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Bell numbers

From Rosetta Code
Task
Bell numbers
You are encouraged to solve this task according to the task description, using any language you may know.

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 Bn 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.


A simple 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. There are other generating algorithms though, and you are free to choose the best / most appropriate for your case.


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

Ada[edit]

Works with: GNAT version 8.3.0
 
with Ada.Text_IO; use Ada.Text_IO;
procedure Main is
type Bell_Triangle is array(Positive range <>, Positive range <>) of Natural;
 
 
procedure Print_Rows (Row : in Positive; Triangle : in Bell_Triangle) is
begin
if Row in Triangle'Range(1) then
for I in Triangle'First(1) .. Row loop
Put_Line (Triangle (I, 1)'Image);
end loop;
end if;
end Print_Rows;
 
procedure Print_Triangle (Num : in Positive; Triangle : in Bell_Triangle) is
begin
if Num in Triangle'Range then
for I in Triangle'First(1) .. Num loop
for J in Triangle'First(2) .. Num loop
if Triangle (I, J) /= 0 then
Put (Triangle (I, J)'Image);
end if;
end loop;
New_Line;
end loop;
end if;
end Print_Triangle;
 
procedure Bell_Numbers is
Triangle : Bell_Triangle(1..15, 1..15) := (Others => (Others => 0));
Temp  : Positive := 1;
begin
 
Triangle (1, 1) := 1;
 
for I in Triangle'First(1) + 1 .. Triangle'Last(1) loop
Triangle (I, 1) := Temp;
 
for J in Triangle'First(2) .. Triangle'Last(2) - 1 loop
if Triangle (I - 1, J) /= 0 then
Triangle (I, J + 1) := Triangle (I, J) + Triangle (I - 1, J);
else
Temp := Triangle (I, J);
exit;
end if;
end loop;
end loop;
 
Put_Line ("First 15 Bell numbers:");
Print_Rows (15, Triangle);
 
New_Line;
 
Put_Line ("First 10 rows of the Bell triangle:");
Print_Triangle (10, Triangle);
end Bell_Numbers;
begin
Bell_Numbers;
end Main;
 
Output:
First 15 Bell numbers:
 1
 1
 2
 5
 15
 52
 203
 877
 4140
 21147
 115975
 678570
 4213597
 27644437
 190899322

First 10 rows of the Bell 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
 

ALGOL-M[edit]

begin
integer function index(row, col);
integer row, col;
index := row * (row-1)/ 2 + col;
 
integer ROWS; ROWS := 15;
begin
decimal(11) array bell[0:ROWS*(ROWS+1)/2];
integer i, j;
 
bell[index(1, 0)] := 1.;
for i := 2 step 1 until ROWS do
begin
bell[index(i, 0)] := bell[index(i-1, i-2)];
for j := 1 step 1 until i-1 do
bell[index(i,j)] := bell[index(i,j-1)] + bell[index(i-1,j-1)];
end;
 
write("First fifteen Bell numbers:");
for i := 1 step 1 until ROWS do
begin
write(i);
writeon(": ");
writeon(bell[index(i,0)]);
end;
 
write("");
write("First ten rows of Bell's triangle:");
for i := 1 step 1 until 10 do
begin
write("");
for j := 0 step 1 until i-1 do
writeon(bell[index(i,j)]);
end;
end;
end
Output:
First fifteen Bell numbers:
     1:  1.0
     2:  1.0
     3:  2.0
     4:  5.0
     5:  15.0
     6:  52.0
     7:  203.0
     8:  877.0
     9:  4140.0
    10:  21147.0
    11:  115975.0
    12:  678570.0
    13:  4213597.0
    14:  27644437.0
    15:  190899322.0

First ten rows of Bell's triangle:
 1.0
 1.0 2.0
 2.0 3.0 5.0
 5.0 7.0 10.0 15.0
 15.0 20.0 27.0 37.0 52.0
 52.0 67.0 87.0 114.0 151.0 203.0
 203.0 255.0 322.0 409.0 523.0 674.0 877.0
 877.0 1080.0 1335.0 1657.0 2066.0 2589.0 3263.0 4140.0
 4140.0 5017.0 6097.0 7432.0 9089.0 11155.0 13744.0 17007.0 21147.0
 21147.0 25287.0 30304.0 36401.0 43833.0 52922.0 64077.0 77821.0 94828.0 115975.0

APL[edit]

Works with: Dyalog APL
bell←{
tr←↑(⊢,(⊂⊃∘⌽+0,+\)∘⊃∘⌽)⍣14⊢,⊂,1
⎕←'First 15 Bell numbers:'
⎕←tr[;1]
⎕←'First 10 rows of Bell''s triangle:'
⎕←tr[⍳10;⍳10]
}
Output:
First 15 Bell numbers:
1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322
First 10 rows of Bell's triangle:
    1     0     0     0     0     0     0     0     0      0
    1     2     0     0     0     0     0     0     0      0
    2     3     5     0     0     0     0     0     0      0
    5     7    10    15     0     0     0     0     0      0
   15    20    27    37    52     0     0     0     0      0
   52    67    87   114   151   203     0     0     0      0
  203   255   322   409   523   674   877     0     0      0
  877  1080  1335  1657  2066  2589  3263  4140     0      0
 4140  5017  6097  7432  9089 11155 13744 17007 21147      0
21147 25287 30304 36401 43833 52922 64077 77821 94828 115975

Arturo[edit]

Translation of: D
bellTriangle: function[n][
tri: map 0..n-1 'x [ map 0..n 'y -> 0 ]
set get tri 1 0 1
loop 2..n-1 'i [
set get tri i 0 get (get tri i-1) i-2
loop 1..i-1 'j [
set get tri i j (get (get tri i) j-1) + ( get (get tri i-1) j-1)
]
]
return tri
]
bt: bellTriangle 51
 
loop 1..15 'x ->
print [x "=>" first bt\[x]]
print ["50 =>" first last bt]
print ""
print "The first ten rows of Bell's triangle:"
 
loop 1..10 'i ->
print filter bt\[i] => zero?
Output:
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 

The 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 

AutoHotkey[edit]

;-----------------------------------
Bell_triangle(maxRows){
row := 1, col := 1, Arr := []
Arr[row, col] := 1
while (Arr.Count() < maxRows){
row++
max := Arr[row-1].Count()
Loop % max{
if (col := A_Index) =1
Arr[row, col] := Arr[row-1, max]
Arr[row, col+1] := Arr[row, col] + Arr[row-1, col]
}
}
return Arr
}
;-----------------------------------
Show_Bell_Number(Arr){
for i, obj in Arr{
res .= obj.1 "`n"
}
return Trim(res, "`n")
}
;-----------------------------------
Show_Bell_triangle(Arr){
for i, obj in Arr{
for j, v in obj
res .= v ", "
res := Trim(res, ", ") . "`n"
}
return Trim(res, "`n")
}
;-----------------------------------
Examples:
MsgBox % Show_Bell_Number(Bell_triangle(15))
MsgBox % Show_Bell_triangle(Bell_triangle(10))
return
Output:
1
1
2
5
15
52
203
877
4140
21147
115975
678570
4213597
27644437
190899322
---------------------------
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

C[edit]

