Faulhaber's formula: Difference between revisions
(→{{header|Sidef}}: better solution, using Math::Polynomial) |
|||
| Line 2,112: | Line 2,112: | ||
'''Note:''' a number of the formulae above are invisible to the majority of browsers, including Chrome, IE/Edge, Safari and Opera. They may (subject to the installation of necessary fronts) be visible to Firefox. |
'''Note:''' a number of the formulae above are '''invisible''' to the majority of browsers, including Chrome, IE/Edge, Safari and Opera. They may (subject to the installation of necessary fronts) be visible to some Firefox installations. |
||
Revision as of 19:23, 2 February 2020
In mathematics, Faulhaber's formula, named after Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers as a (p + 1)th-degree polynomial function of n, the coefficients involving Bernoulli numbers.
- Task
Generate the first 10 closed-form expressions, starting with p = 0.
- Related tasks
- See also
-
- The Wikipedia entry: Faulhaber's formula.
- The Wikipedia entry: Bernoulli numbers.
- The Wikipedia entry: binomial coefficients.
C
<lang c>#include <stdbool.h>
- include <stdio.h>
- include <stdlib.h>
int binomial(int n, int k) {
int num, denom, i;
if (n < 0 || k < 0 || n < k) return -1; if (n == 0 || k == 0) return 1;
num = 1;
for (i = k + 1; i <= n; ++i) {
num = num * i;
}
denom = 1;
for (i = 2; i <= n - k; ++i) {
denom *= i;
}
return num / denom;
}
int gcd(int a, int b) {
int temp;
while (b != 0) {
temp = a % b;
a = b;
b = temp;
}
return a;
}
typedef struct tFrac {
int num, denom;
} Frac;
Frac makeFrac(int n, int d) {
Frac result; int g;
if (d == 0) {
result.num = 0;
result.denom = 0;
return result;
}
if (n == 0) {
d = 1;
} else if (d < 0) {
n = -n;
d = -d;
}
g = abs(gcd(n, d));
if (g > 1) {
n = n / g;
d = d / g;
}
result.num = n; result.denom = d; return result;
}
Frac negateFrac(Frac f) {
return makeFrac(-f.num, f.denom);
}
Frac subFrac(Frac lhs, Frac rhs) {
return makeFrac(lhs.num * rhs.denom - lhs.denom * rhs.num, rhs.denom * lhs.denom);
}
Frac multFrac(Frac lhs, Frac rhs) {
return makeFrac(lhs.num * rhs.num, lhs.denom * rhs.denom);
}
bool equalFrac(Frac lhs, Frac rhs) {
return (lhs.num == rhs.num) && (lhs.denom == rhs.denom);
}
bool lessFrac(Frac lhs, Frac rhs) {
return (lhs.num * rhs.denom) < (rhs.num * lhs.denom);
}
void printFrac(Frac f) {
printf("%d", f.num);
if (f.denom != 1) {
printf("/%d", f.denom);
}
}
Frac bernoulli(int n) {
Frac a[16]; int j, m;
if (n < 0) {
a[0].num = 0;
a[0].denom = 0;
return a[0];
}
for (m = 0; m <= n; ++m) {
a[m] = makeFrac(1, m + 1);
for (j = m; j >= 1; --j) {
a[j - 1] = multFrac(subFrac(a[j - 1], a[j]), makeFrac(j, 1));
}
}
if (n != 1) {
return a[0];
}
return negateFrac(a[0]);
}
void faulhaber(int p) {
Frac coeff, q; int j, pwr, sign;
printf("%d : ", p);
q = makeFrac(1, p + 1);
sign = -1;
for (j = 0; j <= p; ++j) {
sign = -1 * sign;
coeff = multFrac(multFrac(multFrac(q, makeFrac(sign, 1)), makeFrac(binomial(p + 1, j), 1)), bernoulli(j));
if (equalFrac(coeff, makeFrac(0, 1))) {
continue;
}
if (j == 0) {
if (!equalFrac(coeff, makeFrac(1, 1))) {
if (equalFrac(coeff, makeFrac(-1, 1))) {
printf("-");
} else {
printFrac(coeff);
}
}
} else {
if (equalFrac(coeff, makeFrac(1, 1))) {
printf(" + ");
} else if (equalFrac(coeff, makeFrac(-1, 1))) {
printf(" - ");
} else if (lessFrac(makeFrac(0, 1), coeff)) {
printf(" + ");
printFrac(coeff);
} else {
printf(" - ");
printFrac(negateFrac(coeff));
}
}
pwr = p + 1 - j;
if (pwr > 1) {
printf("n^%d", pwr);
} else {
printf("n");
}
}
printf("\n");
}
int main() {
int i;
for (i = 0; i < 10; ++i) {
faulhaber(i);
}
return 0;
}</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
C++
Uses C++17 <lang cpp>#include <iostream>
- include <numeric>
- include <sstream>
- include <vector>
class Frac { public: Frac(long n, long d) { if (d == 0) { throw new std::runtime_error("d must not be zero"); }
long nn = n; long dd = d; if (nn == 0) { dd = 1; } else if (dd < 0) { nn = -nn; dd = -dd; }
long g = abs(std::gcd(nn, dd)); if (g > 1) { nn /= g; dd /= g; }
num = nn; denom = dd; }
Frac operator-() const { return Frac(-num, denom); }
Frac operator+(const Frac& rhs) const { return Frac(num*rhs.denom + denom * rhs.num, rhs.denom*denom); }
Frac operator-(const Frac& rhs) const { return Frac(num*rhs.denom - denom * rhs.num, rhs.denom*denom); }
Frac operator*(const Frac& rhs) const { return Frac(num*rhs.num, denom*rhs.denom); }
bool operator==(const Frac& rhs) const { return num == rhs.num && denom == rhs.denom; }
bool operator!=(const Frac& rhs) const { return num != rhs.num || denom != rhs.denom; }
bool operator<(const Frac& rhs) const { if (denom == rhs.denom) { return num < rhs.num; } return num * rhs.denom < rhs.num * denom; }
friend std::ostream& operator<<(std::ostream&, const Frac&);
static Frac ZERO() { return Frac(0, 1); }
static Frac ONE() { return Frac(1, 1); }
private: long num; long denom; };
std::ostream & operator<<(std::ostream & os, const Frac &f) { if (f.num == 0 || f.denom == 1) { return os << f.num; }
std::stringstream ss; ss << f.num << "/" << f.denom; return os << ss.str(); }
Frac bernoulli(int n) { if (n < 0) { throw new std::runtime_error("n may not be negative or zero"); }
std::vector<Frac> a; for (int m = 0; m <= n; m++) { a.push_back(Frac(1, m + 1)); for (int j = m; j >= 1; j--) { a[j - 1] = (a[j - 1] - a[j]) * Frac(j, 1); } }
// returns 'first' Bernoulli number if (n != 1) return a[0]; return -a[0]; }
int binomial(int n, int k) { if (n < 0 || k < 0 || n < k) { throw new std::runtime_error("parameters are invalid"); } if (n == 0 || k == 0) return 1;
int num = 1; for (int i = k + 1; i <= n; i++) { num *= i; }
int denom = 1; for (int i = 2; i <= n - k; i++) { denom *= i; }
return num / denom; }
void faulhaber(int p) { using namespace std; cout << p << " : ";
auto q = Frac(1, p + 1); int sign = -1; for (int j = 0; j < p + 1; j++) { sign *= -1; auto coeff = q * Frac(sign, 1) * Frac(binomial(p + 1, j), 1) * bernoulli(j); if (coeff == Frac::ZERO()) { continue; } if (j == 0) { if (coeff == -Frac::ONE()) { cout << "-"; } else if (coeff != Frac::ONE()) { cout << coeff; } } else { if (coeff == Frac::ONE()) { cout << " + "; } else if (coeff == -Frac::ONE()) { cout << " - "; } else if (coeff < Frac::ZERO()) { cout << " - " << -coeff; } else { cout << " + " << coeff; } } int pwr = p + 1 - j; if (pwr > 1) { cout << "n^" << pwr; } else { cout << "n"; } } cout << endl; }
int main() { for (int i = 0; i < 10; i++) { faulhaber(i); }
return 0; }</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
C#
<lang csharp>using System;
namespace FaulhabersFormula {
internal class Frac {
private long num;
private long denom;
public static readonly Frac ZERO = new Frac(0, 1);
public static readonly Frac ONE = new Frac(1, 1);
public Frac(long n, long d) {
if (d == 0) {
throw new ArgumentException("d must not be zero");
}
long nn = n;
long dd = d;
if (nn == 0) {
dd = 1;
}
else if (dd < 0) {
nn = -nn;
dd = -dd;
}
long g = Math.Abs(Gcd(nn, dd));
if (g > 1) {
nn /= g;
dd /= g;
}
num = nn;
denom = dd;
}
private static long Gcd(long a, long b) {
if (b == 0) {
return a;
}
return Gcd(b, a % b);
}
