190) 38: (2929993913841559/6) 40: (-261082718496449122051/13530) 42: (1520097643918070802691/1806) 44: (-27833269579301024235023/690) 46: (596451111593912163277961/282) 48: (-5609403368997817686249127547/46410) 50: (495057205241079648212477525/66) 52: (-801165718135489957347924991853/1590) 54: (29149963634884862421418123812691/798) 56: (-2479392929313226753685415739663229/870) 58: (84483613348880041862046775994036021/354) 60: (-1215233140483755572040304994079820246041491/56786730) ->  ## J Implementation: B=:3 :0"0 +/,(<:*(_1^[)*!*(y^~1+[)%1+])"0/~i.1x+y) Task:  require'strings' 'B',.rplc&'r/_-'"1": (#~ 0 ~: {:"1)(,.B) i.61B 0 1B 1 1/2B 2 1/6B 4 -1/30B 6 1/42B 8 -1/30B10 5/66B12 -691/2730B14 7/6B16 -3617/510B18 43867/798B20 -174611/330B22 854513/138B24 -236364091/2730B26 8553103/6B28 -23749461029/870B30 8615841276005/14322B32 -7709321041217/510B34 2577687858367/6B36 -26315271553053477373/1919190B38 2929993913841559/6B40 -261082718496449122051/13530B42 1520097643918070802691/1806B44 -27833269579301024235023/690B46 596451111593912163277961/282B48 -5609403368997817686249127547/46410B50 495057205241079648212477525/66B52 -801165718135489957347924991853/1590B54 29149963634884862421418123812691/798B56 -2479392929313226753685415739663229/870B58 84483613348880041862046775994036021/354B60 -1215233140483755572040304994079820246041491/56786730 ## Java import org.apache.commons.math3.fraction.BigFraction; public class BernoulliNumbers { public static void main(String[] args) { for (int n = 0; n <= 60; n++) { BigFraction b = bernouilli(n); if (!b.equals(BigFraction.ZERO)) System.out.printf("B(%-2d) = %-1s%n", n , b); } } static BigFraction bernouilli(int n) { BigFraction[] A = new BigFraction[n + 1]; for (int m = 0; m <= n; m++) { A[m] = new BigFraction(1, (m + 1)); for (int j = m; j >= 1; j--) A[j - 1] = (A[j - 1].subtract(A[j])).multiply(new BigFraction(j)); } return A[0]; }} B(0 ) = 1 B(1 ) = 1 / 2 B(2 ) = 1 / 6 B(4 ) = -1 / 30 B(6 ) = 1 / 42 B(8 ) = -1 / 30 B(10) = 5 / 66 B(12) = -691 / 2730 B(14) = 7 / 6 B(16) = -3617 / 510 B(18) = 43867 / 798 B(20) = -174611 / 330 B(22) = 854513 / 138 B(24) = -236364091 / 2730 B(26) = 8553103 / 6 B(28) = -23749461029 / 870 B(30) = 8615841276005 / 14322 B(32) = -7709321041217 / 510 B(34) = 2577687858367 / 6 B(36) = -26315271553053477373 / 1919190 B(38) = 2929993913841559 / 6 B(40) = -261082718496449122051 / 13530 B(42) = 1520097643918070802691 / 1806 B(44) = -27833269579301024235023 / 690 B(46) = 596451111593912163277961 / 282 B(48) = -5609403368997817686249127547 / 46410 B(50) = 495057205241079648212477525 / 66 B(52) = -801165718135489957347924991853 / 1590 B(54) = 29149963634884862421418123812691 / 798 B(56) = -2479392929313226753685415739663229 / 870 B(58) = 84483613348880041862046775994036021 / 354 B(60) = -1215233140483755572040304994079820246041491 / 56786730 ## jq Works with: jq version 1.4 This section uses the Akiyama–Tanigawa algorithm for the second Bernoulli numbers, Bn. Therefore, the sign of B(1) differs from the modern definition. The implementation presented here is intended for use with a "BigInt" library that uses string representations of decimal integers. Such a library is at BigInt.jq. To make the code in this section self-contained, stubs for the "BigInt" operations are provided in the first subsection. BigInt Stubs: # def negate:# def lessOrEqual(x; y): # x <= y# def long_add(x;y): # x+y# def long_minus(x;y): # x-y# def long_multiply(x;y) # x*y# def long_divide(x;y): # x/y => [q,r]# def long_div(x;y) # integer division# def long_mod(x;y) # % # In all cases, x and y must be strings def negate: (- tonumber) | tostring; def lessOrEqual(num1; num2): (num1|tonumber) <= (num2|tonumber); def long_add(num1; num2): ((num1|tonumber) + (num2|tonumber)) | tostring; def long_minus(x;y): ((num1|tonumber) - (num2|tonumber)) | tostring; # multiply two decimal strings, which may be signed (+ or -)def long_multiply(num1; num2): ((num1|tonumber) * (num2|tonumber)) | tostring; # return [quotient, remainder] # 0/0 = 1; n/0 => errordef long_divide(xx;yy): # x/y => [q,r] imples x == (y * q) + r def ld(x;y): def abs: if . < 0 then -. else . end; (x|abs) asx | (y|abs) as $y | (if (x >= 0 and y > 0) or (x < 0 and y < 0) then 1 else -1 end) as$sign    | (if x >= 0 then 1 else -1 end) as $sx | [$sign * ($x /$y | floor), $sx * ($x % $y)]; ld( xx|tonumber; yy|tonumber) | map(tostring); def long_div(x;y): long_divide(x;y) | .[0]; def long_mod(x;y): ((x|tonumber) % (y|tonumber)) | tostring; Fractions:  # A fraction is represented by [numerator, denominator] in reduced form, with the sign on top # a and b should be BigInt; return a BigIntdef gcd(a; b): def long_abs: . as$in | if lessOrEqual("0"; $in) then$in else negate end;   # subfunction rgcd expects [a,b] as input  # i.e. a ~ .[0] and b ~ .[1]  def rgcd:    .[0] as $a | .[1] as$b     | if $b == "0" then$a      else [$b, long_mod($a ; $b ) ] | rgcd end; a as$a | b as $b | [$a,$b] | rgcd | long_abs ; def normalize: .[0] as$p | .[1] as $q | if$p == "0" then ["0", "1"]    elif lessOrEqual($q ; "0") then [ ($p|negate), ($q|negate)] | normalize else gcd($p; $q) as$g    | [ long_div($p;$g), long_div($q;$g) ]    end ; # a and b should be fractions expressed in the form [p, q]def add(a; b):  a as $a | b as$b  | if $a[1] == "1" and$b[1] == "1" then [ long_add($a[0];$b[0]) , "1"]    elif $a[1] ==$b[1] then [ long_add( $a[0];$b[0]), $a[1] ] | normalize elif$a[0] == "0" then $b elif$b[0] == "0" then $a else [ long_add( long_multiply($a[0]; $b[1]) ; long_multiply($b[0]; $a[1])), long_multiply($a[1]; $b[1]) ] | normalize end ; # a and/or b may be BigInts, or [p,q] fractionsdef multiply(a; b): a as$a | b as $b | if ($a|type) == "string" and ($b|type) == "string" then [ long_multiply($a; $b), "1"] else if$a|type == "string" then [ long_multiply( $a;$b[0]), $b[1] ] elif$b|type == "string" then [ long_multiply( $b;$a[0]), $a[1] ] else [ long_multiply($a[0]; $b[0]), long_multiply($a[1]; $b[1]) ] end | normalize end ; def minus(a; b): a as$a | b as $b | if$a == $b then ["0", "1"] else add($a; [ ($b[0]|negate),$b[1] ] )    end ;

Bernoulli Numbers:

# Using the algorithm in the task description:def bernoulli(n):  reduce range(0; n+1) as $m ( []; .[$m] = ["1", long_add($m|tostring; "1")] # i.e. 1 / ($m+1)      | reduce ($m - range(0 ;$m)) as $j (.; .[$j-1] = multiply( [($j|tostring), "1"]; minus( .[$j-1] ; .[$j]) ) )) | .[0] # (which is Bn) ; The task: range(0;61)| if . % 2 == 0 or . == 1 then "\(.): \(bernoulli(.) )" else empty end Output: The following output was obtained using the previously mentioned BigInt library. $ jq -n -r -f Bernoulli.jq0: ["1","1"]1: ["1","2"]2: ["1","6"]4: ["-1","30"]6: ["1","42"]8: ["-1","30"]10: ["5","66"]12: ["-691","2730"]14: ["7","6"]16: ["-3617","510"]18: ["43867","798"]20: ["-174611","330"]22: ["854513","138"]24: ["-236364091","2730"]26: ["8553103","6"]28: ["-23749461029","870"]30: ["8615841276005","14322"]32: ["-7709321041217","510"]34: ["2577687858367","6"]36: ["-26315271553053477373","1919190"]38: ["2929993913841559","6"]40: ["-261082718496449122051","13530"]42: ["1520097643918070802691","1806"]44: ["-27833269579301024235023","690"]46: ["596451111593912163277961","282"]48: ["-5609403368997817686249127547","46410"]50: ["495057205241079648212477525","66"]52: ["-801165718135489957347924991853","1590"]54: ["29149963634884862421418123812691","798"]56: ["-2479392929313226753685415739663229","870"]58: ["84483613348880041862046775994036021","354"]60: ["-1215233140483755572040304994079820246041491","56786730"]

## Julia

function bernoulli(n)    A = Vector{Rational{BigInt}}(n + 1)    for m = 0 : n        A[m + 1] = 1 // (m + 1)        for j = m : -1 : 1            A[j] = j * (A[j] - A[j + 1])        end    end    return A[1]end function display(n)    B = map(bernoulli, 0 : n)    pad = mapreduce(x -> ndigits(num(x)) + Int(x < 0), max, B)    argdigits = ndigits(n)    for i = 0 : n        if num(B[i + 1]) & 1 == 1            println(                "B(", lpad(i, argdigits), ") = ",                lpad(num(B[i + 1]), pad), " / ", den(B[i + 1])            )        end    endend display(60)

Produces virtually the same output as the Python version.