Translation of: D
#include <stdio.h>
#include <stdlib.h>
 
// row starts with 1; col < row
size_t bellIndex(int row, int col) {
return row * (row - 1) / 2 + col;
}
 
int getBell(int *bellTri, int row, int col) {
size_t index = bellIndex(row, col);
return bellTri[index];
}
 
void setBell(int *bellTri, int row, int col, int value) {
size_t index = bellIndex(row, col);
bellTri[index] = value;
}
 
int *bellTriangle(int n) {
size_t length = n * (n + 1) / 2;
int *tri = calloc(length, sizeof(int));
int i, j;
 
setBell(tri, 1, 0, 1);
for (i = 2; i <= n; ++i) {
setBell(tri, i, 0, getBell(tri, i - 1, i - 2));
for (j = 1; j < i; ++j) {
int value = getBell(tri, i, j - 1) + getBell(tri, i - 1, j - 1);
setBell(tri, i, j, value);
}
}
 
return tri;
}
 
int main() {
const int rows = 15;
int *bt = bellTriangle(rows);
int i, j;
 
printf("First fifteen Bell numbers:\n");
for (i = 1; i <= rows; ++i) {
printf("%2d: %d\n", i, getBell(bt, i, 0));
}
 
printf("\nThe first ten rows of Bell's triangle:\n");
for (i = 1; i <= 10; ++i) {
printf("%d", getBell(bt, i, 0));
for (j = 1; j < i; ++j) {
printf(", %d", getBell(bt, i, j));
}
printf("\n");
}
 
free(bt);
return 0;
}
Output:
First fifteen 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

The 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

C#[edit]

Translation of: D
using System;
using System.Numerics;
 
namespace BellNumbers {
public static class Utility {
public static void Init<T>(this T[] array, T value) {
if (null == array) return;
for (int i = 0; i < array.Length; ++i) {
array[i] = value;
}
}
}
 
class Program {
static BigInteger[][] BellTriangle(int n) {
BigInteger[][] tri = new BigInteger[n][];
for (int i = 0; i < n; ++i) {
tri[i] = new BigInteger[i];
tri[i].Init(BigInteger.Zero);
}
tri[1][0] = 1;
for (int i = 2; i < n; ++i) {
tri[i][0] = tri[i - 1][i - 2];
for (int j = 1; j < i; ++j) {
tri[i][j] = tri[i][j - 1] + tri[i - 1][j - 1];
}
}
return tri;
}
 
static void Main(string[] args) {
var bt = BellTriangle(51);
Console.WriteLine("First fifteen and fiftieth Bell numbers:");
for (int i = 1; i < 16; ++i) {
Console.WriteLine("{0,2}: {1}", i, bt[i][0]);
}
Console.WriteLine("50: {0}", bt[50][0]);
Console.WriteLine();
Console.WriteLine("The first ten rows of Bell's triangle:");
for (int i = 1; i < 11; ++i) {
//Console.WriteLine(bt[i]);
var it = bt[i].GetEnumerator();
Console.Write("[");
if (it.MoveNext()) {
Console.Write(it.Current);
}
while (it.MoveNext()) {
Console.Write(", ");
Console.Write(it.Current);
}
Console.WriteLine("]");
}
}
}
}
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

The 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]

C++[edit]

Library: Boost

Requires C++14 or later. If HAVE_BOOST is defined, we use the cpp_int class from Boost so we can display the 50th Bell number, as shown in the output section below.

#include <iostream>
#include <vector>
 
#ifdef HAVE_BOOST
#include <boost/multiprecision/cpp_int.hpp>
typedef boost::multiprecision::cpp_int integer;
#else
typedef unsigned int integer;
#endif
 
auto make_bell_triangle(int n) {
std::vector<std::vector<integer>> bell(n);
for (int i = 0; i < n; ++i)
bell[i].assign(i + 1, 0);
bell[0][0] = 1;
for (int i = 1; i < n; ++i) {
std::vector<integer>& row = bell[i];
std::vector<integer>& prev_row = bell[i - 1];
row[0] = prev_row[i - 1];
for (int j = 1; j <= i; ++j)
row[j] = row[j - 1] + prev_row[j - 1];
}
return bell;
}
 
int main() {
#ifdef HAVE_BOOST
const int size = 50;
#else
const int size = 15;
#endif
auto bell(make_bell_triangle(size));
 
const int limit = 15;
std::cout << "First " << limit << " Bell numbers:\n";
for (int i = 0; i < limit; ++i)
std::cout << bell[i][0] << '\n';
 
#ifdef HAVE_BOOST
std::cout << "\n50th Bell number is " << bell[49][0] << "\n\n";
#endif
 
std::cout << "First 10 rows of the Bell triangle:\n";
for (int i = 0; i < 10; ++i) {
std::cout << bell[i][0];
for (int j = 1; j <= i; ++j)
std::cout << ' ' << bell[i][j];
std::cout << '\n';
}
return 0;
}
Output:
First 15 Bell numbers:
1
1
2
5
15
52
203
877
4140
21147
115975
678570
4213597
27644437
190899322

50th Bell number is 10726137154573358400342215518590002633917247281

First 10 rows of the Bell 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

Common Lisp[edit]

via Bell triangle[edit]

;; The triangle is a list of arrays; each array is a
;; triangle's row; the last row is at the head of the list.
(defun grow-triangle (triangle)
(if (null triangle)
'(#(1))
(let* ((last-array (car triangle))
(last-length (length last-array))
(new-array (make-array (1+ last-length)
:element-type 'integer)))
;; copy over the last element of the last array
(setf (aref new-array 0) (aref last-array (1- last-length)))
;; fill in the rest of the array
(loop for i from 0
;; the last index of the new array is the length
;; of the last array, which is 1 unit shorter
for j from 1 upto last-length
for sum = (+ (aref last-array i) (aref new-array i))
do (setf (aref new-array j) sum))
;; return the grown list
(cons new-array triangle))))
 
(defun make-triangle (num)
(if (<= num 1)
(grow-triangle nil)
(grow-triangle (make-triangle (1- num)))))
 
(defun bell (num)
(cond ((< num 0) nil)
((= num 0) 1)
(t (aref (first (make-triangle num)) (1- num)))))
 
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; Printing section
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
 
(defparameter *numbers-to-print*
(append
(loop for i upto 19 collect i)
'(49 50)))
 
(defun array->list (array)
(loop for i upto (1- (length array))
collect (aref array i)))
 
(defun print-bell-number (index bell-number)
(format t "B_~d (~:r Bell number) = ~:d~%"
index (1+ index) bell-number))
 
 
(defun print-bell-triangle (triangle)
(loop for row in (reverse triangle)
do (format t "~{~d~^, ~}~%" (array->list row))))
 
;; Final invocation
(loop for n in *numbers-to-print* do
(print-bell-number n (bell n)))
 
(princ #\newline)
 
(format t "The first 10 rows of Bell triangle:~%")
(print-bell-triangle (make-triangle 10))
Output:
B_0 (first Bell number) = 1
B_1 (second Bell number) = 1
B_2 (third Bell number) = 2
B_3 (fourth Bell number) = 5
B_4 (fifth Bell number) = 15
B_5 (sixth Bell number) = 52
B_6 (seventh Bell number) = 203
B_7 (eighth Bell number) = 877
B_8 (ninth Bell number) = 4,140
B_9 (tenth Bell number) = 21,147
B_10 (eleventh Bell number) = 115,975
B_11 (twelfth Bell number) = 678,570
B_12 (thirteenth Bell number) = 4,213,597
B_13 (fourteenth Bell number) = 27,644,437
B_14 (fifteenth Bell number) = 190,899,322
B_15 (sixteenth Bell number) = 1,382,958,545
B_16 (seventeenth Bell number) = 10,480,142,147
B_17 (eighteenth Bell number) = 82,864,869,804
B_18 (nineteenth Bell number) = 682,076,806,159
B_19 (twentieth Bell number) = 5,832,742,205,057
B_49 (fiftieth Bell number) = 10,726,137,154,573,358,400,342,215,518,590,002,633,917,247,281
B_50 (fifty-first Bell number) = 185,724,268,771,078,270,438,257,767,181,908,917,499,221,852,770

The first 10 rows of Bell 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

via Stirling numbers of the second kind[edit]

This solution's algorithm is substantially slower than the algorithm based on the Bell triangle, because of the many nested loops.