public static Frac operator -(Frac self) {
return new Frac(-self.num, self.denom);
}
public static Frac operator +(Frac lhs, Frac rhs) {
return new Frac(lhs.num * rhs.denom + lhs.denom * rhs.num, rhs.denom * lhs.denom);
}
public static Frac operator -(Frac lhs, Frac rhs) {
return lhs + -rhs;
}
public static Frac operator *(Frac lhs, Frac rhs) {
return new Frac(lhs.num * rhs.num, lhs.denom * rhs.denom);
}
public static bool operator <(Frac lhs, Frac rhs) {
double x = (double)lhs.num / lhs.denom;
double y = (double)rhs.num / rhs.denom;
return x < y;
}
public static bool operator >(Frac lhs, Frac rhs) {
double x = (double)lhs.num / lhs.denom;
double y = (double)rhs.num / rhs.denom;
return x > y;
}
public static bool operator ==(Frac lhs, Frac rhs) {
return lhs.num == rhs.num && lhs.denom == rhs.denom;
}
public static bool operator !=(Frac lhs, Frac rhs) {
return lhs.num != rhs.num || lhs.denom != rhs.denom;
}
public override string ToString() {
if (denom == 1) {
return num.ToString();
}
return string.Format("{0}/{1}", num, denom);
}
public override bool Equals(object obj) {
var frac = obj as Frac;
return frac != null &&
num == frac.num &&
denom == frac.denom;
}
public override int GetHashCode() {
var hashCode = 1317992671;
hashCode = hashCode * -1521134295 + num.GetHashCode();
hashCode = hashCode * -1521134295 + denom.GetHashCode();
return hashCode;
}
}
class Program {
static Frac Bernoulli(int n) {
if (n < 0) {
throw new ArgumentException("n may not be negative or zero");
}
Frac[] a = new Frac[n + 1];
for (int m = 0; m <= n; m++) {
a[m] = new Frac(1, m + 1);
for (int j = m; j >= 1; j--) {
a[j - 1] = (a[j - 1] - a[j]) * new Frac(j, 1);
}
}
// returns 'first' Bernoulli number
if (n != 1) return a[0];
return -a[0];
}
static int Binomial(int n, int k) {
if (n < 0 || k < 0 || n < k) {
throw new ArgumentException();
}
if (n == 0 || k == 0) return 1;
int num = 1;
for (int i = k + 1; i <= n; i++) {
num = num * i;
}
int denom = 1;
for (int i = 2; i <= n - k; i++) {
denom = denom * i;
}
return num / denom;
}
static void Faulhaber(int p) {
Console.Write("{0} : ", p);
Frac q = new Frac(1, p + 1);
int sign = -1;
for (int j = 0; j <= p; j++) {
sign *= -1;
Frac coeff = q * new Frac(sign, 1) * new Frac(Binomial(p + 1, j), 1) * Bernoulli(j);
if (Frac.ZERO == coeff) continue;
if (j == 0) {
if (Frac.ONE != coeff) {
if (-Frac.ONE == coeff) {
Console.Write("-");
}
else {
Console.Write(coeff);
}
}
}
else {
if (Frac.ONE == coeff) {
Console.Write(" + ");
}
else if (-Frac.ONE == coeff) {
Console.Write(" - ");
}
else if (Frac.ZERO < coeff) {
Console.Write(" + {0}", coeff);
}
else {
Console.Write(" - {0}", -coeff);
}
}
int pwr = p + 1 - j;
if (pwr > 1) {
Console.Write("n^{0}", pwr);
}
else {
Console.Write("n");
}
}
Console.WriteLine();
}
static void Main(string[] args) {
for (int i = 0; i < 10; i++) {
Faulhaber(i);
}
}
}
}</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
D
<lang D>import std.algorithm : fold; import std.exception : enforce; import std.format : formattedWrite; import std.numeric : cmp, gcd; import std.range : iota; import std.stdio; import std.traits;
auto abs(T)(T val) if (isNumeric!T) {
if (val < 0) {
return -val;
}
return val;
}
struct Frac {
long num; long denom;
enum ZERO = Frac(0, 1); enum ONE = Frac(1, 1);
this(long n, long d) in {
enforce(d != 0, "Parameter d may not be zero.");
} body {
auto nn = n;
auto dd = d;
if (nn == 0) {
dd = 1;
} else if (dd < 0) {
nn = -nn;
dd = -dd;
}
auto g = gcd(abs(nn), abs(dd));
if (g > 1) {
nn /= g;
dd /= g;
}
num = nn;
denom = dd;
}
auto opBinary(string op)(Frac rhs) const {
static if (op == "+" || op == "-") {
return mixin("Frac(num*rhs.denom"~op~"denom*rhs.num, rhs.denom*denom)");
} else if (op == "*") {
return Frac(num*rhs.num, denom*rhs.denom);
}
}
auto opUnary(string op : "-")() const {
return Frac(-num, denom);
}
int opCmp(Frac rhs) const {
return cmp(cast(real) this, cast(real) rhs);
}
bool opEquals(Frac rhs) const {
return num == rhs.num && denom == rhs.denom;
}
void toString(scope void delegate(const(char)[]) sink) const {
if (denom == 1) {
formattedWrite(sink, "%d", num);
} else {
formattedWrite(sink, "%d/%s", num, denom);
}
}
T opCast(T)() const if (isFloatingPoint!T) {
return cast(T) num / denom;
}
}
auto abs(Frac f) {
if (f.num >= 0) {
return f;
}
return -f;
}
auto bernoulli(int n) in {
enforce(n >= 0, "Parameter n must not be negative.");
} body {
Frac[] a;
a.length = n+1;
a[0] = Frac.ZERO;
foreach (m; 0..n+1) {
a[m] = Frac(1, m+1);
foreach_reverse (j; 1..m+1) {
a[j-1] = (a[j-1] - a[j]) * Frac(j, 1);
}
}
if (n != 1) {
return a[0];
}
return -a[0];
}
auto binomial(int n, int k) in {
enforce(n>=0 && k>=0 && n>=k);
} body {
if (n==0 || k==0) return 1; auto num = iota(k+1, n+1).fold!"a*b"(1); auto den = iota(2, n-k+1).fold!"a*b"(1); return num / den;
}
auto faulhaber(int p) {
write(p, " : ");
auto q = Frac(1, p+1);
auto sign = -1;
foreach (j; 0..p+1) {
sign *= -1;
auto coeff = q * Frac(sign, 1) * Frac(binomial(p+1, j), 1) * bernoulli(j);
if (coeff == Frac.ZERO) continue;
if (j == 0) {
if (coeff == -Frac.ONE) {
write("-");
} else if (coeff != Frac.ONE) {
write(coeff);
}
} else {
if (coeff == Frac.ONE) {
write(" + ");
} else if (coeff == -Frac.ONE) {
write(" - ");
} else if (coeff > Frac.ZERO) {
write(" + ", coeff);
} else {
write(" - ", -coeff);
}
}
auto pwr = p + 1 - j;
if (pwr > 1) {
write("n^", pwr);
} else {
write("n");
}
}
writeln;
}
void main() {
foreach (i; 0..10) {
faulhaber(i);
}
}</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
EchoLisp
<lang scheme> (lib 'math) ;; for bernoulli numbers (string-delimiter "")
- returns list of polynomial coefficients
(define (Faulhaber p) (cons 0 (for/list ([k (in-range p -1 -1)]) (* (Cnp (1+ p) k) (bernoulli k)))))
- prints formal polynomial
(define (task (pmax 10))
(for ((p pmax)) (writeln p '→ (/ 1 (1+ p)) '* (poly->string 'n (Faulhaber p)))))
- extra credit - compute sums
(define (Faulcomp n p) (printf "Σ(1..%d) n^%d = %d" n p (/ (poly n (Faulhaber p)) (1+ p) ))) </lang>
- Output:
(task)
0 → 1 * n
1 → 1/2 * n^2 + n
2 → 1/3 * n^3 + 3/2 n^2 + 1/2 n
3 → 1/4 * n^4 + 2 n^3 + n^2
4 → 1/5 * n^5 + 5/2 n^4 + 5/3 n^3 -1/6 n
5 → 1/6 * n^6 + 3 n^5 + 5/2 n^4 -1/2 n^2
6 → 1/7 * n^7 + 7/2 n^6 + 7/2 n^5 -7/6 n^3 + 1/6 n
7 → 1/8 * n^8 + 4 n^7 + 14/3 n^6 -7/3 n^4 + 2/3 n^2
8 → 1/9 * n^9 + 9/2 n^8 + 6 n^7 -21/5 n^5 + 2 n^3 -3/10 n
9 → 1/10 * n^10 + 5 n^9 + 15/2 n^8 -7 n^6 + 5 n^4 -3/2 n^2
(Faulcomp 100 2)
Σ(1..100) n^2 = 338350
(Faulcomp 100 1)
Σ(1..100) n^1 = 5050
(lib 'bigint)
(Faulcomp 100 9)
Σ(1..100) n^9 = 10507499300049998000
;; check it ...
(for/sum ((n 101)) (expt n 9))
→ 10507499300049998500
Factor
<lang factor>USING: formatting kernel math math.combinatorics math.extras math.functions regexp sequences ;
- faulhaber ( p -- seq )
1 + dup recip swap dup <iota> [ [ nCk ] [ -1 swap ^ ] [ bernoulli ] tri * * * ] 2with map ;
- (poly>str) ( seq -- str )
reverse [ 1 + "%un^%d" sprintf ] map-index reverse " + " join ;
- clean-up ( str -- str' )
R/ n\^1\z/ "n" re-replace ! Change n^1 to n. R/ 1n/ "n" re-replace ! Change 1n to n. R/ \+ -/ "- " re-replace ! Change + - to - . R/ [+-] 0n(\^\d+ )?/ "" re-replace ; ! Remove terms of zero.