## Kotlin

Translation of: Java
Works with: Commons Math version 3.3.5
import org.apache.commons.math3.fraction.BigFraction object Bernoulli {    operator fun invoke(n: Int) : BigFraction {        val A = Array(n + 1, init)        for (m in 0..n)            for (j in m downTo 1)                A[j - 1] = A[j - 1].subtract(A[j]).multiply(integers[j])        return A.first()    }     val max = 60     private val init = { m: Int -> BigFraction(1, m + 1) }    private val integers = Array(max + 1, { m: Int -> BigFraction(m) } )} fun main(args: Array<String>) {    for (n in 0..Bernoulli.max)        if (n % 2 == 0 || n == 1)            System.out.printf("B(%-2d) = %-1s%n", n, Bernoulli(n))}
Output:

Produces virtually the same output as the Java version.

## Maple

print(select(n->n[2]<>0,[seq([n,bernoulli(n,1)],n=0..60)]));
Output:
[[0, 1], [1, 1/2], [2, 1/6], [4, -1/30], [6, 1/42], [8, -1/30], [10, 5/66], [12, -691/2730], [14, 7/6], [16, -3617/510], [18, 43867/798], [20, -174611/330], [22, 854513/138], [24, -236364091/2730], [26, 8553103/6], [28, -23749461029/870], [30, 8615841276005/14322], [32, -7709321041217/510], [34, 2577687858367/6], [36, -26315271553053477373/1919190], [38, 2929993913841559/6], [40, -261082718496449122051/13530], [42, 1520097643918070802691/1806], [44, -27833269579301024235023/690], [46, 596451111593912163277961/282], [48, -5609403368997817686249127547/46410], [50, 495057205241079648212477525/66], [52, -801165718135489957347924991853/1590], [54, 29149963634884862421418123812691/798], [56, -2479392929313226753685415739663229/870], [58, 84483613348880041862046775994036021/354], [60, -1215233140483755572040304994079820246041491/56786730]]

## Mathematica / Wolfram Language

Mathematica has no native way for starting an array at index 0. I therefore had to build the array from 1 to n+1 instead of from 0 to n, adjusting the formula accordingly.

bernoulli[n_] := Module[{a = ConstantArray[0, n + 2]},  Do[    a[[m]] = 1/m;    If[m == 1 && a[[1]] != 0, Print[{m - 1, a[[1]]}]];    Do[     a[[j - 1]] = (j - 1)*(a[[j - 1]] - a[[j]]);     If[j == 2 && a[[1]] != 0, Print[{m - 1, a[[1]]}]];     , {j, m, 2, -1}];    , {m, 1, n + 1}];  ]bernoulli[60]
Output:
{0,1}
{1,1/2}
{2,1/6}
{4,-(1/30)}
{6,1/42}
{8,-(1/30)}
{10,5/66}
{12,-(691/2730)}
{14,7/6}
{16,-(3617/510)}
{18,43867/798}
{20,-(174611/330)}
{22,854513/138}
{24,-(236364091/2730)}
{26,8553103/6}
{28,-(23749461029/870)}
{30,8615841276005/14322}
{32,-(7709321041217/510)}
{34,2577687858367/6}
{36,-(26315271553053477373/1919190)}
{38,2929993913841559/6}
{40,-(261082718496449122051/13530)}
{42,1520097643918070802691/1806}
{44,-(27833269579301024235023/690)}
{46,596451111593912163277961/282}
{48,-(5609403368997817686249127547/46410)}
{50,495057205241079648212477525/66}
{52,-(801165718135489957347924991853/1590)}
{54,29149963634884862421418123812691/798}
{56,-(2479392929313226753685415739663229/870)}
{58,84483613348880041862046775994036021/354}
{60,-(1215233140483755572040304994079820246041491/56786730)}

Or, it's permissible to use the native Bernoulli number function instead of being forced to use the specified algorithm, we very simply have:

(Note from task's author: nobody is forced to use any specific algorithm, the one shown is just a suggestion.)

Table[{i, BernoulliB[i]}, {i, 0, 60}];Select[%, #[[2]] != 0 &] // TableForm
Output:
0	1
1	-(1/2)
2	1/6
4	-(1/30)
6	1/42
8	-(1/30)
10	5/66
12	-(691/2730)
14	7/6
16	-(3617/510)
18	43867/798
20	-(174611/330)
22	854513/138
24	-(236364091/2730)
26	8553103/6
28	-(23749461029/870)
30	8615841276005/14322
32	-(7709321041217/510)
34	2577687858367/6
36	-(26315271553053477373/1919190)
38	2929993913841559/6
40	-(261082718496449122051/13530)
42	1520097643918070802691/1806
44	-(27833269579301024235023/690)
46	596451111593912163277961/282
48	-(5609403368997817686249127547/46410)
50	495057205241079648212477525/66
52	-(801165718135489957347924991853/1590)
54	29149963634884862421418123812691/798
56	-(2479392929313226753685415739663229/870)
58	84483613348880041862046775994036021/354
60	-(1215233140483755572040304994079820246041491/56786730)

## PARI/GP

for(n=0,60,t=bernfrac(n);if(t,print(n" "t)))
Output:
0 1
1 -1/2
2 1/6
4 -1/30
6 1/42
8 -1/30
10 5/66
12 -691/2730
14 7/6
16 -3617/510
18 43867/798
20 -174611/330
22 854513/138
24 -236364091/2730
26 8553103/6
28 -23749461029/870
30 8615841276005/14322
32 -7709321041217/510
34 2577687858367/6
36 -26315271553053477373/1919190
38 2929993913841559/6
40 -261082718496449122051/13530
42 1520097643918070802691/1806
44 -27833269579301024235023/690
46 596451111593912163277961/282
48 -5609403368997817686249127547/46410
50 495057205241079648212477525/66
52 -801165718135489957347924991853/1590
54 29149963634884862421418123812691/798
56 -2479392929313226753685415739663229/870
58 84483613348880041862046775994036021/354
60 -1215233140483755572040304994079820246041491/56786730

## Pascal

 This example is incorrect. Please fix the code and remove this message.Details: Bernoulli numbers that are equal to zero are to be suppressed. The index numbers are not correct. B0 isn't shown, B30 through B60 isn't shown.
 (* Taken from the 'Ada 99' project, https://marquisdegeek.com/code_ada99 *) program BernoulliForAda99;  type  Fraction = object  private    numerator, denominator: Int64;   public    procedure assign(n, d: Int64);    procedure subtract(rhs: Fraction);    procedure multiply(value: Int64);    procedure reduce();    procedure writeOutput();end;  function gcd(a, b: Int64):Int64;begin  if (b = 0) then    gcd := a  else    gcd := gcd(b, a mod b)    end;  procedure Fraction.writeOutput();begin  write(numerator);  if (numerator <> 0) then  begin    write('/');    write(denominator);  end;end;  procedure Fraction.assign(n, d: Int64);begin  numerator := n;  denominator := d;end;  procedure Fraction.subtract(rhs: Fraction);begin  numerator := numerator * rhs.denominator;  numerator := numerator - (rhs.numerator * denominator);  denominator := denominator * rhs.denominator;end;  procedure Fraction.multiply(value: Int64);begin  numerator := numerator * value;end;  procedure Fraction.reduce();var gcdResult: Int64;begin  gcdResult := gcd(numerator, denominator);   begin    numerator := numerator div gcdResult;     (* div is Int64 division *)    denominator := denominator div gcdResult; (* could also use round(d/r) *)  end;  end;  function calculateBernoulli(n: Int64) : Fraction;var  m, j: Int64;  results: array of Fraction;   begin    setlength(results, n);     for m:= 0 to n do    begin      results[m].assign(1, m+1);       for j:= m downto 1 do        begin          results[j-1].subtract(results[j]);          results[j-1].multiply(j);          results[j-1].reduce();        end;      end;     calculateBernoulli := results[0];end;  (* Main program starts here *) var  b: Int64;  result: Fraction; begin  writeln('Calculating Bernoulli numbers...');   for b:= 1 to 25 do    begin      write(b);      write(' : ');      result := calculateBernoulli(b);      result.writeOutput();      writeln;  end; end.
Output:
Calculating Bernoulli numbers...
1 : 1/2
2 : 1/6
3 : 0
4 : 1/-30
5 : 0
6 : 1/42
7 : 0
8 : 1/-30
9 : 0
10 : 5/66
11 : 0
12 : 691/-2730
13 : 0
14 : 7/6
15 : 0
16 : -3617/510
17 : 0
18 : 43867/798
19 : 0
20 : 174611/-330
21 : 0
22 : 854513/138
23 : 0
24 : 236364091/-2730
25 : 0


## Perl

The only thing in the suggested algorithm which depends on N is the number of times through the inner block. This means that all but the last iteration through the loop produce the exact same values of A.