;;; Compute bell numbers analytically
 
;; Compute the factorial
(defun fact (n)
(cond ((< n 0) nil)
((< n 2) 1)
(t (* n (fact (1- n))))))
 
;; Compute the binomial coefficient (n choose k)
(defun binomial (n k)
(loop for i from 1 upto k
collect (/ (- (1+ n) i) i) into lst
finally (return (reduce #'* lst))))
 
;; Compute the Stirling number of the second kind
(defun stirling (n k)
(/
(loop for i upto k summing
(* (expt -1 i) (binomial k i) (expt (- k i) n)))
(fact k)))
 
;; Compute the Bell number
(defun bell (n)
(loop for k upto n summing (stirling n k)))
 
 
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; Printing section
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
 
(defparameter *numbers-to-print*
(append
(loop for i upto 19 collect i)
'(49 50)))
 
 
(defun print-bell-number (index bell-number)
(format t "B_~d (~:r Bell number) = ~:d~%"
index (1+ index) bell-number))
 
;; Final invocation
(loop for n in *numbers-to-print* do
(print-bell-number n (bell n)))
Output:
B_0 (first Bell number) = 1
B_1 (second Bell number) = 1
B_2 (third Bell number) = 2
B_3 (fourth Bell number) = 5
B_4 (fifth Bell number) = 15
B_5 (sixth Bell number) = 52
B_6 (seventh Bell number) = 203
B_7 (eighth Bell number) = 877
B_8 (ninth Bell number) = 4,140
B_9 (tenth Bell number) = 21,147
B_10 (eleventh Bell number) = 115,975
B_11 (twelfth Bell number) = 678,570
B_12 (thirteenth Bell number) = 4,213,597
B_13 (fourteenth Bell number) = 27,644,437
B_14 (fifteenth Bell number) = 190,899,322
B_15 (sixteenth Bell number) = 1,382,958,545
B_16 (seventeenth Bell number) = 10,480,142,147
B_17 (eighteenth Bell number) = 82,864,869,804
B_18 (nineteenth Bell number) = 682,076,806,159
B_19 (twentieth Bell number) = 5,832,742,205,057
B_49 (fiftieth Bell number) = 10,726,137,154,573,358,400,342,215,518,590,002,633,917,247,281
B_50 (fifty-first Bell number) = 185,724,268,771,078,270,438,257,767,181,908,917,499,221,852,770

Cowgol[edit]

Translation of: C
include "cowgol.coh";
 
typedef B is uint32;
typedef I is intptr;
 
sub bellIndex(row: I, col: I): (addr: I) is
addr := (row * (row - 1) / 2 + col) * @bytesof B;
end sub;
 
sub getBell(row: I, col: I): (bell: B) is
bell := [LOMEM as [B] + bellIndex(row, col)];
end sub;
 
sub setBell(row: I, col: I, bell: B) is
[LOMEM as [B] + bellIndex(row, col)] := bell;
end sub;
 
sub bellTriangle(n: I) is
var length := n * (n + 1) / 2;
var bytes := length * @bytesof B;
 
if HIMEM - LOMEM < bytes then
print("not enough memory\n");
ExitWithError();
end if;
 
MemZero(LOMEM, bytes);
 
setBell(1, 0, 1);
var i: I := 2;
while i <= n loop
setBell(i, 0, getBell(i-1, i-2));
var j: I := 1;
while j < i loop
var value := getBell(i, j-1) + getBell(i-1, j-1);
setBell(i, j, value);
j := j + 1;
end loop;
i := i + 1;
end loop;
end sub;
 
const ROWS := 15;
bellTriangle(ROWS);
print("First fifteen Bell numbers:\n");
var i: I := 1;
while i <= ROWS loop
print_i32(i as uint32);
print(": ");
print_i32(getBell(i, 0) as uint32);
print_nl();
i := i + 1;
end loop;
 
print("\nThe first ten rows of Bell's triangle:\n");
i := 1;
while i <= 10 loop
var j: I := 0;
loop
print_i32(getBell(i, j) as uint32);
j := j + 1;
if j == i then break;
else print(", ");
end if;
end loop;
i := i + 1;
print_nl();
end loop;
Output:
First fifteen 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

The 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

D[edit]

Translation of: Go
import std.array : uninitializedArray;
import std.bigint;
import std.stdio : writeln, writefln;
 
auto bellTriangle(int n) {
auto tri = uninitializedArray!(BigInt[][])(n);
foreach (i; 0..n) {
tri[i] = uninitializedArray!(BigInt[])(i);
tri[i][] = BigInt(0);
}
tri[1][0] = 1;
foreach (i; 2..n) {
tri[i][0] = tri[i - 1][i - 2];
foreach (j; 1..i) {
tri[i][j] = tri[i][j - 1] + tri[i - 1][j - 1];
}
}
return tri;
}
 
void main() {
auto bt = bellTriangle(51);
writeln("First fifteen and fiftieth Bell numbers:");
foreach (i; 1..16) {
writefln("%2d: %d", i, bt[i][0]);
}
writeln("50: ", bt[50][0]);
writeln;
writeln("The first ten rows of Bell's triangle:");
foreach (i; 1..11) {
writeln(bt[i]);
}
}
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

The 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]

Delphi[edit]

A console application written in Delphi 7. It shows a way of calculating Bell numbers without using a triangle. Output numbering is as in the statement of the task, namely B_0 = 1, B_1 = 1, B_2 = 2, ....

 
program BellNumbers;
 
// For Rosetta Code.
// Delphi console application to display the Bell numbers B_0, ..., B_25.
// Uses signed 64-bit integers, the largest integer type available in Delphi.
 
{$APPTYPE CONSOLE}
 
uses SysUtils; // only for the display
 
const
MAX_INDEX = 25; // maximum index within the limits of int64
var
n : integer; // index of Bell number
j : integer; // loop variable
a : array [0..MAX_INDEX - 1] of int64; // working array to build up B_n
 
{ Subroutine to display that a[0] is the Bell number B_n }
procedure Display();
begin
WriteLn( SysUtils.Format( 'B_%-2d = %d', [n, a[0]]));
end;
 
(* Main program *)
begin
n := 0;
a[0] := 1;
Display(); // some programmers would prefer Display;
while (n < MAX_INDEX) do begin // and give begin a line to itself
a[n] := a[0];
for j := n downto 1 do inc( a[j - 1], a[j]);
inc(n);
Display();
end;
end.
 