- poly>str ( seq -- str ) (poly>str) clean-up ;
10 [ dup faulhaber poly>str "%d: %s\n" printf ] each-integer</lang>
- Output:
0: n 1: 1/2n^2 + 1/2n 2: 1/3n^3 + 1/2n^2 + 1/6n 3: 1/4n^4 + 1/2n^3 + 1/4n^2 4: 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5: 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6: 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7: 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8: 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9: 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
Fōrmulæ
In this page you can see the solution of this task.
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text (more info). Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition.
The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.
GAP
Straightforward implementation using GAP polynomials, and two different formulas: one based on Stirling numbers of the second kind (sum1, see Python implementation below in this page), and the usual Faulhaber formula (sum2). No optimization is made (one could compute Stirling numbers row by row, or the product in sum1 may be kept from one call to the other). Notice the Bernoulli term in the first formula is here only to correct the value of sum1(0), which is off by one because sum1 computes sums from 0 to n.
<lang gap>n := X(Rationals, "n"); sum1 := p -> Sum([0 .. p], k -> Stirling2(p, k) * Product([0 .. k], j -> n + 1 - j) / (k + 1)) + 2 * Bernoulli(2 * p + 1); sum2 := p -> Sum([0 .. p], j -> (-1)^j * Binomial(p + 1, j) * Bernoulli(j) * n^(p + 1 - j)) / (p + 1); ForAll([0 .. 20], k -> sum1(k) = sum2(k));
for p in [0 .. 9] do
Print(sum1(p), "\n");
od;
n 1/2*n^2+1/2*n 1/3*n^3+1/2*n^2+1/6*n 1/4*n^4+1/2*n^3+1/4*n^2 1/5*n^5+1/2*n^4+1/3*n^3-1/30*n 1/6*n^6+1/2*n^5+5/12*n^4-1/12*n^2 1/7*n^7+1/2*n^6+1/2*n^5-1/6*n^3+1/42*n 1/8*n^8+1/2*n^7+7/12*n^6-7/24*n^4+1/12*n^2 1/9*n^9+1/2*n^8+2/3*n^7-7/15*n^5+2/9*n^3-1/30*n 1/10*n^10+1/2*n^9+3/4*n^8-7/10*n^6+1/2*n^4-3/20*n^2</lang>
Go
<lang Go>package main
import ( "fmt" "math/big" )
func bernoulli(z *big.Rat, n int64) *big.Rat { if z == nil { z = new(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(z.SetInt64(int64(j)), d.Sub(d, &a[j])) } } return z.Set(&a[0]) }
func main() { // allocate needed big.Rat's once q := new(big.Rat) c := new(big.Rat) // coefficients be := new(big.Rat) // for Bernoulli numbers bi := big.NewRat(1, 1) // for binomials
for p := int64(0); p < 10; p++ { fmt.Print(p, " : ") q.SetFrac64(1, p+1) neg := true for j := int64(0); j <= p; j++ { neg = !neg if neg { c.Neg(q) } else { c.Set(q) } bi.Num().Binomial(p+1, j) bernoulli(be, j) c.Mul(c, bi) c.Mul(c, be) if c.Num().BitLen() == 0 { continue } if j == 0 { fmt.Printf(" %4s", c.RatString()) } else { fmt.Printf(" %+2d/%-2d", c.Num(), c.Denom()) } fmt.Print("×n") if exp := p + 1 - j; exp > 1 { fmt.Printf("^%-2d", exp) } } fmt.Println() } }</lang>
- Output:
0 : 1×n 1 : 1/2×n^2 -1/2 ×n 2 : 1/3×n^3 -1/2 ×n^2 +1/6 ×n 3 : 1/4×n^4 -1/2 ×n^3 +1/4 ×n^2 4 : 1/5×n^5 -1/2 ×n^4 +1/3 ×n^3 -1/30×n 5 : 1/6×n^6 -1/2 ×n^5 +5/12×n^4 -1/12×n^2 6 : 1/7×n^7 -1/2 ×n^6 +1/2 ×n^5 -1/6 ×n^3 +1/42×n 7 : 1/8×n^8 -1/2 ×n^7 +7/12×n^6 -7/24×n^4 +1/12×n^2 8 : 1/9×n^9 -1/2 ×n^8 +2/3 ×n^7 -7/15×n^5 +2/9 ×n^3 -1/30×n 9 : 1/10×n^10 -1/2 ×n^9 +3/4 ×n^8 -7/10×n^6 +1/2 ×n^4 -3/20×n^2
Haskell
Bernouilli polynomials
<lang Haskell>import Data.Ratio ((%), numerator, denominator) import Data.List (intercalate, transpose) import Control.Arrow ((&&&), (***)) import Data.Char (isSpace) import Data.Monoid ((<>)) import Data.Bool (bool)
-- FAULHABER ---------------------------------------------- faulhaber :: Rational faulhaber =
tail $
scanl
(\rs n ->
let xs = zipWith ((*) . (n %)) [2 ..] rs
in 1 - sum xs : xs)
[]
[0 ..]
-- EXPRESSION STRINGS ------------------------------------- polynomials :: (String, String) polynomials = fmap ((ratioPower =<<) . reverse . flip zip [1 ..]) faulhaber
-- Rows of (Power string, Ratio string) tuples -> Printable lines expressionTable :: (String, String) -> [String] expressionTable ps =
let cols = transpose (fullTable ps)
in expressionRow <$>
zip
[0 ..]
(transpose $
zipWith
(\(lw, rw) col ->
(fmap (justifyLeft lw ' ' *** justifyLeft rw ' ') col))
(colWidths cols)
cols)
-- Value pair -> String pair (lifted into list for use with >>=) ratioPower :: (Rational, Integer) -> [(String, String)] ratioPower (nd, j) =
let (num, den) = (numerator &&& denominator) nd
sn
| num == 0 = []
| (j /= 1) = ("n^" <> show j)
| otherwise = "n"
sr
| num == 0 = []
| den == 1 && num == 1 = []
| den == 1 = show num <> "n"
| otherwise = intercalate "/" [show num, show den]
s = sr <> sn
in bool [(sn, sr)] [] (null s)
-- Rows of uneven length -> All rows padded to length of longest fullTable :: (String, String) -> (String, String) fullTable xs =
let lng = maximum $ length <$> xs in (<>) <*> (flip replicate ([], []) . (-) lng . length) <$> xs
justifyLeft :: Int -> Char -> String -> String justifyLeft n c s = take n (s <> replicate n c)
-- (Row index, Expression pairs) -> String joined by conjunctions expressionRow :: (Int, [(String, String)]) -> String expressionRow (i, row) =
concat
[ show i
, " -> "
, foldr
(\s a -> concat [s, bool " + " " " (blank a || head a == '-'), a])
[]
(polyTerm <$> row)
]
-- (Power string, Ratio String) -> Combined string with possible '*' polyTerm :: (String, String) -> String polyTerm (l, r)
| blank l || blank r = l <> r | head r == '-' = concat ["- ", l, " * ", tail r] | otherwise = intercalate " * " [l, r]
blank :: String -> Bool blank = all isSpace
-- Maximum widths of power and ratio elements in each column colWidths :: (String, String) -> [(Int, Int)] colWidths =
fmap
(foldr
(\(ls, rs) (lMax, rMax) -> (max (length ls) lMax, max (length rs) rMax))
(0, 0))
-- Length of string excluding any leading '-' unsignedLength :: String -> Int unsignedLength xs =
let l = length xs
in bool (bool l (l - 1) ('-' == head xs)) 0 (0 == l)
-- TEST --------------------------------------------------- main :: IO () main = (putStrLn . unlines . expressionTable . take 10) polynomials</lang>
- Output:
0 -> n 1 -> n^2 * 1/2 + n * 1/2 2 -> n^3 * 1/3 + n^2 * 1/2 + n * 1/6 3 -> n^4 * 1/4 + n^3 * 1/2 + n^2 * 1/4 4 -> n^5 * 1/5 + n^4 * 1/2 + n^3 * 1/3 - n * 1/30 5 -> n^6 * 1/6 + n^5 * 1/2 + n^4 * 5/12 - n^2 * 1/12 6 -> n^7 * 1/7 + n^6 * 1/2 + n^5 * 1/2 - n^3 * 1/6 + n * 1/42 7 -> n^8 * 1/8 + n^7 * 1/2 + n^6 * 7/12 - n^4 * 7/24 + n^2 * 1/12 8 -> n^9 * 1/9 + n^8 * 1/2 + n^7 * 2/3 - n^5 * 7/15 + n^3 * 2/9 - n * 1/30 9 -> n^10 * 1/10 + n^9 * 1/2 + n^8 * 3/4 - n^6 * 7/10 + n^4 * 1/2 - n^2 * 3/20
J
Implementation:
<lang J>Bsecond=:verb define"0
+/,(<:*(_1^[)*!*(y^~1+[)%1+])"0/~i.1x+y
)
Bfirst=: Bsecond - 1&=
Faul=:adverb define
(0,|.(%m+1x) * (_1x&^ * !&(m+1) * Bfirst) i.1+m)&p.
)</lang>
Task example:
<lang J> 0 Faul 0 1x&p.