Instead of doing the same calculations over and over again, I retain the A array until the final Bernoulli number is produced.

#!perluse strict;use warnings;use List::Util qw(max);use Math::BigRat; my $one = Math::BigRat->new(1);sub bernoulli_print { my @a; for my$m ( 0 .. 60 ) {		push @a, $one / ($m + 1);		for my $j ( reverse 1 ..$m ) {				# This line:				( $a[$j-1] -= $a[$j] ) *= $j; # is a faster version of the following line: #$a[$j-1] =$j * ($a[$j-1] - $a[$j]);				# since it avoids unnecessary object creation.		}		next unless $a[0]; printf "B(%2d) = %44s/%s\n",$m, $a[0]->parts; }} bernoulli_print();  The output is exactly the same as the Python entry. We can also use modules for faster results. E.g. Library: ntheory use ntheory qw/bernfrac/; for my$n (0 .. 60) {  my($num,$den) = bernfrac($n); printf "B(%2d) = %44s/%s\n",$n, $num,$den if $num != 0;} with identical output. Or: use Math::Pari qw/bernfrac/; for my$n (0 .. 60) {  my($num,$den) = split "/", bernfrac($n); printf("B(%2d) = %44s/%s\n",$n, $num,$den||1) if $num != 0;} with the difference being that Pari chooses ${\displaystyle B_{1}}$ = -½. ## Perl 6 ### Simple First, a straighforward implementation of the naïve algorithm in the task description. Works with: Rakudo version 2015.12 sub bernoulli($n) {    my @a;    for 0..$n ->$m {        @a[$m] = FatRat.new(1,$m + 1);        for reverse 1..$m ->$j {          @a[$j - 1] =$j * (@a[$j - 1] - @a[$j]);        }    }    return @a[0];} constant @bpairs = grep *.value.so, ($_ => bernoulli($_) for 0..60); my $width = [max] @bpairs.map: *.value.numerator.chars;my$form = "B(%2d) = \%{$width}d/%d\n"; printf$form, .key, .value.nude for @bpairs;
Output:
B( 0) =                                            1/1
B( 1) =                                            1/2
B( 2) =                                            1/6
B( 4) =                                           -1/30
B( 6) =                                            1/42
B( 8) =                                           -1/30
B(10) =                                            5/66
B(12) =                                         -691/2730
B(14) =                                            7/6
B(16) =                                        -3617/510
B(18) =                                        43867/798
B(20) =                                      -174611/330
B(22) =                                       854513/138
B(24) =                                   -236364091/2730
B(26) =                                      8553103/6
B(28) =                                 -23749461029/870
B(30) =                                8615841276005/14322
B(32) =                               -7709321041217/510
B(34) =                                2577687858367/6
B(36) =                        -26315271553053477373/1919190
B(38) =                             2929993913841559/6
B(40) =                       -261082718496449122051/13530
B(42) =                       1520097643918070802691/1806
B(44) =                     -27833269579301024235023/690
B(46) =                     596451111593912163277961/282
B(48) =                -5609403368997817686249127547/46410
B(50) =                  495057205241079648212477525/66
B(52) =              -801165718135489957347924991853/1590
B(54) =             29149963634884862421418123812691/798
B(56) =          -2479392929313226753685415739663229/870
B(58) =          84483613348880041862046775994036021/354
B(60) = -1215233140483755572040304994079820246041491/56786730

### With memoization

Here is a much faster way, following the Perl solution that avoids recalculating previous values each time through the function. We do this in Perl 6 by not defining it as a function at all, but by defining it as an infinite sequence that we can read however many values we like from (52, in this case, to get up to B(100)). In this solution we've also avoided subscripting operations; rather we use a sequence operator (...) iterated over the list of the previous solution to find the next solution. We reverse the array in this case to make reference to the previous value in the list more natural, which means we take the last value of the list rather than the first value, and do so conditionally to avoid 0 values.