Output:
B_0  = 1
B_1  = 1
B_2  = 2
B_3  = 5
B_4  = 15
B_5  = 52
B_6  = 203
B_7  = 877
B_8  = 4140
B_9  = 21147
B_10 = 115975
B_11 = 678570
B_12 = 4213597
B_13 = 27644437
B_14 = 190899322
B_15 = 1382958545
B_16 = 10480142147
B_17 = 82864869804
B_18 = 682076806159
B_19 = 5832742205057
B_20 = 51724158235372
B_21 = 474869816156751
B_22 = 4506715738447323
B_23 = 44152005855084346
B_24 = 445958869294805289
B_25 = 4638590332229999353

Elixir[edit]

 
defmodule Bell do
def triangle(), do: Stream.iterate([1], fn l -> bell_row l, [List.last l] end)
def numbers(), do: triangle() |> Stream.map(&List.first/1)
 
defp bell_row([], r), do: Enum.reverse r
defp bell_row([a|a_s], r = [r0|_]), do: bell_row(a_s, [a + r0|r])
end
 
:io.format "The first 15 bell numbers are ~p~n~n",
[Bell.numbers() |> Enum.take(15)]
 
IO.puts "The 50th Bell number is #{Bell.numbers() |> Enum.take(50) |> List.last}"
IO.puts ""
 
IO.puts "THe first 10 rows of Bell's triangle:"
IO.inspect(Bell.triangle() |> Enum.take(10))
 
Output:
The first 15 bell numbers are [1,1,2,5,15,52,203,877,4140,21147,115975,678570,
                               4213597,27644437,190899322]

The 50th Bell number is 10726137154573358400342215518590002633917247281

THe first 10 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]
]

F#[edit]

The function[edit]

 
// Generate bell triangle. Nigel Galloway: July 6th., 2019
let bell=Seq.unfold(fun g->Some(g,List.scan(+) (List.last g) g))[1I]
 

The Task[edit]

 
bell|>Seq.take 10|>Seq.iter(printfn "%A")
 
Output:
[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]
 
bell|>Seq.take 15|>Seq.iter(fun n->printf "%A " (List.head n));printfn ""
 
Output:
1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322 
 
printfn "%A" (Seq.head (Seq.item 49 bell))
 
Output:
10726137154573358400342215518590002633917247281

Factor[edit]

via Aitken's array[edit]

Works with: Factor version 0.98
USING: formatting io kernel math math.matrices sequences vectors ;
 
: next-row ( prev -- next )
[ 1 1vector ]
[ dup last [ + ] accumulate swap suffix! ] if-empty ;
 
: aitken ( n -- seq )
V{ } clone swap [ next-row dup ] replicate nip ;
 
0 50 aitken col [ 15 head ] [ last ] bi
"First 15 Bell numbers:\n%[%d, %]\n\n50th: %d\n\n" printf
"First 10 rows of the Bell triangle:" print
10 aitken [ "%[%d, %]\n" printf ] each
Output:
First 15 Bell numbers:
{ 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322 }

50th: 10726137154573358400342215518590002633917247281

First 10 rows of the Bell 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 }

via recurrence relation[edit]

This solution makes use of a recurrence relation involving binomial coefficients.

Works with: Factor version 0.98
USING: formatting kernel math math.combinatorics sequences ;
 
: next-bell ( seq -- n )
dup length 1 - [ swap nCk * ] curry map-index sum ;
 
: bells ( n -- seq )
V{ 1 } clone swap 1 - [ dup next-bell suffix! ] times ;
 
50 bells [ 15 head ] [ last ] bi
"First 15 Bell numbers:\n%[%d, %]\n\n50th: %d\n" printf
Output:
First 15 Bell numbers:
{ 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322 }

50th: 10726137154573358400342215518590002633917247281

via Stirling sums[edit]

This solution defines Bell numbers in terms of sums of Stirling numbers of the second kind.

Works with: Factor version 0.99 development release 2019-07-10
USING: formatting kernel math math.extras math.ranges sequences ;
 
: bell ( m -- n )
[ 1 ] [ dup [1,b] [ stirling ] with map-sum ] if-zero ;
 
50 [ bell ] { } map-integers [ 15 head ] [ last ] bi
"First 15 Bell numbers:\n%[%d, %]\n\n50th: %d\n" printf
Output:

As above.

FreeBASIC[edit]

#define MAX 21
 
#macro ncp(n, p)
(fact(n)/(fact(p))/(fact(n-p)))
#endmacro
 
dim as ulongint fact(0 to MAX), bell(0 to MAX)
dim as uinteger n=0, k
 
fact(0) = 1
for k=1 to MAX
fact(k) = k*fact(k-1)
next k
 
bell(n) = 1
print n, bell(n)
for n=0 to MAX-1
for k=0 to n
bell(n+1) += ncp(n, k)*bell(k)
next k
print n+1, bell(n+1)
next n

Go[edit]

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(51)
fmt.Println("First fifteen and fiftieth Bell numbers:")
for i := 1; i <= 15; i++ {
fmt.Printf("%2d: %d\n", i, bt[i][0])
}
fmt.Println("50:", bt[50][0])
fmt.Println("\nThe first ten rows of Bell's triangle:")
for i := 1; i <= 10; i++ {
fmt.Println(bt[i])
}
}
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]

Groovy[edit]

Translation of: Java
class Bell {
private static class BellTriangle {
private List<Integer> arr
 
BellTriangle(int n) {
int length = (int) (n * (n + 1) / 2)
arr = new ArrayList<>(length)
for (int i = 0; i < length; ++i) {
arr.add(0)
}
 
set(1, 0, 1)
for (int i = 2; i <= n; ++i) {
set(i, 0, get(i - 1, i - 2))
for (int j = 1; j < i; ++j) {
int value = get(i, j - 1) + get(i - 1, j - 1)
set(i, j, value)
}
}
}
 
private static int index(int row, int col) {
if (row > 0 && col >= 0 && col < row) {
return row * (row - 1) / 2 + col
} else {
throw new IllegalArgumentException()
}
}
 
int get(int row, int col) {
int i = index(row, col)
return arr.get(i)
}
 
void set(int row, int col, int value) {
int i = index(row, col)
arr.set(i, value)
}
}
 
static void main(String[] args) {
final int rows = 15
BellTriangle bt = new BellTriangle(rows)
 
System.out.println("First fifteen Bell numbers:")
for (int i = 0; i < rows; ++i) {
System.out.printf("%2d: %d\n", i + 1, bt.get(i + 1, 0))
}
 
for (int i = 1; i <= 10; ++i) {
System.out.print(bt.get(i, 0))
for (int j = 1; j < i; ++j) {
System.out.printf(", %d", bt.get(i, j))
}
System.out.println()
}
}
}
Output:
First fifteen 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
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

Haskell[edit]

bellTri :: [[Integer]]
bellTri = map snd (iterate (f . uncurry (scanl (+))) (1,[1]))
where
f xs = (last xs, xs)
 
bell :: [Integer]
bell = map head bellTri
 
main = do
putStrLn "First 10 rows of Bell's Triangle"
mapM_ print (take 10 bellTri)
putStrLn "First 15 Bell numbers"
mapM_ print (take 15 bell)
putStrLn "50th Bell number"
print (bell !! 49)
 
Output:
First 10 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]
First 15 Bell numbers
1
1
2
5
15
52
203
877
4140
21147
115975
678570
4213597
27644437
190899322
50th Bell number
10726137154573358400342215518590002633917247281

And, of course, in terms of Control.Arrow or Control.Applicative, the triangle function could also be written as:

import Control.Arrow
 
bellTri :: [[Integer]]
bellTri = map snd (iterate ((last &&& id) . uncurry (scanl (+))) (1,[1]))

or:

import Control.Applicative
 
bellTri :: [[Integer]]
bellTri = map snd (iterate ((liftA2 (,) last id) . uncurry (scanl (+))) (1,[1]))

or, as an applicative without the need for an import:

bellTri :: [[Integer]]
bellTri = map snd (iterate (((,) . last <*> id) . uncurry (scanl (+))) (1, [1]))

J[edit]

 
bell=: ([: +/\ (,~ {:))&.>@:{:
 
,. bell^:(<5) <1
+--------------+
|1 |
+--------------+
|1 2 |
+--------------+
|2 3 5 |
+--------------+
|5 7 10 15 |
+--------------+
|15 20 27 37 52|
+--------------+
 
{.&> bell^:(<15) <1
1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322
 
{:>bell^:49<1x
185724268771078270438257767181908917499221852770
 

Java[edit]