1 Faul
0 1r2 1r2&p.
2 Faul
0 1r6 1r2 1r3&p.
3 Faul
0 0 1r4 1r2 1r4&p.
4 Faul
0 _1r30 0 1r3 1r2 1r5&p.
5 Faul
0 0 _1r12 0 5r12 1r2 1r6&p.
6 Faul
0 1r42 0 _1r6 0 1r2 1r2 1r7&p.
7 Faul
0 0 1r12 0 _7r24 0 7r12 1r2 1r8&p.
8 Faul
0 _1r30 0 2r9 0 _7r15 0 2r3 1r2 1r9&p.
9 Faul
0 0 _3r20 0 1r2 0 _7r10 0 3r4 1r2 1r10&p.</lang>
Double checking our work:
<lang J> Fcheck=: dyad def'+/(1+i.y)^x'"0
9 Faul i.5
0 1 513 20196 282340
9 Fcheck i.5
0 1 513 20196 282340
2 Faul i.5
0 1 5 14 30
2 Fcheck i.5
0 1 5 14 30</lang>
Java
<lang Java>import java.util.Arrays; import java.util.stream.IntStream;
public class FaulhabersFormula {
private static long gcd(long a, long b) {
if (b == 0) {
return a;
}
return gcd(b, a % b);
}
private static class Frac implements Comparable<Frac> {
private long num;
private long denom;
public static final Frac ZERO = new Frac(0, 1);
public static final Frac ONE = new Frac(1, 1);
public Frac(long n, long d) {
if (d == 0) throw new IllegalArgumentException("d must not be zero");
long nn = n;
long dd = d;
if (nn == 0) {
dd = 1;
} else if (dd < 0) {
nn = -nn;
dd = -dd;
}
long g = Math.abs(gcd(nn, dd));
if (g > 1) {
nn /= g;
dd /= g;
}
num = nn;
denom = dd;
}
public Frac plus(Frac rhs) {
return new Frac(num * rhs.denom + denom * rhs.num, rhs.denom * denom);
}
public Frac unaryMinus() {
return new Frac(-num, denom);
}
public Frac minus(Frac rhs) {
return this.plus(rhs.unaryMinus());
}
public Frac times(Frac rhs) {
return new Frac(this.num * rhs.num, this.denom * rhs.denom);
}
@Override
public int compareTo(Frac o) {
double diff = toDouble() - o.toDouble();
return Double.compare(diff, 0.0);
}
@Override
public boolean equals(Object obj) {
return null != obj && obj instanceof Frac && this.compareTo((Frac) obj) == 0;
}
@Override
public String toString() {
if (denom == 1) {
return Long.toString(num);
}
return String.format("%d/%d", num, denom);
}
private double toDouble() {
return (double) num / denom;
}
}
private static Frac bernoulli(int n) {
if (n < 0) throw new IllegalArgumentException("n may not be negative or zero");
Frac[] a = new Frac[n + 1];
Arrays.fill(a, Frac.ZERO);
for (int m = 0; m <= n; ++m) {
a[m] = new Frac(1, m + 1);
for (int j = m; j >= 1; --j) {
a[j - 1] = a[j - 1].minus(a[j]).times(new Frac(j, 1));
}
}
// returns 'first' Bernoulli number
if (n != 1) return a[0];
return a[0].unaryMinus();
}
private static int binomial(int n, int k) {
if (n < 0 || k < 0 || n < k) throw new IllegalArgumentException();
if (n == 0 || k == 0) return 1;
int num = IntStream.rangeClosed(k + 1, n).reduce(1, (a, b) -> a * b);
int den = IntStream.rangeClosed(2, n - k).reduce(1, (acc, i) -> acc * i);
return num / den;
}
private static void faulhaber(int p) {
System.out.printf("%d : ", p);
Frac q = new Frac(1, p + 1);
int sign = -1;
for (int j = 0; j <= p; ++j) {
sign *= -1;
Frac coeff = q.times(new Frac(sign, 1)).times(new Frac(binomial(p + 1, j), 1)).times(bernoulli(j));
if (Frac.ZERO.equals(coeff)) continue;
if (j == 0) {
if (!Frac.ONE.equals(coeff)) {
if (Frac.ONE.unaryMinus().equals(coeff)) {
System.out.print("-");
} else {
System.out.print(coeff);
}
}
} else {
if (Frac.ONE.equals(coeff)) {
System.out.print(" + ");
} else if (Frac.ONE.unaryMinus().equals(coeff)) {
System.out.print(" - ");
} else if (coeff.compareTo(Frac.ZERO) > 0) {
System.out.printf(" + %s", coeff);
} else {
System.out.printf(" - %s", coeff.unaryMinus());
}
}
int pwr = p + 1 - j;
if (pwr > 1)
System.out.printf("n^%d", pwr);
else
System.out.print("n");
}
System.out.println();
}
public static void main(String[] args) {
for (int i = 0; i <= 9; ++i) {
faulhaber(i);
}
}
}</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
Julia
Module: <lang julia>module Faulhaber
function bernoulli(n::Integer)
n ≥ 0 || throw(DomainError(n, "n must be a positive-or-0 number"))
a = fill(0 // 1, n + 1)
for m in 1:n
a[m] = 1 // (m + 1)
for j in m:-1:2
a[j - 1] = (a[j - 1] - a[j]) * j
end
end
return ifelse(n != 1, a[1], -a[1])
end
const _exponents = collect(Char, "⁰¹²³⁴⁵⁶⁷⁸⁹") toexponent(n) = join(_exponents[reverse(digits(n)) .+ 1])
function formula(p::Integer)
print(p, ": ")
q = 1 // (p + 1)
s = -1
for j in 0:p
s *= -1
coeff = q * s * binomial(p + 1, j) * bernoulli(j)
iszero(coeff) && continue
if iszero(j)
print(coeff == 1 ? "" : coeff == -1 ? "-" : "$coeff")
else
print(coeff == 1 ? " + " : coeff == -1 ? " - " : coeff > 0 ? " + $coeff " : " - $(-coeff) ")
end
pwr = p + 1 - j
if pwr > 1
print("n", toexponent(pwr))
else
print("n")
end
end
println()
end
end # module Faulhaber</lang>
Main: <lang julia>Faulhaber.formula.(1:10)</lang>
- Output:
1: + 1//2 n 2: + 1//2 n² + 1//3 n 3: + 1//2 n³ + 1//2 n² - 1//6 n 4: + 1//2 n⁴ + 2//3 n³ - 1//3 n² + 1//30 n 5: + 1//2 n⁵ + 5//6 n⁴ - 5//9 n³ + 1//12 n² + 1//30 n 6: + 1//2 n⁶ + n⁵ - 5//6 n⁴ + 1//6 n³ + 1//10 n² - 1//42 n 7: + 1//2 n⁷ + 7//6 n⁶ - 7//6 n⁵ + 7//24 n⁴ + 7//30 n³ - 1//12 n² - 1//42 n 8: + 1//2 n⁸ + 4//3 n⁷ - 14//9 n⁶ + 7//15 n⁵ + 7//15 n⁴ - 2//9 n³ - 2//21 n² + 1//30 n 9: + 1//2 n⁹ + 3//2 n⁸ - 2//1 n⁷ + 7//10 n⁶ + 21//25 n⁵ - 1//2 n⁴ - 2//7 n³ + 3//20 n² + 1//30 n 10: + 1//2 n¹⁰ + 5//3 n⁹ - 5//2 n⁸ + n⁷ + 7//5 n⁶ - n⁵ - 5//7 n⁴ + 1//2 n³ + 1//6 n² - 5//66 n
Kotlin
As Kotlin doesn't have support for rational numbers built in, a cut-down version of the Frac class in the Arithmetic/Rational task has been used in order to express the polynomial coefficients as fractions. <lang scala>// version 1.1.2
fun gcd(a: Long, b: Long): Long = if (b == 0L) a else gcd(b, a % b)
class Frac : Comparable<Frac> {
val num: Long val denom: Long
companion object {
val ZERO = Frac(0, 1)
val ONE = Frac(1, 1)
}
constructor(n: Long, d: Long) {
require(d != 0L)
var nn = n
var dd = d
if (nn == 0L) {
dd = 1
}
else if (dd < 0) {
nn = -nn
dd = -dd
}
val g = Math.abs(gcd(nn, dd))
if (g > 1) {
nn /= g
dd /= g
}
num = nn
denom = dd
}
constructor(n: Int, d: Int) : this(n.toLong(), d.toLong())
operator fun plus(other: Frac) =
Frac(num * other.denom + denom * other.num, other.denom * denom)
operator fun unaryMinus() = Frac(-num, denom)
operator fun minus(other: Frac) = this + (-other)
operator fun times(other: Frac) = Frac(this.num * other.num, this.denom * other.denom)
fun abs() = if (num >= 0) this else -this
override fun compareTo(other: Frac): Int {
val diff = this.toDouble() - other.toDouble()
return when {
diff < 0.0 -> -1