Works with: Rakudo version 2015.12
constant bernoulli = gather {    my @a;    for 0..* -> $m { @a = FatRat.new(1,$m + 1),                -> $prev { my$j = @a.elems;                    $j * (@a.shift -$prev);                } ... { not @a.elems }        take $m => @a[*-1] if @a[*-1]; }} constant @bpairs = bernoulli[^52]; my$width = [max] @bpairs.map: *.value.numerator.chars;my $form = "B(%d)\t= \%{$width}d/%d\n"; printf $form, .key, .value.nude for @bpairs; Output: B(0) = 1/1 B(1) = 1/2 B(2) = 1/6 B(4) = -1/30 B(6) = 1/42 B(8) = -1/30 B(10) = 5/66 B(12) = -691/2730 B(14) = 7/6 B(16) = -3617/510 B(18) = 43867/798 B(20) = -174611/330 B(22) = 854513/138 B(24) = -236364091/2730 B(26) = 8553103/6 B(28) = -23749461029/870 B(30) = 8615841276005/14322 B(32) = -7709321041217/510 B(34) = 2577687858367/6 B(36) = -26315271553053477373/1919190 B(38) = 2929993913841559/6 B(40) = -261082718496449122051/13530 B(42) = 1520097643918070802691/1806 B(44) = -27833269579301024235023/690 B(46) = 596451111593912163277961/282 B(48) = -5609403368997817686249127547/46410 B(50) = 495057205241079648212477525/66 B(52) = -801165718135489957347924991853/1590 B(54) = 29149963634884862421418123812691/798 B(56) = -2479392929313226753685415739663229/870 B(58) = 84483613348880041862046775994036021/354 B(60) = -1215233140483755572040304994079820246041491/56786730 B(62) = 12300585434086858541953039857403386151/6 B(64) = -106783830147866529886385444979142647942017/510 B(66) = 1472600022126335654051619428551932342241899101/64722 B(68) = -78773130858718728141909149208474606244347001/30 B(70) = 1505381347333367003803076567377857208511438160235/4686 B(72) = -5827954961669944110438277244641067365282488301844260429/140100870 B(74) = 34152417289221168014330073731472635186688307783087/6 B(76) = -24655088825935372707687196040585199904365267828865801/30 B(78) = 414846365575400828295179035549542073492199375372400483487/3318 B(80) = -4603784299479457646935574969019046849794257872751288919656867/230010 B(82) = 1677014149185145836823154509786269900207736027570253414881613/498 B(84) = -2024576195935290360231131160111731009989917391198090877281083932477/3404310 B(86) = 660714619417678653573847847426261496277830686653388931761996983/6 B(88) = -1311426488674017507995511424019311843345750275572028644296919890574047/61410 B(90) = 1179057279021082799884123351249215083775254949669647116231545215727922535/272118 B(92) = -1295585948207537527989427828538576749659341483719435143023316326829946247/1410 B(94) = 1220813806579744469607301679413201203958508415202696621436215105284649447/6 B(96) = -211600449597266513097597728109824233673043954389060234150638733420050668349987259/4501770 B(98) = 67908260672905495624051117546403605607342195728504487509073961249992947058239/6 B(100) = -94598037819122125295227433069493721872702841533066936133385696204311395415197247711/33330 ### Functional And if you're a pure enough FP programmer to dislike destroying and reconstructing the array each time, here's the same algorithm without side effects. We use zip with the pair constructor => to keep values associated with their indices. This provides sufficient local information that we can define our own binary operator "bop" to reduce between each two terms, using the "triangle" form (called "scan" in Haskell) to return the intermediate results that will be important to compute the next Bernoulli number. Works with: Rakudo version 2016.12 sub infix:<bop>(\prev, \this) { this.key => this.key * (this.value - prev.value)} sub next-bernoulli ( (:key($pm), :value(@pa)) ) {    $pm + 1 => [ map *.value, [\bop] ($pm + 2 ... 1) Z=> FatRat.new(1, $pm + 2), |@pa ]} constant bernoulli = grep *.value, map { .key => .value[*-1] }, (0 => [FatRat.new(1,1)], &next-bernoulli ... *); constant @bpairs = bernoulli[^52]; my$width = [max] @bpairs.map: *.value.numerator.chars;my $form = "B(%d)\t= \%{$width}d/%d\n"; printf form, .key, .value.nude for @bpairs; Same output as memoization example ## Phix Library: bigatom Using routines from Arithmetic/Rational, adapted to use bigatoms Note: this exposed a bug in ba_trunc, so you may want to check for a mod dated 19/11/15 in that routine (in builtins\bigatom.e). include builtins\bigatom.e constant NUM = 1, DEN = 2 type ba_frac(object r) return sequence(r) and length(r)=2 and bigatom(r[NUM]) and bigatom(r[DEN])end type function ba_gcd(bigatom u, bigatom v)bigatom t u = ba_floor(ba_abs(u)) v = ba_floor(ba_abs(v)) while v!=BA_ZERO do t = u u = v v = ba_remainder(t, v) end while return uend function function ba_frac_normalise(bigatom n, bigatom d)bigatom g if ba_compare(d,BA_ZERO)<0 then n = ba_sub(0,n) d = ba_sub(0,d) end if g = ba_gcd(n,d) return {ba_idivide(n,g),ba_idivide(d,g)}end function function ba_frac_sub(ba_frac a, ba_frac b)bigatom {an,ad} = a, {bn,bd} = b return ba_frac_normalise(ba_sub(ba_multiply(an,bd),ba_multiply(bn,ad)),ba_multiply(ad,bd))end function function ba_frac_mul(ba_frac a, ba_frac b)bigatom {an,ad} = a, {bn,bd} = b return ba_frac_normalise(ba_multiply(an,bn),ba_multiply(ad,bd))end function After which the code itself is pretty trivial sequence a = {}for m=0 to 60 do a = append(a,{ba_new(1),ba_new(m+1)}) for j=m to 1 by -1 do a[j] = ba_frac_mul({ba_new(j),ba_new(1)},ba_frac_sub(a[j+1],a[j])) end for if a[1][1]!=BA_ZERO then printf(1,"B(%2d) = %44s / %s\n",{m,ba_sprint(a[1][1]),ba_sprint(a[1][2])}) end ifend for Output: B( 0) = 1 / 1 B( 1) = -1 / 2 B( 2) = 1 / 6 B( 4) = -1 / 30 B( 6) = 1 / 42 B( 8) = -1 / 30 B(10) = 5 / 66 B(12) = -691 / 2730 B(14) = 7 / 6 B(16) = -3617 / 510 B(18) = 43867 / 798 B(20) = -174611 / 330 B(22) = 854513 / 138 B(24) = -236364091 / 2730 B(26) = 8553103 / 6 B(28) = -23749461029 / 870 B(30) = 8615841276005 / 14322 B(32) = -7709321041217 / 510 B(34) = 2577687858367 / 6 B(36) = -26315271553053477373 / 1919190 B(38) = 2929993913841559 / 6 B(40) = -261082718496449122051 / 13530 B(42) = 1520097643918070802691 / 1806 B(44) = -27833269579301024235023 / 690 B(46) = 596451111593912163277961 / 282 B(48) = -5609403368997817686249127547 / 46410 B(50) = 495057205241079648212477525 / 66 B(52) = -801165718135489957347924991853 / 1590 B(54) = 29149963634884862421418123812691 / 798 B(56) = -2479392929313226753685415739663229 / 870 B(58) = 84483613348880041862046775994036021 / 354 B(60) = -1215233140483755572040304994079820246041491 / 56786730  ## PicoLisp Brute force and method by Srinivasa Ramanujan. (load "@lib/frac.l") (de fact (N) (cache '(NIL) N (if (=0 N) 1 (apply * (range 1 N))) ) ) (de binomial (N K) (frac (/ (fact N) (* (fact (- N K)) (fact K)) ) 1 ) ) (de A (N M) (let Sum (0 . 1) (for X M (setq Sum (f+ Sum (f* (binomial (+ N 3) (- N (* X 6))) (berno (- N (* X 6)) ) ) ) ) ) Sum ) ) (de berno (N) (cache '(NIL) N (cond ((=0 N) (1 . 1)) ((= 1 N) (-1 . 2)) ((bit? 1 N) (0 . 1)) (T (case (% N 6) (0 (f/ (f- (frac (+ N 3) 3) (A N (/ N 6)) ) (binomial (+ N 3) N) ) ) (2 (f/ (f- (frac (+ N 3) 3) (A N (/ (- N 2) 6)) ) (binomial (+ N 3) N) ) ) (4 (f/ (f- (f* (-1 . 1) (frac (+ N 3) 6)) (A N (/ (- N 4) 6)) ) (binomial (+ N 3) N) ) ) ) ) ) ) ) (de berno-brute (N) (cache '(NIL) N (let Sum (0 . 1) (cond ((=0 N) (1 . 1)) ((= 1 N) (-1 . 2)) ((bit? 1 N) (0 . 1)) (T (for (X 0 (> N X) (inc X)) (setq Sum (f+ Sum (f* (binomial (inc N) X) (berno-brute X)) ) ) ) (f/ (f* (-1 . 1) Sum) (binomial (inc N) N)) ) ) ) ) ) (for (N 0 (> 62 N) (inc N)) (if (or (= N 1) (not (bit? 1 N))) (tab (2 4 -60) N " => " (sym (berno N))) ) ) (for (N 0 (> 400 N) (inc N)) (test (berno N) (berno-brute N)) ) (bye) ## PL/I Bern: procedure options (main); /* 4 July 2014 */ declare i fixed binary; declare B complex fixed (31); Bernoulli: procedure (n) returns (complex fixed (31)); declare n fixed binary; declare anum(0:n) fixed (31), aden(0:n) fixed (31); declare (j, m) fixed; declare F fixed (31); do m = 0 to n; anum(m) = 1; aden(m) = m+1; do j = m to 1 by -1; anum(j-1) = j*( aden(j)*anum(j-1) - aden(j-1)*anum(j) ); aden(j-1) = ( aden(j-1) * aden(j) ); F = gcd(abs(anum(j-1)), abs(aden(j-1)) ); if F ^= 1 then do; anum(j-1) = anum(j-1) / F; aden(j-1) = aden(j-1) / F; end; end; end; return ( complex(anum(0), aden(0)) );end Bernoulli; do i = 0, 1, 2 to 36 by 2; /* 36 is upper limit imposed by hardware. */ B = Bernoulli(i); put skip edit ('B(' , trim(i) , ')=' , real(B) , '/' , trim(imag(B)) ) (3 A, column(10), F(32), 2 A); end;end Bern; The above uses GCD (see Rosetta Code) extended for 31-digit working. Results obtained by this program are limited to the entries shown below due to the restrictions imposed by storing numbers in fixed decimal (31 digits). B(0)= 1/1 B(1)= 1/2 B(2)= 1/6 B(4)= -1/30 B(6)= 1/42 B(8)= -1/30 B(10)= 5/66 B(12)= -691/2730 B(14)= 7/6 B(16)= -3617/510 B(18)= 43867/798 B(20)= -174611/330 B(22)= 854513/138 B(24)= -236364091/2730 B(26)= 8553103/6 B(28)= -23749461029/870 B(30)= 8615841276005/14322 B(32)= -7709321041217/510 B(34)= 2577687858367/6 B(36)= -26315271553053477373/1919190  ## Python ### Python: Using task algorithm from fractions import Fraction as Fr def bernoulli(n): A = [0] * (n+1) for m in range(n+1): A[m] = Fr(1, m+1) for j in range(m, 0, -1): A[j-1] = j*(A[j-1] - A[j]) return A[0] # (which is Bn) bn = [(i, bernoulli(i)) for i in range(61)]bn = [(i, b) for i,b in bn if b]width = max(len(str(b.numerator)) for i,b in bn)for i,b in bn: print('B(%2i) = %*i/%i' % (i, width, b.numerator, b.denominator)) Output: B( 0) = 1/1 B( 1) = 1/2 B( 2) = 1/6 B( 4) = -1/30 B( 6) = 1/42 B( 8) = -1/30 B(10) = 5/66 B(12) = -691/2730 B(14) = 7/6 B(16) = -3617/510 B(18) = 43867/798 B(20) = -174611/330 B(22) = 854513/138 B(24) = -236364091/2730 B(26) = 8553103/6 B(28) = -23749461029/870 B(30) = 8615841276005/14322 B(32) = -7709321041217/510 B(34) = 2577687858367/6 B(36) = -26315271553053477373/1919190 B(38) = 2929993913841559/6 B(40) = -261082718496449122051/13530 B(42) = 1520097643918070802691/1806 B(44) = -27833269579301024235023/690 B(46) = 596451111593912163277961/282 B(48) = -5609403368997817686249127547/46410 B(50) = 495057205241079648212477525/66 B(52) = -801165718135489957347924991853/1590 B(54) = 29149963634884862421418123812691/798 B(56) = -2479392929313226753685415739663229/870 B(58) = 84483613348880041862046775994036021/354 B(60) = -1215233140483755572040304994079820246041491/56786730 ### Python: Optimised task algorithm Using the optimization mentioned in the Perl entry to reduce intermediate calculations we create and use the generator bernoulli2(): def bernoulli2(): A, m = [], 0 while True: A.append(Fr(1, m+1)) for j in range(m, 0, -1): A[j-1] = j*(A[j-1] - A[j]) yield A[0] # (which is Bm) m += 1 bn2 = [ix for ix in zip(range(61), bernoulli2())]bn2 = [(i, b) for i,b in bn2 if b]width = max(len(str(b.numerator)) for i,b in bn2)for i,b in bn2: print('B(%2i) = %*i/%i' % (i, width, b.numerator, b.denominator)) Output is exactly the same as before. ## R  This example is incorrect. Please fix the code and remove this message.Details: The index numbers are not correct. B0 isn't shown. The Bernoulli numbers are not shown as (reduced) fractions. Bernoulli numbers equal to zero are to be suppressed. R has the built-in function bernoulli(n), where n is the index, a whole number greater or equal to 0. It returns the first n+1 Bernoulli numbers, that are defined as a sequence of rational numbers. Works with: R version 3.3.2 and above  # Bernoulli numbers. 12/8/16 aevrequire(pracma)bernoulli(60)  Output: > require(pracma) Loading required package: pracma > bernoulli(60) [1] 1.000000e+00 -5.000000e-01 1.666667e-01 0.000000e+00 -3.333333e-02 [6] 0.000000e+00 2.380952e-02 0.000000e+00 -3.333333e-02 0.000000e+00 [11] 7.575758e-02 0.000000e+00 -2.531136e-01 0.000000e+00 1.166667e+00 [16] 0.000000e+00 -7.092157e+00 0.000000e+00 5.497118e+01 0.000000e+00 [21] -5.291242e+02 0.000000e+00 6.192123e+03 0.000000e+00 -8.658025e+04 [26] 0.000000e+00 1.425517e+06 0.000000e+00 -2.729823e+07 0.000000e+00 [31] 6.015809e+08 0.000000e+00 -1.511632e+10 0.000000e+00 4.296146e+11 [36] 0.000000e+00 -1.371166e+13 0.000000e+00 4.883323e+14 0.000000e+00 [41] -1.929658e+16 0.000000e+00 8.416930e+17 0.000000e+00 -4.033807e+19 [46] 0.000000e+00 2.115075e+21 0.000000e+00 -1.208663e+23 0.000000e+00 [51] 7.500867e+24 0.000000e+00 -5.038778e+26 0.000000e+00 3.652878e+28 [56] 0.000000e+00 -2.849877e+30 0.000000e+00 2.386543e+32 0.000000e+00 [61] -2.139995e+34 >  ## Racket This implements, firstly, the algorithm specified with the task... then the better performing bernoulli.3, which uses the "double sum formula" listed under REXX. The number generators all (there is also a bernoulli.2) use the same emmitter... it's just a matter of how long to wait for the emission. #lang racket;; For: http://rosettacode.org/wiki/Bernoulli_numbers ;; As described in task...