Translation of: Kotlin
import java.util.ArrayList;
import java.util.List;
 
public class Bell {
private static class BellTriangle {
private List<Integer> arr;
 
BellTriangle(int n) {
int length = n * (n + 1) / 2;
arr = new ArrayList<>(length);
for (int i = 0; i < length; ++i) {
arr.add(0);
}
 
set(1, 0, 1);
for (int i = 2; i <= n; ++i) {
set(i, 0, get(i - 1, i - 2));
for (int j = 1; j < i; ++j) {
int value = get(i, j - 1) + get(i - 1, j - 1);
set(i, j, value);
}
}
}
 
private int index(int row, int col) {
if (row > 0 && col >= 0 && col < row) {
return row * (row - 1) / 2 + col;
} else {
throw new IllegalArgumentException();
}
}
 
public int get(int row, int col) {
int i = index(row, col);
return arr.get(i);
}
 
public void set(int row, int col, int value) {
int i = index(row, col);
arr.set(i, value);
}
}
 
public static void main(String[] args) {
final int rows = 15;
BellTriangle bt = new BellTriangle(rows);
 
System.out.println("First fifteen Bell numbers:");
for (int i = 0; i < rows; ++i) {
System.out.printf("%2d: %d\n", i + 1, bt.get(i + 1, 0));
}
 
for (int i = 1; i <= 10; ++i) {
System.out.print(bt.get(i, 0));
for (int j = 1; j < i; ++j) {
System.out.printf(", %d", bt.get(i, j));
}
System.out.println();
}
}
}
Output:
First fifteen 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
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

jq[edit]

Translation of: Julia
# nth Bell number
def bell:
. as $n
| if $n < 0 then "non-negative integer expected"
elif $n < 2 then 1
else
reduce range(1; $n) as $i ([1];
reduce range(1; $i) as $j (.;
.[$i - $j] as $x
| .[$i - $j - 1] += $x )
| .[$i] = .[0] + .[$i - 1] )
| .[$n - 1]
end;
 
# The task
range(1;51) | bell
Output:

For displaying the results, we will first use gojq, the Go implementation of jq, as it supports unbounded-precision integer arithmetic.

1
2
5
15
...
37450059502461511196505342096431510120174682
628919796303118415420210454071849537746015761
10726137154573358400342215518590002633917247281
185724268771078270438257767181908917499221852770

Using the C-based implementation of jq, the results become inexact from bell(23) onwards:

[1,1]
[2,2]
...
[21,474869816156751]
[22,4506715738447323]
# inexact 
[23,44152005855084344]
[24,445958869294805300]
...
[49,1.0726137154573358e+46]
[50,1.8572426877107823e+47]

Julia[edit]

Source: Combinatorics at https://github.com/JuliaMath/Combinatorics.jl/blob/master/src/numbers.jl

"""
bellnum(n)
Compute the ``n``th Bell number.
"""
function bellnum(n::Integer)
if n < 0
throw(DomainError(n))
elseif n < 2
return 1
end
list = Vector{BigInt}(undef, n)
list[1] = 1
for i = 2:n
for j = 1:i - 2
list[i - j - 1] += list[i - j]
end
list[i] = list[1] + list[i - 1]
end
return list[n]
end
 
for i in 1:50
println(bellnum(i))
end
Output:
1
2
5
15
52
203
877
4140
21147
115975
678570
4213597
27644437
190899322
1382958545
10480142147
82864869804
682076806159
5832742205057
51724158235372
474869816156751
4506715738447323
44152005855084346
445958869294805289
4638590332229999353
49631246523618756274
545717047936059989389
6160539404599934652455
71339801938860275191172
846749014511809332450147
10293358946226376485095653
128064670049908713818925644
1629595892846007606764728147
21195039388640360462388656799
281600203019560266563340426570
3819714729894818339975525681317
52868366208550447901945575624941
746289892095625330523099540639146
10738823330774692832768857986425209
157450588391204931289324344702531067
2351152507740617628200694077243788988
35742549198872617291353508656626642567
552950118797165484321714693280737767385
8701963427387055089023600531855797148876
139258505266263669602347053993654079693415
2265418219334494002928484444705392276158355
37450059502461511196505342096431510120174682
628919796303118415420210454071849537746015761
10726137154573358400342215518590002633917247281
185724268771078270438257767181908917499221852770

Kotlin[edit]

Translation of: C
class BellTriangle(n: Int) {
private val arr: Array<Int>
 
init {
val length = n * (n + 1) / 2
arr = Array(length) { 0 }
 
set(1, 0, 1)
for (i in 2..n) {
set(i, 0, get(i - 1, i - 2))
for (j in 1 until i) {
val value = get(i, j - 1) + get(i - 1, j - 1)
set(i, j, value)
}
}
}
 
private fun index(row: Int, col: Int): Int {
require(row > 0)
require(col >= 0)
require(col < row)
return row * (row - 1) / 2 + col
}
 
operator fun get(row: Int, col: Int): Int {
val i = index(row, col)
return arr[i]
}
 
private operator fun set(row: Int, col: Int, value: Int) {
val i = index(row, col)
arr[i] = value
}
}
 
fun main() {
val rows = 15
val bt = BellTriangle(rows)
 
println("First fifteen Bell numbers:")
for (i in 1..rows) {
println("%2d: %d".format(i, bt[i, 0]))
}
 
for (i in 1..10) {
print("${bt[i, 0]}")
for (j in 1 until i) {
print(", ${bt[i, j]}")
}
println()
}
}
Output:
First fifteen 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
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

Lua[edit]

-- Bell numbers in Lua
-- db 6/11/2020 (to replace missing original)
 
local function bellTriangle(n)
local tri = { {1} }
for i = 2, n do
tri[i] = { tri[i-1][i-1] }
for j = 2, i do
tri[i][j] = tri[i][j-1] + tri[i-1][j-1]
end
end
return tri
end
 
local N = 25 -- in lieu of 50, practical limit with double precision floats
local tri = bellTriangle(N)
 
print("First 15 and "..N.."th Bell numbers:")
for i = 1, 15 do
print(i, tri[i][1])
end
print(N, tri[N][1])
 
print()
 
print("First 10 rows of Bell triangle:")
for i = 1, 10 do
print("[ " .. table.concat(tri[i],", ") .. " ]")
end
Output:
First 15 and 25th 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
25      445958869294805289

First 10 rows of Bell 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 ]

Maple[edit]

bell1:=proc(n)
option remember;
add(binomial(n-1,k)*bell1(k),k=0..n-1)
end:
bell1(0):=1:
 
bell1(50);
# 185724268771078270438257767181908917499221852770
 
combinat[bell](50);
# 185724268771078270438257767181908917499221852770
 
bell1~([$0..20]);
# [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570,
# 4213597, 27644437, 190899322, 1382958545, 10480142147,
# 82864869804, 682076806159, 5832742205057, 51724158235372]

Mathematica / Wolfram Language[edit]

Function definition:

 
BellTriangle[n_Integer?Positive] := NestList[Accumulate[# /. {a___, b_} :> {b, a, b}] &, {1}, n - 1];
BellNumber[n_Integer] := BellTriangle[n][[n, 1]];
 

Output:

 
In[51]:= Array[BellNumber, 25]
 
Out[51]= {1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, \
4213597, 27644437, 190899322, 1382958545, 10480142147, 82864869804, \
682076806159, 5832742205057, 51724158235372, 474869816156751, \
4506715738447323, 44152005855084346, 445958869294805289}
 
In[52]:= BellTriangle[10]
 
Out[52]= {{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}}
 

Nim[edit]