diff > 0.0 -> +1
else -> 0
}
}
override fun equals(other: Any?): Boolean {
if (other == null || other !is Frac) return false
return this.compareTo(other) == 0
}
override fun toString() = if (denom == 1L) "$num" else "$num/$denom" fun toDouble() = num.toDouble() / denom
}
fun bernoulli(n: Int): Frac {
require(n >= 0)
val a = Array<Frac>(n + 1) { Frac.ZERO }
for (m in 0..n) {
a[m] = Frac(1, m + 1)
for (j in m downTo 1) a[j - 1] = (a[j - 1] - a[j]) * Frac(j, 1)
}
return if (n != 1) a[0] else -a[0] // returns 'first' Bernoulli number
}
fun binomial(n: Int, k: Int): Int {
require(n >= 0 && k >= 0 && n >= k)
if (n == 0 || k == 0) return 1
val num = (k + 1..n).fold(1) { acc, i -> acc * i }
val den = (2..n - k).fold(1) { acc, i -> acc * i }
return num / den
}
fun faulhaber(p: Int) {
print("$p : ")
val q = Frac(1, p + 1)
var sign = -1
for (j in 0..p) {
sign *= -1
val coeff = q * Frac(sign, 1) * Frac(binomial(p + 1, j), 1) * bernoulli(j)
if (coeff == Frac.ZERO) continue
if (j == 0) {
print(when {
coeff == Frac.ONE -> ""
coeff == -Frac.ONE -> "-"
else -> "$coeff"
})
}
else {
print(when {
coeff == Frac.ONE -> " + "
coeff == -Frac.ONE -> " - "
coeff > Frac.ZERO -> " + $coeff"
else -> " - ${-coeff}"
})
}
val pwr = p + 1 - j
if (pwr > 1)
print("n^${p + 1 - j}")
else
print("n")
}
println()
}
fun main(args: Array<String>) {
for (i in 0..9) faulhaber(i)
}</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
Lua
<lang lua>function binomial(n,k)
if n<0 or k<0 or n<k then return -1 end if n==0 or k==0 then return 1 end
local num = 1
for i=k+1,n do
num = num * i
end
local denom = 1
for i=2,n-k do
denom = denom * i
end
return num / denom
end
function gcd(a,b)
while b ~= 0 do
local temp = a % b
a = b
b = temp
end
return a
end
function makeFrac(n,d)
local result = {}
if d==0 then
result.num = 0
result.denom = 0
return result
end
if n==0 then
d = 1
elseif d < 0 then
n = -n
d = -d
end
local g = math.abs(gcd(n, d))
if g>1 then
n = n / g
d = d / g
end
result.num = n result.denom = d return result
end
function negateFrac(f)
return makeFrac(-f.num, f.denom)
end
function subFrac(lhs, rhs)
return makeFrac(lhs.num * rhs.denom - lhs.denom * rhs.num, rhs.denom * lhs.denom)
end
function multFrac(lhs, rhs)
return makeFrac(lhs.num * rhs.num, lhs.denom * rhs.denom)
end
function equalFrac(lhs, rhs)
return (lhs.num == rhs.num) and (lhs.denom == rhs.denom)
end
function lessFrac(lhs, rhs)
return (lhs.num * rhs.denom) < (rhs.num * lhs.denom)
end
function printFrac(f)
io.write(f.num)
if f.denom ~= 1 then
io.write("/"..f.denom)
end
return nil
end
function bernoulli(n)
if n<0 then
return {num=0, denom=0}
end
local a = {}
for m=0,n do
a[m] = makeFrac(1, m+1)
for j=m,1,-1 do
a[j-1] = multFrac(subFrac(a[j-1], a[j]), makeFrac(j, 1))
end
end
if n~=1 then
return a[0]
end
return negateFrac(a[0])
end
function faulhaber(p)
io.write(p.." : ")
local q = makeFrac(1, p+1)
local sign = -1
for j=0,p do
sign = -1 * sign
local coeff = multFrac(multFrac(multFrac(q, makeFrac(sign, 1)), makeFrac(binomial(p + 1, j), 1)), bernoulli(j))
if not equalFrac(coeff, makeFrac(0, 1)) then
if j==0 then
if not equalFrac(coeff, makeFrac(1, 1)) then
if equalFrac(coeff, makeFrac(-1, 1)) then
io.write("-")
else
printFrac(coeff)
end
end
else
if equalFrac(coeff, makeFrac(1, 1)) then
io.write(" + ")
elseif equalFrac(coeff, makeFrac(-1, 1)) then
io.write(" - ")
elseif lessFrac(makeFrac(0, 1), coeff) then
io.write(" + ")
printFrac(coeff)
else
io.write(" - ")
printFrac(negateFrac(coeff))
end
end
local pwr = p + 1 - j
if pwr>1 then
io.write("n^"..pwr)
else
io.write("n")
end
end
end
print()
return nil
end
-- main for i=0,9 do
faulhaber(i)
end</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
Maxima
<lang maxima>sum1(p):=sum(stirling2(p,k)*pochhammer(n-k+1,k+1)/(k+1),k,0,p)$ sum2(p):=sum((-1)^j*binomial(p+1,j)*bern(j)*n^(p-j+1),j,0,p)/(p+1)$
makelist(expand(sum1(p)-sum2(p)),p,1,10); [0,0,0,0,0,0,0,0,0,0]
for p from 0 thru 9 do print(expand(sum2(p)));</lang>
- Output:
n n^2/2+n/2 n^3/3+n^2/2+n/6 n^4/4+n^3/2+n^2/4 n^5/5+n^4/2+n^3/3-n/30 n^6/6+n^5/2+(5*n^4)/12-n^2/12 n^7/7+n^6/2+n^5/2-n^3/6+n/42 n^8/8+n^7/2+(7*n^6)/12-(7*n^4)/24+n^2/12 n^9/9+n^8/2+(2*n^7)/3-(7*n^5)/15+(2*n^3)/9-n/30 n^10/10+n^9/2+(3*n^8)/4-(7*n^6)/10+n^4/2-(3*n^2)/20
Modula-2
<lang modula2>MODULE Faulhaber; FROM EXCEPTIONS IMPORT AllocateSource,ExceptionSource,GetMessage,RAISE; FROM FormatString IMPORT FormatString; FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
VAR TextWinExSrc : ExceptionSource;
(* Helper Functions *) PROCEDURE Abs(n : INTEGER) : INTEGER; BEGIN
IF n < 0 THEN
RETURN -n
END;
RETURN n
END Abs;
PROCEDURE Binomial(n,k : INTEGER) : INTEGER; VAR i,num,denom : INTEGER; BEGIN
IF (n < 0) OR (k < 0) OR (n < k) THEN
RAISE(TextWinExSrc, 0, "Argument Exception.")
END;
IF (n = 0) OR (k = 0) THEN
RETURN 1
END;
num := 1;
FOR i:=k+1 TO n DO
num := num * i
END;
denom := 1;
FOR i:=2 TO n - k DO
denom := denom * i
END;
RETURN num / denom
END Binomial;
PROCEDURE GCD(a,b : INTEGER) : INTEGER; BEGIN
IF b = 0 THEN
RETURN a
END;
RETURN GCD(b, a MOD b)
END GCD;
PROCEDURE WriteInteger(n : INTEGER); VAR buf : ARRAY[0..15] OF CHAR; BEGIN
FormatString("%i", buf, n);
WriteString(buf)
END WriteInteger;
(* Fraction Handling *) TYPE Frac = RECORD
num,denom : INTEGER;
END;
PROCEDURE InitFrac(n,d : INTEGER) : Frac; VAR nn,dd,g : INTEGER; BEGIN
IF d = 0 THEN
RAISE(TextWinExSrc, 0, "The denominator must not be zero.")
END;
IF n = 0 THEN
d := 1
ELSIF d < 0 THEN
n := -n;
d := -d
END;
g := Abs(GCD(n, d));
IF g > 1 THEN
n := n / g;
d := d / g
END;
RETURN Frac{n, d}
END InitFrac;
PROCEDURE EqualFrac(a,b : Frac) : BOOLEAN; BEGIN
RETURN (a.num = b.num) AND (a.denom = b.denom)
END EqualFrac;
PROCEDURE LessFrac(a,b : Frac) : BOOLEAN; BEGIN
RETURN a.num * b.denom < b.num * a.denom
END LessFrac;
PROCEDURE NegateFrac(f : Frac) : Frac; BEGIN
RETURN Frac{-f.num, f.denom}
END NegateFrac;
PROCEDURE SubFrac(lhs,rhs : Frac) : Frac; BEGIN
RETURN InitFrac(lhs.num * rhs.denom - lhs.denom * rhs.num, rhs.denom * lhs.denom)
END SubFrac;
PROCEDURE MultFrac(lhs,rhs : Frac) : Frac; BEGIN
RETURN InitFrac(lhs.num * rhs.num, lhs.denom * rhs.denom)
END MultFrac;
PROCEDURE Bernoulli(n : INTEGER) : Frac; VAR
a : ARRAY[0..15] OF Frac; i,j,m : INTEGER;
BEGIN
IF n < 0 THEN
RAISE(TextWinExSrc, 0, "n may not be negative or zero.")