(define (bernoulli.1 n) (define A (make-vector (add1 n))) (for ((m (in-range 0 (add1 n)))) (vector-set! A m (/ (add1 m))) (for ((j (in-range m (sub1 1) -1))) (define new-A_j-1 (* j (- (vector-ref A (sub1 j)) (vector-ref A j)))) (vector-set! A (sub1 j) new-A_j-1))) (vector-ref A 0)) (define (non-zero-bernoulli-indices s) (sequence-filter (λ (n) (or (even? n) (= n 1))) s))(define (bernoulli_0..n B N) (for/list ((n (non-zero-bernoulli-indices (in-range (add1 N))))) (B n))) ;; From REXX description / http://mathworld.wolfram.com/BernoulliNumber.html #33;; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~;; bernoulli.2 is for illustrative purposes, binomial is very costly if there is no memoisation;; (which math/number-theory doesn't do)(require (only-in math/number-theory binomial))(define (bernoulli.2 n) (for/sum ((k (in-range 0 (add1 n)))) (* (/ (add1 k)) (for/sum ((r (in-range 0 (add1 k)))) (* (expt -1 r) (binomial k r) (expt r n)))))) ;; Three things to do:;; 1. (expt -1 r): is 1 for even r, -1 for odd r... split the sum between those two.;; 2. splitting the sum might has arithmetic advantages, too. We're using rationals, so the smaller;; summations should require less normalisation of intermediate, fractional results;; 3. a memoised binomial... although the one from math/number-theory is fast, it is (and its;; factorials are) computed every time which is redundant(define kCr-memo (make-hasheq))(define !-memo (make-vector 1000 #f))(vector-set! !-memo 0 1) ;; seed the memo(define (! k) (cond [(vector-ref !-memo k) => values] [else (define k! (* k (! (- k 1)))) (vector-set! !-memo k k!) k!]))(define (kCr k r) ; If we want (kCr ... r>1000000) we'll have to reconsider this. However, until then... (define hash-key (+ (* 1000000 k) r)) (hash-ref! kCr-memo hash-key (λ () (/ (! k) (! r) (! (- k r)))))) (define (bernoulli.3 n) (for/sum ((k (in-range 0 (add1 n)))) (define k+1 (add1 k)) (* (/ k+1) (- (for/sum ((r (in-range 0 k+1 2))) (* (kCr k r) (expt r n))) (for/sum ((r (in-range 1 k+1 2))) (* (kCr k r) (expt r n))))))) (define (display/align-fractions caption/idx-fmt Bs) ;; widths are one more than the order of magnitude (define oom+1 (compose add1 order-of-magnitude)) (define-values (I-width N-width D-width) (for/fold ((I 0) (N 0) (D 0)) ((b Bs) (n (non-zero-bernoulli-indices (in-naturals)))) (define +b (abs b)) (values (max I (oom+1 (max n 1))) (max N (+ (oom+1 (numerator +b)) (if (negative? b) 1 0))) (max D (oom+1 (denominator +b)))))) (define (~a/w/a n w a) (~a n #:width w #:align a)) (for ((n (non-zero-bernoulli-indices (in-naturals))) (b Bs)) (printf "~a ~a/~a~%" (format caption/idx-fmt (~a/w/a n I-width 'right)) (~a/w/a (numerator b) N-width 'right) (~a/w/a (denominator b) D-width 'left)))) (module+ main (display/align-fractions "B(~a) =" (bernoulli_0..n bernoulli.3 60))) (module+ test (require rackunit) ; correctness and timing tests (check-match (time (bernoulli_0..n bernoulli.1 60)) (list 1/1 (app abs 1/2) 1/6 -1/30 1/42 -1/30 _ ...)) (check-match (time (bernoulli_0..n bernoulli.2 60)) (list 1/1 (app abs 1/2) 1/6 -1/30 1/42 -1/30 _ ...)) (check-match (time (bernoulli_0..n bernoulli.3 60)) (list 1/1 (app abs 1/2) 1/6 -1/30 1/42 -1/30 _ ...)) ; timing only ... (void (time (bernoulli_0..n bernoulli.3 100)))) Output: B( 0) = 1/1 B( 1) = -1/2 B( 2) = 1/6 B( 4) = -1/30 B( 6) = 1/42 B( 8) = -1/30 B(10) = 5/66 B(12) = -691/2730 B(14) = 7/6 B(16) = -3617/510 B(18) = 43867/798 B(20) = -174611/330 B(22) = 854513/138 B(24) = -236364091/2730 B(26) = 8553103/6 B(28) = -23749461029/870 B(30) = 8615841276005/14322 B(32) = -7709321041217/510 B(34) = 2577687858367/6 B(36) = -26315271553053477373/1919190 B(38) = 2929993913841559/6 B(40) = -261082718496449122051/13530 B(42) = 1520097643918070802691/1806 B(44) = -27833269579301024235023/690 B(46) = 596451111593912163277961/282 B(48) = -5609403368997817686249127547/46410 B(50) = 495057205241079648212477525/66 B(52) = -801165718135489957347924991853/1590 B(54) = 29149963634884862421418123812691/798 B(56) = -2479392929313226753685415739663229/870 B(58) = 84483613348880041862046775994036021/354 B(60) = -1215233140483755572040304994079820246041491/56786730 ## REXX The double sum formula used is number (33) from the entry Bernoulli number on Wolfram MathWorldTM. ${\displaystyle B_{n}=\sum _{k=0}^{n}{\frac {1}{k+1}}\sum _{r=0}^{k}(-1)^{r}{\binom {k}{r}}r^{n}}$ where ${\displaystyle {\binom {k}{r}}}$ is a binomial coefficient. /*REXX program calculates N number of Bernoulli numbers expressed as fractions. */parse arg N .; if N=='' | N=="," then N= 60 /*Not specified? Then use the default.*/d= n*2; if d>digits() then numeric digits d /*increase the decimal digits if needed*/!.=0; w= max(length(N), 4); Nw= N + w + N % 4 /*used for aligning (output) fractions.*/say 'B(n)' center("Bernoulli number expressed as a fraction", max(78-w, Nw)) /*title.*/say copies('─',w) copies("─", max(78-w,Nw+2*w)) /*display 2nd line of title, separators*/ do #=0 to N /*process the numbers from 0 ──► N. */ b= bern(#); if b==0 then iterate /*calculate Bernoulli number, skip if 0*/ indent= max(0, nW - pos('/', b) ) /*calculate the alignment (indentation)*/ say right(#, w) left('', indent) b /*display the indented Bernoulli number*/ end /*#*/ /* [↑] align the Bernoulli fractions. */exit /*stick a fork in it, we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/bern: parse arg x; if x==0 then return '1/1' /*handle the special case of zero. */ if x==1 then return '-1/2' /* " " " " " one. */ if x//2 then return 0 /* " " " " " odds > 1.*/ do j=2 to x by 2; jp= j+1; d= j+j /*process the positive integers up to X*/ sn= 1 - j /*define the numerator. */ sd= 2 /* " " denominator. */ do k=2 to j-1 by 2 /*calculate a SN/SD sequence. */ parse var @.k bn '/' ad /*get a previously calculated fraction.*/ an= comb(jp, k) * bn /*use COMBination for the next term. */lcm= lcm(sd, ad)             /*use Least Common Denominator function*/                   sn= $lcm % sd * sn; sd=$lcm /*calculate the   current  numerator.  */                   an= $lcm % ad * an; ad=$lcm /*    "      "      next      "        */                   sn= sn + an                   /*    "      "    current     "        */                   end   /*k*/                   /* [↑]  calculate the  SN/SD  sequence.*/        sn= -sn                                  /*adjust the sign for the numerator.   */        sd= sd * jp                              /*calculate           the denominator. */        if sn\==1  then do;  _= gcd(sn, sd)      /*get the  Greatest Common Denominator.*/                            sn= sn%_;  sd= sd%_  /*reduce the numerator and denominator.*/                        end                      /* [↑]   done with the reduction(s).   */        @.j= sn'/'sd                             /*save the result for the next round.  */        end     /*j*/                            /* [↑]  done calculating Bernoulli #'s.*/      return sn'/'sd/*──────────────────────────────────────────────────────────────────────────────────────*/comb: procedure expose !.; parse arg x,y;        if x==y  then return 1      if !.c.x.y \== 0  then return !.c.x.y               /*combination computed before?*/      if x-y < y   then y= x-y;     z= perm(x, y);    do j=2  to y;   z= z % j;  end /*J*/      !.c.x.y= z;     return z                            /*assign memoization & return.*//*──────────────────────────────────────────────────────────────────────────────────────*/gcd:  procedure;           parse arg x,y;       x= abs(x)        do  until y==0;    parse value  x//y  y    with    y  x;  end;            return x/*──────────────────────────────────────────────────────────────────────────────────────*/lcm:  procedure; parse arg x,y  /*x=abs(x);  y=abs(y)   not needed for Bernoulli numbers*/                 if y==0  then return 0               /*if zero, then LCM is also zero. */                 d= x * y                             /*calculate part of the LCM here. */                          do  until y==0;   parse  value   x//y  y      with      y  x                          end   /*until*/             /* [↑]  this is a short & fast GCD*/                 return d % x;                        /*divide the pre─calculated value.*//*──────────────────────────────────────────────────────────────────────────────────────*/perm: procedure expose !.;  parse arg x,y;          if !.p.x.y \== 0  then return !.p.x.y      z= 1;       do j=x-y+1  to x;     z= z*j;     end;        !.p.x.y= z;       return z
output   when using the default input:
B(n)                    Bernoulli number expressed as a fraction
──── ───────────────────────────────────────────────────────────────────────────────────────
0                                                                               1/1
1                                                                              -1/2
2                                                                               1/6
4                                                                              -1/30
6                                                                               1/42
8                                                                              -1/30
10                                                                               5/66
12                                                                            -691/2730
14                                                                               7/6
16                                                                           -3617/510
18                                                                           43867/798
20                                                                         -174611/330
22                                                                          854513/138
24                                                                      -236364091/2730
26                                                                         8553103/6
28                                                                    -23749461029/870
30                                                                   8615841276005/14322
32                                                                  -7709321041217/510
34                                                                   2577687858367/6
36                                                           -26315271553053477373/1919190
38                                                                2929993913841559/6
40                                                          -261082718496449122051/13530
42                                                          1520097643918070802691/1806
44                                                        -27833269579301024235023/690
46                                                        596451111593912163277961/282
48                                                   -5609403368997817686249127547/46410
50                                                     495057205241079648212477525/66
52                                                 -801165718135489957347924991853/1590
54                                                29149963634884862421418123812691/798
56                                             -2479392929313226753685415739663229/870
58                                             84483613348880041862046775994036021/354
60                                    -1215233140483755572040304994079820246041491/56786730


