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Bell or exponential numbers are enumerations of the number of different ways to partition a set that has exactly n elements. Each element of the sequence B_n is the number of partitions of a set of size n where order of the elements and order of the partitions are non-significant. E.G.: {a b} is the same as {b a} and {a} {b} is the same as {b} {a}.

So

B₀ = 1 trivially. There is only one way to partition a set with zero elements. { }

B₁ = 1 There is only one way to partition a set with one element. {a}

B₂ = 2 Two elements may be partitioned in two ways. {a} {b}, {a b}

B₃ = 5 Three elements may be partitioned in five ways {a} {b} {c}, {a b} {c}, {a} {b c}, {a c} {b}, {a b c}

and so on.

In general, the easiest way to find the Bell numbers is construct a Bell triangle, also known as an Aitken's array or Peirce triangle, and read off the numbers in the first column of each row.

Task

Write a routine (function, generator, whatever) to generate the Bell number sequence and call the routine to show here, on this page at least the first 15 and (if your language supports big Integers) 50th elements of the sequence.

If you do use the Bell triangle method to generate the numbers, also show the first ten rows of the Bell triangle.

See also

Go

<lang go>package main

import (

   "fmt"
   "math/big"

)

func bellTriangle(n int) [][]*big.Int {

   tri := make([][]*big.Int, n)
   for i := 0; i < n; i++ {
       tri[i] = make([]*big.Int, i)
       for j := 0; j < i; j++ {
           tri[i][j] = new(big.Int)
       }
   }
   tri[1][0].SetUint64(1)
   for i := 2; i < n; i++ {
       tri[i][0].Set(tri[i-1][i-2])
       for j := 1; j < i; j++ {
           tri[i][j].Add(tri[i][j-1], tri[i-1][j-1])
       }
   }
   return tri

}

func main() {

   bt := bellTriangle(50)
   fmt.Println("First fifteen and fiftieth Bell numbers:")
   fmt.Println(" 1: 1")
   for i := 1; i < 15; i++ {
       fmt.Printf("%2d: %d\n", i+1, bt[i][i-1])
   }
   fmt.Println("50:", bt[49][48])
   fmt.Println("\nFirst ten rows of Bell's triangle:")
   for i := 1; i < 11; i++ {
       fmt.Println(bt[i])
   }

}</lang>

Output:

First fifteen and fiftieth Bell numbers:
 1: 1
 2: 1
 3: 2
 4: 5
 5: 15
 6: 52
 7: 203
 8: 877
 9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
50: 10726137154573358400342215518590002633917247281

First ten rows of Bell's triangle:
[1]
[1 2]
[2 3 5]
[5 7 10 15]
[15 20 27 37 52]
[52 67 87 114 151 203]
[203 255 322 409 523 674 877]
[877 1080 1335 1657 2066 2589 3263 4140]
[4140 5017 6097 7432 9089 11155 13744 17007 21147]
[21147 25287 30304 36401 43833 52922 64077 77821 94828 115975]

Perl 6

Works with: Rakudo version 2019.03

<lang perl6> my @Aitkens-array = lazy [1], -> @b {

    my @c = @b.tail;
    @c.push: @b[$_] + @c[$_] for ^@b;
    @c
} ... *;

my @Bell-numbers = @Aitkens-array.map: { .head };

say "First fifteen and fiftieth Bell numbers:"; printf "%2d: %s\n", 1+$_, @Bell-numbers[$_] for flat ^15, 49;

say "\nFirst ten rows of Aitken's array:"; .say for @Aitkens-array[^10];</lang>

Output:

First fifteen and fiftieth Bell numbers:
 1: 1
 2: 1
 3: 2
 4: 5
 5: 15
 6: 52
 7: 203
 8: 877
 9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
50: 10726137154573358400342215518590002633917247281

First ten rows of Aitken's array:
[1]
[1 2]
[2 3 5]
[5 7 10 15]
[15 20 27 37 52]
[52 67 87 114 151 203]
[203 255 322 409 523 674 877]
[877 1080 1335 1657 2066 2589 3263 4140]
[4140 5017 6097 7432 9089 11155 13744 17007 21147]
[21147 25287 30304 36401 43833 52922 64077 77821 94828 115975]

REXX

Bell numbers are the number of ways of placing n labeled balls into n indistinguishable boxes. Bell(0) is defined as 1.

This REXX version uses an index of the Bell number (which starts a zero).

A little optimization was added in calculating the factorial of a number. <lang rexx>/*REXX program calculates and displays a range of Bell numbers (index starts at zero).*/ parse arg LO HI . /*obtain optional arguments from the CL*/ if LO== & HI=="" then do; LO=0; HI=14; end /*Not specified? Then use the default.*/ if LO== | LO=="," then LO= 0 /* " " " " " " */ if HI== | HI=="," then HI= 15 /* " " " " " " */ numeric digits max(9, HI*2) /*crudely calculate the # decimal digs.*/ !.=; @.= 1 /*the FACT function uses memoization.*/

    do j=0  for  HI+1;    $= 0;       jm= j-1   /*JM  is used for a shortcut  (below). */
    if j==0  then $= 1                          /*use fiat value for the first Bell #. */
             else do k=0  for j;       _= jm-k  /* [↓]  calculate a Bell # the easy way*/
                  $= $ + comb(jm,k) * @._       /*COMB≡combination or binomial function*/
                  end   /*k*/
    @.j= $                                      /*assign the Jth Bell number to @ array*/
    if j>=LO  &  j<=HI  then say '    bell('right(j, length(HI) )") = "      $
    end   /*j*/

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ comb: procedure expose !.; parse arg x,y; if x==y then return 1; if y>x then return 0

     if x-y<y  then y= x - y
     _= 1;          do j=x-y+1  to x;  _=_*j;  end;          return _ / fact(y)

/*──────────────────────────────────────────────────────────────────────────────────────*/ fact: procedure expose !.; parse arg x; if !.x\== then return !.x

     !=1;  do f=2  to x;  != !*f;  end;    !.x=!;            return !</lang>

output when using the internal default inputs of: 0 14

    Bell( 0) =  1
    Bell( 2) =  2
    Bell( 3) =  5
    Bell( 4) =  15
    Bell( 5) =  52
    Bell( 6) =  203
    Bell( 7) =  877
    Bell( 8) =  4140
    Bell( 9) =  21147
    Bell(10) =  115975
    Bell(11) =  678570
    Bell(12) =  4213597
    Bell(13) =  27644437
    Bell(14) =  190899322

output when using the inputs of: 49 49

    Bell(49) =  10726137154573358400342215518590002633917247281