Using Recurrence relation[edit]

import math
 
iterator b(): int =
## Iterator yielding the bell numbers.
var numbers = @[1]
yield 1
var n = 0
while true:
var next = 0
for k in 0..n:
next += binom(n, k) * numbers[k]
numbers.add(next)
yield next
inc n
 
when isMainModule:
 
import strformat
 
const Limit = 25 # Maximum index beyond which an overflow occurs.
 
echo "Bell numbers from B0 to B25:"
var i = 0
for n in b():
echo fmt"{i:2d}: {n:>20d}"
inc i
if i > Limit:
break
Output:
Bell numbers from B0 to B25:
 0:                    1
 1:                    1
 2:                    2
 3:                    5
 4:                   15
 5:                   52
 6:                  203
 7:                  877
 8:                 4140
 9:                21147
10:               115975
11:               678570
12:              4213597
13:             27644437
14:            190899322
15:           1382958545
16:          10480142147
17:          82864869804
18:         682076806159
19:        5832742205057
20:       51724158235372
21:      474869816156751
22:     4506715738447323
23:    44152005855084346
24:   445958869294805289
25:  4638590332229999353

Using Bell triangle[edit]

iterator b(): int =
## Iterator yielding the bell numbers.
var row = @[1]
yield 1
yield 1
while true:
var newRow = newSeq[int](row.len + 1)
newRow[0] = row[^1]
for i in 1..newRow.high:
newRow[i] = newRow[i - 1] + row[i - 1]
row.shallowCopy(newRow)
yield row[^1] # The last value of the row is one step ahead of the first one.
 
iterator bellTriangle(): seq[int] =
## Iterator yielding the rows of the Bell triangle.
var row = @[1]
yield row
while true:
var newRow = newSeq[int](row.len + 1)
newRow[0] = row[^1]
for i in 1..newRow.high:
newRow[i] = newRow[i - 1] + row[i - 1]
row.shallowCopy(newRow)
yield row
 
when isMainModule:
 
import strformat
import strutils
 
const Limit = 25 # Maximum index beyond which an overflow occurs.
 
echo "Bell numbers from B0 to B25:"
var i = 0
for n in b():
echo fmt"{i:2d}: {n:>20d}"
inc i
if i > Limit:
break
 
echo "\nFirst ten rows of Bell triangle:"
i = 0
for row in bellTriangle():
inc i
var line = ""
for val in row:
line.addSep(" ", 0)
line.add(fmt"{val:6d}")
echo line
if i == 10:
break
Output:
Bell numbers from B0 to B25:
 0:                    1
 1:                    1
 2:                    2
 3:                    5
 4:                   15
 5:                   52
 6:                  203
 7:                  877
 8:                 4140
 9:                21147
10:               115975
11:               678570
12:              4213597
13:             27644437
14:            190899322
15:           1382958545
16:          10480142147
17:          82864869804
18:         682076806159
19:        5832742205057
20:       51724158235372
21:      474869816156751
22:     4506715738447323
23:    44152005855084346
24:   445958869294805289
25:  4638590332229999353

First ten rows of Bell 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[edit]

Translation of: Raku
use strict 'vars';
use warnings;
use feature 'say';
use bigint;
 
my @b = 1;
my @Aitkens = [1];
 
push @Aitkens, do {
my @c = $b[-1];
push @c, $b[$_] + $c[$_] for 0..$#b;
@b = @c;
[@c]
} until (@Aitkens == 50);
 
my @Bell_numbers = map { @$_[0] } @Aitkens;
 
say 'First fifteen and fiftieth Bell numbers:';
printf "%2d: %s\n", 1+$_, $Bell_numbers[$_] for 0..14, 49;
 
say "\nFirst ten rows of Aitken's array:";
printf '%-7d'x@{$Aitkens[$_]}."\n", @{$Aitkens[$_]} for 0..9;
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

Phix[edit]

Library: Phix/mpfr

Started out as a translation of Go, but the main routine has now been completely replaced.

with javascript_semantics
include mpfr.e
function bellTriangle(integer n)
-- nb: returns strings to simplify output
    mpz z = mpz_init(1) 
    string sz = "1"
    sequence tri = {}, line = {}
    for i=1 to n do
        line = prepend(line,mpz_init_set(z))
        tri = append(tri,{sz})
        for j=2 to length(line) do
            mpz_add(z,z,line[j])
            mpz_set(line[j],z)
            sz = mpz_get_str(z)
            tri[$] = append(tri[$],sz)
        end for
    end for
    line = mpz_free(line)
    z = mpz_free(z)
    return tri
end function
 
sequence bt = bellTriangle(50)
printf(1,"First fifteen and fiftieth Bell numbers:\n%s\n50:%s\n\n",
         {join(vslice(bt[1..15],1)),bt[50][1]})
printf(1,"The first ten rows of Bell's triangle:\n")
for i=1 to 10 do
    printf(1,"%s\n",{join(bt[i])})
end for
Output:
First fifteen and fiftieth Bell numbers:
1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322
50:10726137154573358400342215518590002633917247281

The 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

PicoLisp[edit]

(de bell (N)
(make
(setq L (link (list 1)))
(do N
(setq L
(link
(make
(setq A (link (last L)))
(for B L
(setq A (link (+ A B))) ) ) ) ) ) ) )
(setq L (bell 51))
(for N 15
(tab (2 -2 -2) N ":" (get L N 1)) )
(prinl 50 ": " (get L 50 1))
(prinl)
(prinl "First ten rows:")
(for N 10
(println (get L N)) )
Output:
 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:
(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)

Prolog[edit]

Works with: SWI Prolog
bell(N, Bell):-
bell(N, Bell, [], _).
 
bell(0, [[1]|T], T, [1]):-!.
bell(N, Bell, B, Row):-
N1 is N - 1,
bell(N1, Bell, [Row|B], Last),
next_row(Row, Last).
 
next_row([Last|Bell], Bell1):-
last(Bell1, Last),
next_row1(Last, Bell, Bell1).
 
next_row1(_, [], []):-!.
next_row1(X, [Y|Rest], [B|Bell]):-
Y is X + B,
next_row1(Y, Rest, Bell).
 
print_bell_numbers(_, 0):-!.
print_bell_numbers([[Number|_]|Bell], N):-
writef('%w\n', [Number]),
N1 is N - 1,
print_bell_numbers(Bell, N1).
 
print_bell_rows(_, 0):-!.
print_bell_rows([Row|Rows], N):-
print_bell_row(Row),
N1 is N - 1,
print_bell_rows(Rows, N1).
 
print_bell_row([Number]):-
!,
writef('%w\n', [Number]).
print_bell_row([Number|Numbers]):-
writef('%w ', [Number]),
print_bell_row(Numbers).
 
main:-
bell(49, Bell),
writef('First 15 Bell numbers:\n'),
print_bell_numbers(Bell, 15),
last(Bell, [Number|_]),
writef('\n50th Bell number: %w\n', [Number]),
writef('\nFirst 10 rows of Bell triangle:\n'),
print_bell_rows(Bell, 10).
Output:
First 15 Bell numbers:
1
1
2
5
15
52
203
877
4140
21147
115975
678570
4213597
27644437
190899322

50th Bell number: 10726137154573358400342215518590002633917247281

First 10 rows of Bell 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

Python[edit]

Procedural[edit]