## Ruby

Translation of: Python
bernoulli = Enumerator.new do |y|  ar, m = [], 0  loop do    ar << Rational(1, m+1)    m.downto(1){|j| ar[j-1] = j*(ar[j-1] - ar[j]) }    y << ar.first  # yield    m += 1  endend b_nums = bernoulli.take(61)width  = b_nums.map{|b| b.numerator.to_s.size}.maxb_nums.each_with_index {|b,i| puts "B(%2i) = %*i/%i" % [i, width, b.numerator, b.denominator] unless b.zero? }
Output:
B( 0) =                                            1/1
B( 1) =                                            1/2
B( 2) =                                            1/6
B( 4) =                                           -1/30
B( 6) =                                            1/42
B( 8) =                                           -1/30
B(10) =                                            5/66
B(12) =                                         -691/2730
B(14) =                                            7/6
B(16) =                                        -3617/510
B(18) =                                        43867/798
B(20) =                                      -174611/330
B(22) =                                       854513/138
B(24) =                                   -236364091/2730
B(26) =                                      8553103/6
B(28) =                                 -23749461029/870
B(30) =                                8615841276005/14322
B(32) =                               -7709321041217/510
B(34) =                                2577687858367/6
B(36) =                        -26315271553053477373/1919190
B(38) =                             2929993913841559/6
B(40) =                       -261082718496449122051/13530
B(42) =                       1520097643918070802691/1806
B(44) =                     -27833269579301024235023/690
B(46) =                     596451111593912163277961/282
B(48) =                -5609403368997817686249127547/46410
B(50) =                  495057205241079648212477525/66
B(52) =              -801165718135489957347924991853/1590
B(54) =             29149963634884862421418123812691/798
B(56) =          -2479392929313226753685415739663229/870
B(58) =          84483613348880041862046775994036021/354
B(60) = -1215233140483755572040304994079820246041491/56786730