END;
FOR m:=0 TO n DO
a[m] := Frac{1, m + 1};
FOR j:=m TO 1 BY -1 DO
a[j-1] := MultFrac(SubFrac(a[j-1], a[j]), Frac{j, 1})
END
END;
IF n # 1 THEN RETURN a[0] END;
RETURN NegateFrac(a[0])
END Bernoulli;
PROCEDURE WriteFrac(f : Frac); BEGIN
WriteInteger(f.num);
IF f.denom # 1 THEN
WriteString("/");
WriteInteger(f.denom)
END
END WriteFrac;
(* Target *) PROCEDURE Faulhaber(p : INTEGER); VAR
j,pwr,sign : INTEGER; q,coeff : Frac;
BEGIN
WriteInteger(p);
WriteString(" : ");
q := InitFrac(1, p + 1);
sign := -1;
FOR j:=0 TO p DO
sign := -1 * sign;
coeff := MultFrac(MultFrac(MultFrac(q, Frac{sign, 1}), Frac{Binomial(p + 1, j), 1}), Bernoulli(j));
IF EqualFrac(coeff, Frac{0, 1}) THEN CONTINUE END;
IF j = 0 THEN
IF NOT EqualFrac(coeff, Frac{1, 1}) THEN
IF EqualFrac(coeff, Frac{-1, 1}) THEN
WriteString("-")
ELSE
WriteFrac(coeff)
END
END
ELSE
IF EqualFrac(coeff, Frac{1, 1}) THEN
WriteString(" + ")
ELSIF EqualFrac(coeff, Frac{-1, 1}) THEN
WriteString(" - ")
ELSIF LessFrac(Frac{0, 1}, coeff) THEN
WriteString(" + ");
WriteFrac(coeff)
ELSE
WriteString(" - ");
WriteFrac(NegateFrac(coeff))
END
END;
pwr := p + 1 - j;
IF pwr > 1 THEN
WriteString("n^");
WriteInteger(pwr)
ELSE
WriteString("n")
END
END;
WriteLn
END Faulhaber;
(* Main *) VAR i : INTEGER; BEGIN
FOR i:=0 TO 9 DO
Faulhaber(i)
END;
ReadChar
END Faulhaber.</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
PARI/GP
PARI/GP has 2 built in functions: bernfrac(n) for Bernoulli numbers and bernpol(n) for Bernoulli polynomials. Using Bernoulli polynomials in PARI/GP is more simple, clear and much faster. (See version #2).
Version #1. Using Bernoulli numbers.
This version is using "Faulhaber's" formula based on Bernoulli numbers.
It's not worth using Bernoulli numbers in PARI/GP, because too much cleaning if you are avoiding "dirty" (but correct) result.
Note: Find ssubstr() function here on RC.
<lang parigp>
\\ Using "Faulhaber's" formula based on Bernoulli numbers. aev 2/7/17
\\ In str string replace all occurrences of the search string ssrch with the replacement string srepl. aev 3/8/16
sreplace(str,ssrch,srepl)={
my(sn=#str,ssn=#ssrch,srn=#srepl,sres="",Vi,Vs,Vz,vin,vin1,vi,L=List(),tok,zi,js=1);
if(sn==0,return("")); if(ssn==0||ssn>sn,return(str));
\\ Vi - found ssrch indexes
Vi=sfindalls(str,ssrch); vin=#Vi;
if(vin==0,return(str));
vin1=vin+1; Vi=Vec(Vi,vin1); Vi[vin1]=sn+1;
for(i=1,vin1, vi=Vi[i];
for(j=js,sn, \\print("ij:",i,"/",j,": ",sres);
if(j!=vi, sres=concat(sres,ssubstr(str,j,1)),
sres=concat(sres,srepl); js=j+ssn; break)
); \\fend j
); \\fend i
return(sres);
} B(n)=(bernfrac(n)); Comb(n,k)={my(r=0); if(k<=n, r=n!/(n-k)!/k!); return(r)}; Faulhaber2(p)={
my(s="",s1="",s2="",c1=0,c2=0);
for(j=0,p, c1=(-1)^j*Comb(p+1,j)*B(j); c2=(p+1-j);
s2="*n";
if(c1==0, next);
if(c2==1, s1="", s1=Str("^",c2));
s=Str(s,c1,s2,s1,"+") );
s=ssubstr(s,1,#s-1); s=sreplace(s,"1*n","n"); s=sreplace(s,"+-","-");
if(p==0, s="n", s=Str("(",s,")/",p+1)); print(p,": ",s);
} {\\ Testing: for(i=0,9, Faulhaber2(i))} </lang>
- Output:
0: n 1: (n^2+n)/2 2: (n^3+3/2*n^2+1/2*n)/3 3: (n^4+2*n^3+n^2)/4 4: (n^5+5/2*n^4+5/3*n^3-1/6*n)/5 5: (n^6+3*n^5+5/2*n^4-1/2*n^2)/6 6: (n^7+7/2*n^6+7/2*n^5-7/6*n^3+1/6*n)/7 7: (n^8+4*n^7+14/3*n^6-7/3*n^4+2/3*n^2)/8 8: (n^9+9/2*n^8+6*n^7-21/5*n^5+2*n^3-3/10*n)/9 9: (n^10+5*n^9+15/2*n^8-7*n^6+5*n^4-3/2*n^2)/10 time = 16 ms.
Version #2. Using Bernoulli polynomials.
This version is using the sums of pth powers formula from Bernoulli polynomials. It has small, simple and clear code, and produces instant result. <lang parigp> \\ Using a formula based on Bernoulli polynomials. aev 2/5/17 Faulhaber1(m)={
my(B,B1,B2,Bn); Bn=bernpol(m+1); x=n+1; B1=eval(Bn); x=0; B2=eval(Bn); Bn=(B1-B2)/(m+1); if(m==0, Bn=Bn-1); print(m,": ",Bn);
} {\\ Testing:
for(i=0,9, Faulhaber1(i))}
</lang>
- Output:
0: n 1: 1/2*n^2 + 1/2*n 2: 1/3*n^3 + 1/2*n^2 + 1/6*n 3: 1/4*n^4 + 1/2*n^3 + 1/4*n^2 4: 1/5*n^5 + 1/2*n^4 + 1/3*n^3 - 1/30*n 5: 1/6*n^6 + 1/2*n^5 + 5/12*n^4 - 1/12*n^2 6: 1/7*n^7 + 1/2*n^6 + 1/2*n^5 - 1/6*n^3 + 1/42*n 7: 1/8*n^8 + 1/2*n^7 + 7/12*n^6 - 7/24*n^4 + 1/12*n^2 8: 1/9*n^9 + 1/2*n^8 + 2/3*n^7 - 7/15*n^5 + 2/9*n^3 - 1/30*n 9: 1/10*n^10 + 1/2*n^9 + 3/4*n^8 - 7/10*n^6 + 1/2*n^4 - 3/20*n^2 > ## *** last result computed in 0 ms
Perl
<lang perl>use 5.014; use Math::Algebra::Symbols;
sub bernoulli_number {
my ($n) = @_;
return 0 if $n > 1 && $n % 2;
my @A;
for my $m (0 .. $n) {
$A[$m] = symbols(1) / ($m + 1);
for (my $j = $m ; $j > 0 ; $j--) {
$A[$j - 1] = $j * ($A[$j - 1] - $A[$j]);
}
}
return $A[0];
}
sub binomial {
my ($n, $k) = @_; return 1 if $k == 0 || $n == $k; binomial($n - 1, $k - 1) + binomial($n - 1, $k);
}
sub faulhaber_s_formula {
my ($p) = @_;
my $formula = 0;
for my $j (0 .. $p) {
$formula += binomial($p + 1, $j)
* bernoulli_number($j)
* symbols('n')**($p + 1 - $j);
}
(symbols(1) / ($p + 1) * $formula)
=~ s/\$n/n/gr =~ s/\*\*/^/gr =~ s/\*/ /gr;
}
foreach my $i (0 .. 9) {
say "$i: ", faulhaber_s_formula($i);
}</lang>
- Output:
0: n 1: 1/2 n+1/2 n^2 2: 1/6 n+1/2 n^2+1/3 n^3 3: 1/4 n^2+1/2 n^3+1/4 n^4 4: -1/30 n+1/3 n^3+1/2 n^4+1/5 n^5 5: -1/12 n^2+5/12 n^4+1/2 n^5+1/6 n^6 6: 1/42 n-1/6 n^3+1/2 n^5+1/2 n^6+1/7 n^7 7: 1/12 n^2-7/24 n^4+7/12 n^6+1/2 n^7+1/8 n^8 8: -1/30 n+2/9 n^3-7/15 n^5+2/3 n^7+1/2 n^8+1/9 n^9 9: -3/20 n^2+1/2 n^4-7/10 n^6+3/4 n^8+1/2 n^9+1/10 n^10
Perl 6
<lang perl6>sub bernoulli_number($n) {
return 1/2 if $n == 1; return 0/1 if $n % 2;
my @A;
for 0..$n -> $m {
@A[$m] = 1 / ($m + 1);
for $m, $m-1 ... 1 -> $j {
@A[$j - 1] = $j * (@A[$j - 1] - @A[$j]);
}
}
return @A[0];
}
sub binomial($n, $k) {
$k == 0 || $n == $k ?? 1 !! binomial($n-1, $k-1) + binomial($n-1, $k);
}
sub faulhaber_s_formula($p) {
my @formula = gather for 0..$p -> $j {
take '('
~ join('/', (binomial($p+1, $j) * bernoulli_number($j)).Rat.nude)
~ ")*n^{$p+1 - $j}";
}
my $formula = join(' + ', @formula.grep({!m{'(0/1)*'}}));