Translation of: D
Works with: Python version 2.7
def bellTriangle(n):
tri = [None] * n
for i in xrange(n):
tri[i] = [0] * i
tri[1][0] = 1
for i in xrange(2, n):
tri[i][0] = tri[i - 1][i - 2]
for j in xrange(1, i):
tri[i][j] = tri[i][j - 1] + tri[i - 1][j - 1]
return tri
 
def main():
bt = bellTriangle(51)
print "First fifteen and fiftieth Bell numbers:"
for i in xrange(1, 16):
print "%2d: %d" % (i, bt[i][0])
print "50:", bt[50][0]
print
print "The first ten rows of Bell's triangle:"
for i in xrange(1, 11):
print bt[i]
 
main()
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

The 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]

Functional[edit]

Translation of: Haskell
Works with: Python version 3.7
'''Bell numbers'''
 
from itertools import accumulate, chain, islice
from operator import add, itemgetter
from functools import reduce
 
 
# bellNumbers :: [Int]
def bellNumbers():
'''Bell or exponential numbers.
A000110
'''

return map(itemgetter(0), bellTriangle())
 
 
# bellTriangle :: [[Int]]
def bellTriangle():
'''Bell triangle.'''
return map(
itemgetter(1),
iterate(
compose(
bimap(last)(identity),
list, uncurry(scanl(add))
)
)((1, [1]))
)
 
 
# ------------------------- TEST --------------------------
# main :: IO ()
def main():
'''Tests'''
showIndex = compose(repr, succ, itemgetter(0))
showValue = compose(repr, itemgetter(1))
print(
fTable(
'First fifteen Bell numbers:'
)(showIndex)(showValue)(identity)(list(
enumerate(take(15)(bellNumbers()))
))
)
 
print('\nFiftieth Bell number:')
bells = bellNumbers()
drop(49)(bells)
print(
next(bells)
)
 
print(
fTable(
"\nFirst 10 rows of Bell's triangle:"
)(showIndex)(showValue)(identity)(list(
enumerate(take(10)(bellTriangle()))
))
)
 
 
# ------------------------ GENERIC ------------------------
 
# bimap :: (a -> b) -> (c -> d) -> (a, c) -> (b, d)
def bimap(f):
'''Tuple instance of bimap.
A tuple of the application of f and g to the
first and second values respectively.
'''

def go(g):
def gox(x):
return (f(x), g(x))
return gox
return go
 
 
# compose :: ((a -> a), ...) -> (a -> a)
def compose(*fs):
'''Composition, from right to left,
of a series of functions.
'''

def go(f, g):
def fg(x):
return f(g(x))
return fg
return reduce(go, fs, identity)
 
 
# drop :: Int -> [a] -> [a]
# drop :: Int -> String -> String
def drop(n):
'''The sublist of xs beginning at
(zero-based) index n.
'''

def go(xs):
if isinstance(xs, (list, tuple, str)):
return xs[n:]
else:
take(n)(xs)
return xs
return go
 
 
# fTable :: String -> (a -> String) ->
# (b -> String) -> (a -> b) -> [a] -> String
def fTable(s):
'''Heading -> x display function -> fx display function ->
f -> xs -> tabular string.
'''

def gox(xShow):
def gofx(fxShow):
def gof(f):
def goxs(xs):
ys = [xShow(x) for x in xs]
w = max(map(len, ys))
 
def arrowed(x, y):
return y.rjust(w, ' ') + ' -> ' + fxShow(f(x))
return s + '\n' + '\n'.join(
map(arrowed, xs, ys)
)
return goxs
return gof
return gofx
return gox
 
 
# identity :: a -> a
def identity(x):
'''The identity function.'''
return x
 
 
# iterate :: (a -> a) -> a -> Gen [a]
def iterate(f):
'''An infinite list of repeated
applications of f to x.
'''

def go(x):
v = x
while True:
yield v
v = f(v)
return go
 
 
# last :: [a] -> a
def last(xs):
'''The last element of a non-empty list.'''
return xs[-1]
 
 
# scanl :: (b -> a -> b) -> b -> [a] -> [b]
def scanl(f):
'''scanl is like reduce, but returns a succession of
intermediate values, building from the left.
'''

def go(a):
def g(xs):
return accumulate(chain([a], xs), f)
return g
return go
 
 
# succ :: Enum a => a -> a
def succ(x):
'''The successor of a value.
For numeric types, (1 +).
'''

return 1 + x
 
 
# take :: Int -> [a] -> [a]
# take :: Int -> String -> String
def take(n):
'''The prefix of xs of length n,
or xs itself if n > length xs.
'''

def go(xs):
return (
xs[0:n]
if isinstance(xs, (list, tuple))
else list(islice(xs, n))
)
return go
 
 
# uncurry :: (a -> b -> c) -> ((a, b) -> c)
def uncurry(f):
'''A function over a tuple,
derived from a curried function.
'''

def go(tpl):
return f(tpl[0])(tpl[1])
return go
 
 
# MAIN ---
if __name__ == '__main__':
main()
Output:
First fifteen 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

Fiftieth Bell number:
10726137154573358400342215518590002633917247281

First 10 rows of Bell's triangle:
 1 -> [1]
 2 -> [1, 2]
 3 -> [2, 3, 5]
 4 -> [5, 7, 10, 15]
 5 -> [15, 20, 27, 37, 52]
 6 -> [52, 67, 87, 114, 151, 203]
 7 -> [203, 255, 322, 409, 523, 674, 877]
 8 -> [877, 1080, 1335, 1657, 2066, 2589, 3263, 4140]
 9 -> [4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147]
10 -> [21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975]

Quackery[edit]

  [ ' [ [ 1 ] ]  ' [ 1 ] 
rot 1 - times
[ dup -1 peek nested
swap witheach
[ over -1 peek + join ]
tuck nested join swap ]
drop ] is bell's-triangle ( n --> [ )
 
[ bell's-triangle [] swap
witheach [ 0 peek join ] ] is bell-numbers ( n --> [ )
 
say "First fifteen Bell numbers:" cr
15 bell-numbers echo
cr cr
say "Fiftieth Bell number:" cr
50 bell-numbers -1 peek echo
cr cr
say "First ten rows of Bell's triangle:" cr
10 bell's-triangle witheach [ echo cr ]
Output:
First fifteen Bell numbers:
[ 1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322 ]

Fiftieth Bell number:
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 ]

Raku[edit]

(formerly Perl 6)

via Aitken's array[edit]

Works with: Rakudo version 2019.03
 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];
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]

via Recurrence relation[edit]

Works with: Rakudo version 2019.03
sub binomial { [*] ($^n0) Z/ 1 .. $^p }
 
my @bell = 1, -> *@s { [+] @s »*« @s.keys.map: { binomial(@s-1, $_) } }*;
 
.say for @bell[^15], @bell[50 - 1];
Output:
(1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322)
10726137154573358400342215518590002633917247281

via Stirling sums[edit]

Works with: Rakudo version 2019.03
my @Stirling_numbers_of_the_second_kind =
(1,),
{ (0, |@^last) »+« (|(@^last »*« @^last.keys), 0) }*
;
my @bell = @Stirling_numbers_of_the_second_kind.map: *.sum;
 
.say for @bell.head(15), @bell[50 - 1];
Output:
(1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322)
10726137154573358400342215518590002633917247281 

REXX[edit]

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 by using memoization.

Also, see this task's   discussion   to view how the sizes of Bell numbers increase in relation to its index.