## Rust

 This example is incorrect. Please fix the code and remove this message.Details: B1 isn't shown.
// 2.5 implementations presented here:  naive, optimized, and an iterator using// the optimized function. The speeds vary significantly: relative// speeds of optimized:iterator:naive implementations is 625:25:1. #![feature(test)] extern crate num;extern crate test; use num::bigint::{BigInt, ToBigInt};use num::rational::{BigRational};use std::cmp::max;use std::env;use std::ops::{Mul, Sub};use std::process; struct Bn {    value: BigRational,    index: i32} struct Context {    bigone_const: BigInt,    a: Vec<BigRational>,    index: i32              // Counter for iterator implementation} impl Context {    pub fn new() -> Context {        let bigone = 1.to_bigint().unwrap();        let a_vec: Vec<BigRational> = vec![];        Context {            bigone_const: bigone,            a: a_vec,            index: -1        }    }} impl Iterator for Context {    type Item = Bn;     fn next(&mut self) -> Option<Bn> {        self.index += 1;        Some(Bn { value: bernoulli(self.index as usize, self), index: self.index })    }} fn help() {    println!("Usage: bernoulli_numbers <up_to>");} fn main() {    let args: Vec<String> = env::args().collect();    let mut up_to: usize = 60;     match args.len() {        1 => {},        2 => {            up_to = args[1].parse::<usize>().unwrap();        },        _ => {            help();            process::exit(0);        }    }     let context = Context::new();    // Collect the solutions by using the Context iterator    // (this is not as fast as calling the optimized function directly).    let res = context.take(up_to + 1).collect::<Vec<_>>();    let width = res.iter().fold(0, |a, r| max(a, r.value.numer().to_string().len()));     for r in res.iter().filter(|r| r.index % 2 == 0) {        println!("B({:>2}) = {:>2$} / {denom}", r.index, r.value.numer(), width, denom = r.value.denom()); }} // Implementation with no reused calculations.fn _bernoulli_naive(n: usize, c: &mut Context) -> BigRational { for m in 0..n + 1 { c.a.push(BigRational::new(c.bigone_const.clone(), (m + 1).to_bigint().unwrap())); for j in (1..m + 1).rev() { c.a[j - 1] = (c.a[j - 1].clone().sub(c.a[j].clone())).mul( BigRational::new(j.to_bigint().unwrap(), c.bigone_const.clone()) ); } } c.a[0].reduced()} // Implementation with reused calculations (does not require sequential calls).fn bernoulli(n: usize, c: &mut Context) -> BigRational { for i in 0..n + 1 { if i >= c.a.len() { c.a.push(BigRational::new(c.bigone_const.clone(), (i + 1).to_bigint().unwrap())); for j in (1..i + 1).rev() { c.a[j - 1] = (c.a[j - 1].clone().sub(c.a[j].clone())).mul( BigRational::new(j.to_bigint().unwrap(), c.bigone_const.clone()) ); } } } c.a[0].reduced()} #[cfg(test)]mod tests { use super::{Bn, Context, bernoulli, _bernoulli_naive}; use num::rational::{BigRational}; use std::str::FromStr; use test::Bencher; // [tests elided] #[bench] fn bench_bernoulli_naive(b: &mut Bencher) { let mut context = Context::new(); b.iter(|| { let mut res: Vec<Bn> = vec![]; for n in 0..30 + 1 { let b = _bernoulli_naive(n, &mut context); res.push(Bn { value:b.clone(), index: n as i32}); } }); } #[bench] fn bench_bernoulli(b: &mut Bencher) { let mut context = Context::new(); b.iter(|| { let mut res: Vec<Bn> = vec![]; for n in 0..30 + 1 { let b = bernoulli(n, &mut context); res.push(Bn { value:b.clone(), index: n as i32}); } }); } #[bench] fn bench_bernoulli_iter(b: &mut Bencher) { b.iter(|| { let context = Context::new(); let _res = context.take(30 + 1).collect::<Vec<_>>(); }); }}  Output: B( 0) = 1 / 1 B( 2) = 1 / 6 B( 4) = -1 / 30 B( 6) = 1 / 42 B( 8) = -1 / 30 B(10) = 5 / 66 B(12) = -691 / 2730 B(14) = 7 / 6 B(16) = -3617 / 510 B(18) = 43867 / 798 B(20) = -174611 / 330 B(22) = 854513 / 138 B(24) = -236364091 / 2730 B(26) = 8553103 / 6 B(28) = -23749461029 / 870 B(30) = 8615841276005 / 14322 B(32) = -7709321041217 / 510 B(34) = 2577687858367 / 6 B(36) = -26315271553053477373 / 1919190 B(38) = 2929993913841559 / 6 B(40) = -261082718496449122051 / 13530 B(42) = 1520097643918070802691 / 1806 B(44) = -27833269579301024235023 / 690 B(46) = 596451111593912163277961 / 282 B(48) = -5609403368997817686249127547 / 46410 B(50) = 495057205241079648212477525 / 66 B(52) = -801165718135489957347924991853 / 1590 B(54) = 29149963634884862421418123812691 / 798 B(56) = -2479392929313226753685415739663229 / 870 B(58) = 84483613348880041862046775994036021 / 354 B(60) = -1215233140483755572040304994079820246041491 / 56786730  ## Scala With Custom Rational Number Class (code will run in Scala REPL with a cut-and-paste without need for a third-party library) /** Roll our own pared-down BigFraction class just for these Bernoulli Numbers */case class BFraction( numerator:BigInt, denominator:BigInt ) { require( denominator != BigInt(0), "Denominator cannot be zero" ) val gcd = numerator.gcd(denominator) val num = numerator / gcd val den = denominator / gcd def unary_- = BFraction(-num, den) def -( that:BFraction ) = that match { case f if f.num == BigInt(0) => this case f if f.den == this.den => BFraction(this.num - f.num, this.den) case f => BFraction(((this.num * f.den) - (f.num * this.den)), this.den * f.den ) } def *( that:Int ) = BFraction( num * that, den ) override def toString = num + " / " + den} def bernoulliB( n:Int ) : BFraction = { val aa : Array[BFraction] = Array.ofDim(n+1) for( m <- 0 to n ) { aa(m) = BFraction(1,(m+1)) for( n <- m to 1 by -1 ) { aa(n-1) = (aa(n-1) - aa(n)) * n } } aa(0)} assert( {val b12 = bernoulliB(12); b12.num == -691 && b12.den == 2730 } ) val r = for( n <- 0 to 60; b = bernoulliB(n) if b.num != 0 ) yield (n, b) val numeratorSize = r.map(_._2.num.toString.length).max // Print the resultsr foreach{ case (i,b) => { val label = f"b($i)"  val num = (" " * (numeratorSize - b.num.toString.length)) + b.num  println( f"$label%-6s$num / ${b.den}" )}}  Output: b(0) 1 / 1 b(1) 1 / 2 b(2) 1 / 6 b(4) -1 / 30 b(6) 1 / 42 b(8) -1 / 30 b(10) 5 / 66 b(12) -691 / 2730 b(14) 7 / 6 b(16) -3617 / 510 b(18) 43867 / 798 b(20) -174611 / 330 b(22) 854513 / 138 b(24) -236364091 / 2730 b(26) 8553103 / 6 b(28) -23749461029 / 870 b(30) 8615841276005 / 14322 b(32) -7709321041217 / 510 b(34) 2577687858367 / 6 b(36) -26315271553053477373 / 1919190 b(38) 2929993913841559 / 6 b(40) -261082718496449122051 / 13530 b(42) 1520097643918070802691 / 1806 b(44) -27833269579301024235023 / 690 b(46) 596451111593912163277961 / 282 b(48) -5609403368997817686249127547 / 46410 b(50) 495057205241079648212477525 / 66 b(52) -801165718135489957347924991853 / 1590 b(54) 29149963634884862421418123812691 / 798 b(56) -2479392929313226753685415739663229 / 870 b(58) 84483613348880041862046775994036021 / 354 b(60) -1215233140483755572040304994079820246041491 / 56786730  ## Sidef Recursive solution (with auto-memoization): func bernoulli_number{} func bern_helper(n, k) { binomial(n, k) * (bernoulli_number(k) / (n - k + 1))} func bern_diff(n, k, d) { n < k ? d : bern_diff(n, k + 1, d - bern_helper(n + 1, k))} bernoulli_number = func(n) is cached { n.is_one && return 1/2 n.is_odd && return 0 n > 0 ? bern_diff(n - 1, 0, 1) : 1} for i (0..60) { var num = bernoulli_number(i) || next printf("B(%2d) = %44s / %s\n", i, num.nude)} Iterative solution: func bernoulli_print { var a = [] for m (0..60) { a.append(1/(m+1)) for j (flip(1..m)) { (a[j-1] -= a[j]) *= j } a[0] || next printf("B(%2d) = %44s / %s\n", m, a[0].nude) }} bernoulli_print() Output: B( 0) = 1 / 1 B( 1) = 1 / 2 B( 2) = 1 / 6 B( 4) = -1 / 30 B( 6) = 1 / 42 B( 8) = -1 / 30 B(10) = 5 / 66 B(12) = -691 / 2730 B(14) = 7 / 6 B(16) = -3617 / 510 B(18) = 43867 / 798 B(20) = -174611 / 330 B(22) = 854513 / 138 B(24) = -236364091 / 2730 B(26) = 8553103 / 6 B(28) = -23749461029 / 870 B(30) = 8615841276005 / 14322 B(32) = -7709321041217 / 510 B(34) = 2577687858367 / 6 B(36) = -26315271553053477373 / 1919190 B(38) = 2929993913841559 / 6 B(40) = -261082718496449122051 / 13530 B(42) = 1520097643918070802691 / 1806 B(44) = -27833269579301024235023 / 690 B(46) = 596451111593912163277961 / 282 B(48) = -5609403368997817686249127547 / 46410 B(50) = 495057205241079648212477525 / 66 B(52) = -801165718135489957347924991853 / 1590 B(54) = 29149963634884862421418123812691 / 798 B(56) = -2479392929313226753685415739663229 / 870 B(58) = 84483613348880041862046775994036021 / 354 B(60) = -1215233140483755572040304994079820246041491 / 56786730  ## SPAD  for n in 0..60 | (b:=bernoulli(n)$INTHEORY; b~=0) repeat print [n,b]