$formula .= subst(rx{ '(1/1)*' }, , :g);
$formula .= subst(rx{ '^1'» }, , :g);
"1/{$p+1} * ($formula)";
}
for 0..9 -> $p {
say "f($p) = ", faulhaber_s_formula($p);
}</lang>
- Output:
f(0) = 1/1 * (n) f(1) = 1/2 * (n^2 + n) f(2) = 1/3 * (n^3 + (3/2)*n^2 + (1/2)*n) f(3) = 1/4 * (n^4 + (2/1)*n^3 + n^2) f(4) = 1/5 * (n^5 + (5/2)*n^4 + (5/3)*n^3 + (-1/6)*n) f(5) = 1/6 * (n^6 + (3/1)*n^5 + (5/2)*n^4 + (-1/2)*n^2) f(6) = 1/7 * (n^7 + (7/2)*n^6 + (7/2)*n^5 + (-7/6)*n^3 + (1/6)*n) f(7) = 1/8 * (n^8 + (4/1)*n^7 + (14/3)*n^6 + (-7/3)*n^4 + (2/3)*n^2) f(8) = 1/9 * (n^9 + (9/2)*n^8 + (6/1)*n^7 + (-21/5)*n^5 + (2/1)*n^3 + (-3/10)*n) f(9) = 1/10 * (n^10 + (5/1)*n^9 + (15/2)*n^8 + (-7/1)*n^6 + (5/1)*n^4 + (-3/2)*n^2)
Phix
<lang Phix>include builtins\pfrac.e -- (0.8.0+)
function bernoulli(integer n)
sequence a = {}
for m=0 to n do
a = append(a,{1,m+1})
for j=m to 1 by -1 do
a[j] = frac_mul({j,1},frac_sub(a[j+1],a[j]))
end for
end for
if n!=1 then return a[1] end if
return frac_uminus(a[1])
end function
function binomial(integer n, k)
if n<0 or k<0 or n<k then ?9/0 end if
if n=0 or k=0 then return 1 end if
integer num = 1,
denom = 1
for i=k+1 to n do
num *= i
end for
for i=2 to n-k do
denom *= i
end for
return num / denom
end function
procedure faulhaber(integer p)
string res = sprintf("%d : ", p)
frac q = {1, p+1}
for j=0 to p do
frac bj = bernoulli(j)
if frac_ne(bj,frac_zero) then
frac coeff = frac_mul({binomial(p+1,j),p+1},bj)
string s = frac_sprint(coeff)
if j=0 then
if s="1" then
s = ""
end if
else
if s[1]='-' then
s[1..1] = " - "
else
s[1..0] = " + "
end if
end if
res &= s&"n"
integer pwr = p+1-j
if pwr>1 then
res &= sprintf("^%d", pwr)
end if
end if
end for
printf(1,"%s\n",{res})
end procedure
for i=0 to 9 do
faulhaber(i)
end for</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
Python
The following implementation does not use Bernoulli numbers, but Stirling numbers of the second kind, based on the relation: .
Then summing: .
One has then to expand the product in order to get a polynomial in the variable . Also, for the sum of , the sum is too large by one (since we start at ), this has to be taken into account.
Note: a number of the formulae above are invisible to the majority of browsers, including Chrome, IE/Edge, Safari and Opera. They may (subject to the installation of necessary fronts) be visible to some Firefox installations.
<lang python>from fractions import Fraction
def nextu(a):
n = len(a)
a.append(1)
for i in range(n - 1, 0, -1):
a[i] = i * a[i] + a[i - 1]
return a
def nextv(a):
n = len(a) - 1
b = [(1 - n) * x for x in a]
b.append(1)
for i in range(n):
b[i + 1] += a[i]
return b
def sumpol(n):
u = [0, 1]
v = [[1], [1, 1]]
yield [Fraction(0), Fraction(1)]
for i in range(1, n):
v.append(nextv(v[-1]))
t = [0] * (i + 2)
p = 1
for j, r in enumerate(u):
r = Fraction(r, j + 1)
for k, s in enumerate(v[j + 1]):
t[k] += r * s
yield t
u = nextu(u)
def polstr(a):
s = ""
q = False
n = len(a) - 1
for i, x in enumerate(reversed(a)):
i = n - i
if i < 2:
m = "n" if i == 1 else ""
else:
m = "n^%d" % i
c = str(abs(x))
if i > 0:
if c == "1":
c = ""
else:
m = " " + m
if x != 0:
if q:
t = " + " if x > 0 else " - "
s += "%s%s%s" % (t, c, m)
else:
t = "" if x > 0 else "-"
s = "%s%s%s" % (t, c, m)
q = True
if q:
return s
else:
return "0"
for i, p in enumerate(sumpol(10)):
print(i, ":", polstr(p))</lang>
- Output:
0 : n 1 : 1/2 n^2 + 1/2 n 2 : 1/3 n^3 + 1/2 n^2 + 1/6 n 3 : 1/4 n^4 + 1/2 n^3 + 1/4 n^2 4 : 1/5 n^5 + 1/2 n^4 + 1/3 n^3 - 1/30 n 5 : 1/6 n^6 + 1/2 n^5 + 5/12 n^4 - 1/12 n^2 6 : 1/7 n^7 + 1/2 n^6 + 1/2 n^5 - 1/6 n^3 + 1/42 n 7 : 1/8 n^8 + 1/2 n^7 + 7/12 n^6 - 7/24 n^4 + 1/12 n^2 8 : 1/9 n^9 + 1/2 n^8 + 2/3 n^7 - 7/15 n^5 + 2/9 n^3 - 1/30 n 9 : 1/10 n^10 + 1/2 n^9 + 3/4 n^8 - 7/10 n^6 + 1/2 n^4 - 3/20 n^2
Racket
Racket will simplify rational numbers; if this code simplifies the expressions too much for your tastes (e.g. you like 1/1 * (n)) then tweak the simplify... clauses to taste.
<lang racket>#lang racket/base
(require racket/match
racket/string
math/number-theory)
(define simplify-arithmetic-expression
(letrec ((s-a-e
(match-lambda
[(list (and op '+) l ... (list '+ m ...) r ...) (s-a-e `(,op ,@l ,@m ,@r))]
[(list (and op '+) l ... (? number? n1) m ... (? number? n2) r ...) (s-a-e `(,op ,@l ,(+ n1 n2) ,@m ,@r))]
[(list (and op '+) (app s-a-e l _) ... 0 (app s-a-e r _) ...) (s-a-e `(,op ,@l ,@r))]
[(list (and op '+) (app s-a-e x _)) (values x #t)]
[(list (and op '*) l ... (list '* m ...) r ...) (s-a-e `(,op ,@l ,@m ,@r))]
[(list (and op '*) l ... (? number? n1) m ... (? number? n2) r ...) (s-a-e `(,op ,@l ,(* n1 n2) ,@m ,@r))]
[(list (and op '*) (app s-a-e l _) ... 1 (app s-a-e r _) ...) (s-a-e `(,op ,@l ,@r))]
[(list (and op '*) (app s-a-e l _) ... 0 (app s-a-e r _) ...) (values 0 #t)]
[(list (and op '*) (app s-a-e x _)) (values x #t)]
[(list 'expt (app s-a-e x x-simplified?) 1) (values x x-simplified?)]
[(list op (app s-a-e a #f) ...) (values `(,op ,@a) #f)]
[(list op (app s-a-e a _) ...) (s-a-e `(,op ,@a))]
[e (values e #f)])))
s-a-e))
(define (expression->infix-string e)
(define (parenthesise-maybe s p?)
(if p? (string-append "(" s ")") s))
(letrec ((e->is
(lambda (paren?)
(match-lambda
[(list (and op (or '+ '- '* '*)) (app (e->is #t) a p?) ...)
(define bits (map parenthesise-maybe a p?))
(define compound (string-join bits (format " ~a " op)))
(values (if paren? (string-append "(" compound ")") compound) #f)]
[(list 'expt (app (e->is #t) x xp?) (app (e->is #t) n np?))
(values (format "~a^~a" (parenthesise-maybe x xp?) (parenthesise-maybe n np?)) #f)]
[(? number? (app number->string s)) (values s #f)]
[(? symbol? (app symbol->string s)) (values s #f)]))))
(define-values (str needs-parens?) ((e->is #f) e))
str))
(define (faulhaber p)
(define p+1 (add1 p))
(define-values (simpler simplified?)