Package:IntegerNumberTheoryFunctions

Output:
===============
Format: [n,B_n]
===============
[0,1]
1
[1,- -]
2
1
[2,-]
6
1
[4,- --]
30
1
[6,--]
42
1
[8,- --]
30
5
[10,--]
66
691
[12,- ----]
2730
7
[14,-]
6
3617
[16,- ----]
510
43867
[18,-----]
798
174611
[20,- ------]
330
854513
[22,------]
138
236364091
[24,- ---------]
2730
8553103
[26,-------]
6
23749461029
[28,- -----------]
870
8615841276005
[30,-------------]
14322
7709321041217
[32,- -------------]
510
2577687858367
[34,-------------]
6
26315271553053477373
[36,- --------------------]
1919190
2929993913841559
[38,----------------]
6
261082718496449122051
[40,- ---------------------]
13530
1520097643918070802691
[42,----------------------]
1806
27833269579301024235023
[44,- -----------------------]
690
596451111593912163277961
[46,------------------------]
282
5609403368997817686249127547
[48,- ----------------------------]
46410
495057205241079648212477525
[50,---------------------------]
66
801165718135489957347924991853
[52,- ------------------------------]
1590
29149963634884862421418123812691
[54,--------------------------------]
798
2479392929313226753685415739663229
[56,- ----------------------------------]
870
84483613348880041862046775994036021
[58,-----------------------------------]
354
1215233140483755572040304994079820246041491
[60,- -------------------------------------------]
56786730
Type: Void


## Tcl

proc bernoulli {n} {    for {set m 0} {$m <=$n} {incr m} {	lappend A [list 1 [expr {$m + 1}]] for {set j$m} {[set i $j] >= 1} {} { lassign [lindex$A [incr j -1]] a1 b1	    lassign [lindex $A$i] a2 b2	    set x [set p [expr {$i * ($a1*$b2 -$a2*$b1)}]] set y [set q [expr {$b1 * $b2}]] while {$q} {set q [expr {$p % [set p$q]}]}	    lset A $j [list [expr {$x/$p}] [expr {$y/$p}]] } } return [lindex$A 0]} set len 0for {set n 0} {$n <= 60} {incr n} { set b [bernoulli$n]    if {[lindex $b 0]} { lappend result$n {*}$b set len [expr {max($len, [string length [lindex $b 0]])}] }}foreach {n num denom}$result {    puts [format {B_%-2d = %*lld/%lld} $n$len $num$denom]}
Output:
B_0  =                                            1/1
B_1  =                                            1/2
B_2  =                                            1/6
B_4  =                                           -1/30
B_6  =                                            1/42
B_8  =                                           -1/30
B_10 =                                            5/66
B_12 =                                         -691/2730
B_14 =                                            7/6
B_16 =                                        -3617/510
B_18 =                                        43867/798
B_20 =                                      -174611/330
B_22 =                                       854513/138
B_24 =                                   -236364091/2730
B_26 =                                      8553103/6
B_28 =                                 -23749461029/870
B_30 =                                8615841276005/14322
B_32 =                               -7709321041217/510
B_34 =                                2577687858367/6
B_36 =                        -26315271553053477373/1919190
B_38 =                             2929993913841559/6
B_40 =                       -261082718496449122051/13530
B_42 =                       1520097643918070802691/1806
B_44 =                     -27833269579301024235023/690
B_46 =                     596451111593912163277961/282
B_48 =                -5609403368997817686249127547/46410
B_50 =                  495057205241079648212477525/66
B_52 =              -801165718135489957347924991853/1590
B_54 =             29149963634884862421418123812691/798
B_56 =          -2479392929313226753685415739663229/870
B_58 =          84483613348880041862046775994036021/354
B_60 = -1215233140483755572040304994079820246041491/56786730


## Visual Basic .NET

Works with: Visual Basic .NET version 2013
' Bernoulli numbers - vb.net - 06/03/2017Imports System.Numerics 'BigInteger Module Bernoulli_numbers     Function gcd_BigInt(ByVal x As BigInteger, ByVal y As BigInteger) As BigInteger        Dim y2 As BigInteger        x = BigInteger.Abs(x)        Do            y2 = BigInteger.Remainder(x, y)            x = y            y = y2        Loop Until y = 0        Return x    End Function 'gcd_BigInt     Sub bernoul_BigInt(n As Integer, ByRef bnum As BigInteger, ByRef bden As BigInteger)        Dim j, m As Integer        Dim f As BigInteger        Dim anum(), aden() As BigInteger        ReDim anum(n + 1), aden(n + 1)        For m = 0 To n            anum(m + 1) = 1            aden(m + 1) = m + 1            For j = m To 1 Step -1                anum(j) = j * (aden(j + 1) * anum(j) - aden(j) * anum(j + 1))                aden(j) = aden(j) * aden(j + 1)                f = gcd_BigInt(BigInteger.Abs(anum(j)), BigInteger.Abs(aden(j)))                If f <> 1 Then                    anum(j) = anum(j) / f                    aden(j) = aden(j) / f                End If            Next        Next        bnum = anum(1) : bden = aden(1)    End Sub 'bernoul_BigInt     Sub bernoulli_BigInt()        Dim i As Integer        Dim bnum, bden As BigInteger        bnum = 0 : bden = 0        For i = 0 To 60            bernoul_BigInt(i, bnum, bden)            If bnum <> 0 Then                Console.WriteLine("B(" & i & ")=" & bnum.ToString("D") & "/" & bden.ToString("D"))            End If        Next i    End Sub 'bernoulli_BigInt End Module 'Bernoulli_numbers
Output:
B(0)=1/1
B(1)=1/2
B(2)=1/6
B(4)=-1/30
B(6)=1/42
B(8)=-1/30
B(10)=5/66
B(12)=-691/2730
B(14)=7/6
B(16)=-3617/510
B(18)=43867/798
B(20)=-174611/330
B(22)=854513/138
B(24)=-236364091/2730
B(26)=8553103/6
B(28)=-23749461029/870
B(30)=8615841276005/14322
B(32)=-7709321041217/510
B(34)=2577687858367/6
B(36)=-26315271553053477373/1919190
B(38)=2929993913841559/6
B(40)=-261082718496449122051/13530
B(42)=1520097643918070802691/1806
B(44)=-27833269579301024235023/690
B(46)=596451111593912163277961/282
B(48)=-5609403368997817686249127547/46410
B(50)=495057205241079648212477525/66
B(52)=-801165718135489957347924991853/1590
B(54)=29149963634884862421418123812691/798
B(56)=-2479392929313226753685415739663229/870
B(58)=84483613348880041862046775994036021/354
B(60)=-1215233140483755572040304994079820246041491/56786730


## zkl

Translation of: EchoLisp

Uses lib GMP (GNU MP Bignum Library).

class Rational{  // Weenie Rational class, can handle BigInts   fcn init(_a,_b){ var a=_a, b=_b; normalize(); }   fcn toString{ "%50d / %d".fmt(a,b) }   fcn normalize{  // divide a and b by gcd      g:= a.gcd(b);      a/=g; b/=g;      if(b<0){ a=-a; b=-b; } // denominator > 0      self   }   fcn __opAdd(n){      if(Rational.isChildOf(n)) self(a*n.b + b*n.a, b*n.b); // Rat + Rat      else self(b*n + a, b);				    // Rat + Int   }   fcn __opSub(n){ self(a*n.b - b*n.a, b*n.b) }		    // Rat - Rat   fcn __opMul(n){      if(Rational.isChildOf(n)) self(a*n.a, b*n.b);	    // Rat * Rat      else self(a*n, b);				    // Rat * Int   }   fcn __opDiv(n){ self(a*n.b,b*n.a) }			    // Rat / Rat}
var [const] BN=Import.lib("zklBigNum");	// libGMP (GNU MP Bignum Library)fcn B(N){				// calculate Bernoulli(n)   var A=List.createLong(100,0);  // aka static aka not thread safe   foreach m in (N+1){      A[m]=Rational(BN(1),BN(m+1));      foreach j in ([m..1, -1]){ A[j-1]= (A[j-1] - A[j])*j; }   }   A[0]}
foreach b in ([0..1].chain([2..60,2])){ println("B(%2d)%s".fmt(b,B(b))) }
Output:
B( 0)                                                 1 / 1
B( 1)                                                 1 / 2
B( 2)                                                 1 / 6
B( 4)                                                -1 / 30
B( 6)                                                 1 / 42
B( 8)                                                -1 / 30
B(10)                                                 5 / 66
B(12)                                              -691 / 2730
B(14)                                                 7 / 6
B(16)                                             -3617 / 510
B(18)                                             43867 / 798
B(20)                                           -174611 / 330
B(22)                                            854513 / 138
B(24)                                        -236364091 / 2730
B(26)                                           8553103 / 6
B(28)                                      -23749461029 / 870
B(30)                                     8615841276005 / 14322
B(32)                                    -7709321041217 / 510
B(34)                                     2577687858367 / 6
B(36)                             -26315271553053477373 / 1919190
B(38)                                  2929993913841559 / 6
B(40)                            -261082718496449122051 / 13530
B(42)                            1520097643918070802691 / 1806
B(44)                          -27833269579301024235023 / 690
B(46)                          596451111593912163277961 / 282
B(48)                     -5609403368997817686249127547 / 46410
B(50)                       495057205241079648212477525 / 66
B(52)                   -801165718135489957347924991853 / 1590
B(54)                  29149963634884862421418123812691 / 798
B(56)               -2479392929313226753685415739663229 / 870
B(58)               84483613348880041862046775994036021 / 354
B(60)      -1215233140483755572040304994079820246041491 / 56786730