(simplify-arithmetic-expression
`(* ,(/ 1 p+1)
(+ ,@(for/list ((j (in-range p+1)))
`(* ,(* (expt -1 j)
(binomial p+1 j))
(* ,(bernoulli-number j)
(expt n ,(- p+1 j)))))))))
simpler)
(for ((p (in-range 0 (add1 9))))
(printf "f(~a) = ~a~%" p (expression->infix-string (faulhaber p))))
</lang>
- Output:
f(0) = n f(1) = 1/2 * (n^2 + n) f(2) = 1/3 * (n^3 + (3/2 * n^2) + (1/2 * n)) f(3) = 1/4 * (n^4 + (2 * n^3) + n^2) f(4) = 1/5 * (n^5 + (5/2 * n^4) + (5/3 * n^3) + (-1/6 * n)) f(5) = 1/6 * (n^6 + (3 * n^5) + (5/2 * n^4) + (-1/2 * n^2)) f(6) = 1/7 * (n^7 + (7/2 * n^6) + (7/2 * n^5) + (-7/6 * n^3) + (1/6 * n)) f(7) = 1/8 * (n^8 + (4 * n^7) + (14/3 * n^6) + (-7/3 * n^4) + (2/3 * n^2)) f(8) = 1/9 * (n^9 + (9/2 * n^8) + (6 * n^7) + (-21/5 * n^5) + (2 * n^3) + (-3/10 * n)) f(9) = 1/10 * (n^10 + (5 * n^9) + (15/2 * n^8) + (-7 * n^6) + (5 * n^4) + (-3/2 * n^2))
Sidef
<lang ruby>const AnyNum = require('Math::AnyNum') const Poly = require('Math::Polynomial')
Poly.string_config(Hash(
fold_sign => true, prefix => "", suffix => "", variable => "n"
))
func anynum(n) {
AnyNum.new(n.as_rat)
}
func faulhaber_formula(p) {
(p+1).of { |j|
Poly.monomial(p - j + 1)\
.mul_const(anynum(bernoulli(j)))\
.mul_const(anynum(binomial(p+1, j)))
}.reduce(:add).div_const(p+1)
}
for p in (^10) {
printf("%2d: %s\n", p, faulhaber_formula(p))
}</lang>
- Output:
0: n 1: 1/2 n^2 + 1/2 n 2: 1/3 n^3 + 1/2 n^2 + 1/6 n 3: 1/4 n^4 + 1/2 n^3 + 1/4 n^2 4: 1/5 n^5 + 1/2 n^4 + 1/3 n^3 - 1/30 n 5: 1/6 n^6 + 1/2 n^5 + 5/12 n^4 - 1/12 n^2 6: 1/7 n^7 + 1/2 n^6 + 1/2 n^5 - 1/6 n^3 + 1/42 n 7: 1/8 n^8 + 1/2 n^7 + 7/12 n^6 - 7/24 n^4 + 1/12 n^2 8: 1/9 n^9 + 1/2 n^8 + 2/3 n^7 - 7/15 n^5 + 2/9 n^3 - 1/30 n 9: 1/10 n^10 + 1/2 n^9 + 3/4 n^8 - 7/10 n^6 + 1/2 n^4 - 3/20 n^2
Visual Basic .NET
<lang vbnet>Module Module1
Function Gcd(a As Long, b As Long)
If b = 0 Then
Return a
End If
Return Gcd(b, a Mod b)
End Function
Class Frac
ReadOnly num As Long
ReadOnly denom As Long
Public Shared ReadOnly ZERO As New Frac(0, 1)
Public Shared ReadOnly ONE As New Frac(1, 1)
Public Sub New(n As Long, d As Long)
If d = 0 Then Throw New ArgumentException("d must not be zero")
Dim nn = n
Dim dd = d
If nn = 0 Then
dd = 1
ElseIf dd < 0 Then
nn = -nn
dd = -dd
End If
Dim g = Math.Abs(Gcd(nn, dd))
If g > 1 Then
nn /= g
dd /= g
End If
num = nn
denom = dd
End Sub
Public Shared Operator -(self As Frac) As Frac
Return New Frac(-self.num, self.denom)
End Operator
Public Shared Operator +(lhs As Frac, rhs As Frac) As Frac
Return New Frac(lhs.num * rhs.denom + lhs.denom * rhs.num, rhs.denom * lhs.denom)
End Operator
Public Shared Operator -(lhs As Frac, rhs As Frac) As Frac
Return lhs + -rhs
End Operator
Public Shared Operator *(lhs As Frac, rhs As Frac) As Frac
Return New Frac(lhs.num * rhs.num, lhs.denom * rhs.denom)
End Operator
Public Shared Operator <(lhs As Frac, rhs As Frac) As Boolean
Dim x = lhs.num / lhs.denom
Dim y = rhs.num / rhs.denom
Return x < y
End Operator
Public Shared Operator >(lhs As Frac, rhs As Frac) As Boolean
Dim x = lhs.num / lhs.denom
Dim y = rhs.num / rhs.denom
Return x > y
End Operator
Public Shared Operator =(lhs As Frac, rhs As Frac) As Boolean
Return lhs.num = rhs.num AndAlso lhs.denom = rhs.denom
End Operator
Public Shared Operator <>(lhs As Frac, rhs As Frac) As Boolean
Return lhs.num <> rhs.num OrElse lhs.denom <> rhs.denom
End Operator
Public Overloads Function Equals(obj As Object) As Boolean
Dim frac = CType(obj, Frac)
Return Not IsNothing(frac) AndAlso num = frac.num AndAlso denom = frac.denom
End Function
Public Overloads Function GetHashCode() As Integer
Dim hashCode = 1317992671
hashCode = hashCode * -1521134295 + num.GetHashCode()
hashCode = hashCode * -1521134295 + denom.GetHashCode()
Return hashCode
End Function
Public Overloads Function ToString() As String
If denom = 1 Then Return num.ToString()
Return String.Format("{0}/{1}", num, denom)
End Function
End Class
Function Bernoulli(n As Integer) As Frac
If n < 0 Then Throw New ArgumentException("n may not be negative or zero")
Dim a(n + 1) As Frac
For m = 0 To n
a(m) = New Frac(1, m + 1)
For j = m To 1 Step -1
a(j - 1) = (a(j - 1) - a(j)) * New Frac(j, 1)
Next
Next
'returns the first Bernoulli number
If n <> 1 Then Return a(0)
Return -a(0)
End Function
Function Binomial(n As Integer, k As Integer) As Integer
If n < 0 OrElse k < 0 OrElse n < k Then
Throw New ArgumentException()
End If
If n = 0 OrElse k = 0 Then
Return 1
End If
Dim num = 1
For i = k + 1 To n
num *= i
Next
Dim denom = 1
For i = 2 To n - k
denom *= i
Next
Return num / denom
End Function
Sub Faulhaber(p As Integer)
Console.Write("{0} : ", p)
Dim q As New Frac(1, p + 1)
Dim sign = -1
For j = 0 To p
sign *= -1
Dim coeff = q * New Frac(sign, 1) * New Frac(Binomial(p + 1, j), 1) * Bernoulli(j)
If Frac.ZERO = coeff Then Continue For
If j = 0 Then
If Frac.ONE <> coeff Then
If -Frac.ONE = coeff Then
Console.Write("-")
Else
Console.Write(coeff.ToString())
End If
End If
Else
If Frac.ONE = coeff Then
Console.Write(" + ")
ElseIf -Frac.ONE = coeff Then
Console.Write(" - ")
ElseIf Frac.ZERO < coeff Then
Console.Write(" + {0}", coeff.ToString())
Else
Console.Write(" - {0}", (-coeff).ToString())
End If
End If
Dim pwr = p + 1 - j
If pwr > 1 Then
Console.Write("n^{0}", pwr)
Else
Console.Write("n")
End If
Next
Console.WriteLine()
End Sub
Sub Main()
For i = 0 To 9
Faulhaber(i)
Next
End Sub
End Module</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
zkl
GNU Multiple Precision Arithmetic Library
Uses code from the Bernoulli numbers task (copied here). <lang zkl>var [const] BN=Import("zklBigNum"); // libGMP (GNU MP Bignum Library)
fcn faulhaberFormula(p){ //-->(Rational,Rational...)
[p..0,-1].pump(List(),'wrap(k){ B(k)*BN(p+1).binomial(k) })
.apply('*(Rational(1,p+1)))
}</lang> <lang zkl>foreach p in (10){
println("F(%d) --> %s".fmt(p,polyRatString(faulhaberFormula(p))))
}</lang> <lang zkl>class Rational{ // Weenie Rational class, can handle BigInts
fcn init(_a,_b){ var a=_a, b=_b; normalize(); }
fcn toString{
if(b==1) a.toString()
else "%d/%d".fmt(a,b)
}
var [proxy] isZero=fcn{ a==0 };
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
}</lang> <lang zkl>fcn B(N){ // calculate Bernoulli(n) --> Rational
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]
} fcn polyRatString(terms){ // (a1,a2...)-->"a1n + a2n^2 ..."
str:=[1..].zipWith('wrap(n,a){ if(a.isZero) "" else "+ %sn^%s ".fmt(a,n) },
terms)
.pump(String)
.replace(" 1n"," n").replace("n^1 ","n ").replace("+ -","- ");
if(not str) return(" "); // all zeros
if(str[0]=="+") str[1,*]; // leave leading space
else String("-",str[2,*]);
}</lang>
- Output:
F(0) --> n F(1) --> 1/2n + 1/2n^2 F(2) --> 1/6n + 1/2n^2 + 1/3n^3 F(3) --> 1/4n^2 + 1/2n^3 + 1/4n^4 F(4) --> -1/30n + 1/3n^3 + 1/2n^4 + 1/5n^5 F(5) --> -1/12n^2 + 5/12n^4 + 1/2n^5 + 1/6n^6 F(6) --> 1/42n - 1/6n^3 + 1/2n^5 + 1/2n^6 + 1/7n^7 F(7) --> 1/12n^2 - 7/24n^4 + 7/12n^6 + 1/2n^7 + 1/8n^8 F(8) --> -1/30n + 2/9n^3 - 7/15n^5 + 2/3n^7 + 1/2n^8 + 1/9n^9 F(9) --> -3/20n^2 + 1/2n^4 - 7/10n^6 + 3/4n^8 + 1/2n^9 + 1/10n^10