/*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.*/
!.=.;  !.0= 1;  !.1= 1; @.= 1 /*the FACT function uses memoization.*/
do j=0 for HI+1; $= j==0; jm= j-1 /*JM is used for a shortcut (below). */
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) )") = " commas($)
end /*j*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg _; do c=length(_)-3 to 1 by -3; _=insert(',', _, c); end; return _
/*──────────────────────────────────────────────────────────────────────────────────────*/
comb: procedure expose !.; parse arg x,y; if x==y then return 1
if x-y<y then y= x - y; if !.x.y\==. then return !.x.y / fact(y)
_= 1; do j=x-y+1 to x; _= _*j; end;  !.x.y= _; return _ / fact(y)
/*──────────────────────────────────────────────────────────────────────────────────────*/
fact: procedure expose !.; parse arg x; if !.x\==. then return !.x;  != 1
do f=2 for x-1;  != ! * f; end;  !.x= !; return !
output   when using the internal default inputs of:     0   14
      Bell( 0) =  1
      Bell( 1) =  1
      Bell( 2) =  2
      Bell( 3) =  5
      Bell( 4) =  15
      Bell( 5) =  52
      Bell( 6) =  203
      Bell( 7) =  877
      Bell( 8) =  4,140
      Bell( 9) =  21,147
      Bell(10) =  115,975
      Bell(11) =  678,570
      Bell(12) =  4,213,597
      Bell(13) =  27,644,437
      Bell(14) =  190,899,322
output   when using the inputs of:     49   49
      Bell(49) =  10,726,137,154,573,358,400,342,215,518,590,002,633,917,247,281 

Ruby[edit]

Translation of: D
def bellTriangle(n)
tri = Array.new(n)
for i in 0 .. n - 1 do
tri[i] = Array.new(i)
for j in 0 .. i - 1 do
tri[i][j] = 0
end
end
tri[1][0] = 1
for i in 2 .. n - 1 do
tri[i][0] = tri[i - 1][i - 2]
for j in 1 .. i - 1 do
tri[i][j] = tri[i][j - 1] + tri[i - 1][j - 1]
end
end
return tri
end
 
def main
bt = bellTriangle(51)
puts "First fifteen and fiftieth Bell numbers:"
for i in 1 .. 15 do
puts "%2d: %d" % [i, bt[i][0]]
end
puts "50: %d" % [bt[50][0]]
puts
 
puts "The first ten rows of Bell's triangle:"
for i in 1 .. 10 do
puts bt[i].inspect
end
end
 
main()
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

The 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]

Rust[edit]

Translation of: D
Library: num version 0.2
use num::BigUint;
 
fn main() {
let bt = bell_triangle(51);
// the fifteen first
for i in 1..=15 {
println!("{}: {}", i, bt[i][0]);
}
 
// the fiftieth
println!("50: {}", bt[50][0])
}
 
fn bell_triangle(n: usize) -> Vec<Vec<BigUint>> {
let mut tri: Vec<Vec<BigUint>> = Vec::with_capacity(n);
for i in 0..n {
let v = vec![BigUint::from(0u32); i];
tri.push(v);
}
tri[1][0] = BigUint::from(1u32);
 
for i in 2..n {
tri[i][0] = BigUint::from_bytes_be(&tri[i - 1][i - 2].to_bytes_be());
for j in 1..i {
let added_big_uint = &tri[i][j - 1] + &tri[i - 1][j - 1];
tri[i][j] = BigUint::from_bytes_be(&added_big_uint.to_bytes_be());
}
}
 
tri
}
 
Output:
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

Scala[edit]

Translation of: Java
import scala.collection.mutable.ListBuffer
 
object BellNumbers {
class BellTriangle {
val arr: ListBuffer[Int] = ListBuffer.empty[Int]
 
def this(n: Int) {
this()
 
val length = n * (n + 1) / 2
for (_ <- 0 until length) {
arr += 0
}
 
this (1, 0) = 1
for (i <- 2 to n) {
this (i, 0) = this (i - 1, i - 2)
for (j <- 1 until i) {
this (i, j) = this (i, j - 1) + this (i - 1, j - 1)
}
}
}
 
private def index(row: Int, col: Int): Int = {
require(row > 0, "row must be greater than zero")
require(col >= 0, "col must not be negative")
require(col < row, "col must be less than row")
 
row * (row - 1) / 2 + col
}
 
def apply(row: Int, col: Int): Int = {
val i = index(row, col)
arr(i)
}
 
def update(row: Int, col: Int, value: Int): Unit = {
val i = index(row, col)
arr(i) = value
}
}
 
def main(args: Array[String]): Unit = {
val rows = 15
val bt = new BellTriangle(rows)
 
println("First fifteen Bell numbers:")
for (i <- 0 until rows) {
printf("%2d: %d\n", i + 1, bt(i + 1, 0))
}
 
for (i <- 1 to 10) {
print(bt(i, 0))
for (j <- 1 until i) {
print(s", ${bt(i, j)}")
}
println()
}
}
}
Output:
First fifteen 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
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

Sidef[edit]

Built-in:

say 15.of { .bell }

Formula as a sum of Stirling numbers of the second kind:

func bell(n) { sum(0..n, {|k| stirling2(n, k) }) }

Via Aitken's array (optimized for space):

func bell_numbers (n) {
 
var acc = []
var bell = [1]
 
(n-1).times {
acc.unshift(bell[-1])
acc.accumulate!
bell.push(acc[-1])
}
 
bell
}
 
var B = bell_numbers(50)
say "The first 15 Bell numbers: #{B.first(15).join(', ')}"
say "The fiftieth Bell number : #{B[50-1]}"
Output:
The first 15 Bell numbers: 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322
The fiftieth Bell number : 10726137154573358400342215518590002633917247281

Aitken's array:

func aitken_array (n) {
 
var A = [1]
 
[[1]] + (n-1).of {
A = [A[-1], A...].accumulate
}
}
 
aitken_array(10).each { .say }
Output:
[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]

Aitken's array (recursive definition):

func A((0), (0))       { 1 }
func A(n, (0)) { A(n-1, n-1) }
func A(n, k) is cached { A(n, k-1) + A(n-1, k-1) }
 
for n in (^10) {
say (0..n -> map{|k| A(n, k) })
}

(same output as above)

Swift[edit]

Translation of: Kotlin
public struct BellTriangle<T: BinaryInteger> {
@usableFromInline
var arr: [T]
 
@inlinable
public internal(set) subscript(row row: Int, col col: Int) -> T {
get { arr[row * (row - 1) / 2 + col] }
set { arr[row * (row - 1) / 2 + col] = newValue }
}
 
@inlinable
public init(n: Int) {
arr = Array(repeating: 0, count: n * (n + 1) / 2)
 
self[row: 1, col: 0] = 1
 
for i in 2...n {
self[row: i, col: 0] = self[row: i - 1, col: i - 2]
 
for j in 1..<i {
self[row: i, col: j] = self[row: i, col: j - 1] + self[row: i - 1, col: j - 1]
}
}
}
}
 
let tri = BellTriangle<Int>(n: 15)
 
print("First 15 Bell numbers:")
 
for i in 1...15 {
print("\(i): \(tri[row: i, col: 0])")
}
 
for i in 1...10 {
print(tri[row: i, col: 0], terminator: "")
 
for j in 1..<i {
print(", \(tri[row: i, col: j])", terminator: "")
}
 
print()
}
Output:
First 15 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
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

Visual Basic .NET[edit]

Translation of: C#
Imports System.Numerics
Imports System.Runtime.CompilerServices
 
Module Module1
 
<Extension()>
Sub Init(Of T)(array As T(), value As T)
If IsNothing(array) Then Return
For i = 0 To array.Length - 1
array(i) = value
Next
End Sub
 
Function BellTriangle(n As Integer) As BigInteger()()
Dim tri(n - 1)() As BigInteger
For i = 0 To n - 1
Dim temp(i - 1) As BigInteger
tri(i) = temp
tri(i).Init(0)
Next
tri(1)(0) = 1
For i = 2 To n - 1
tri(i)(0) = tri(i - 1)(i - 2)
For j = 1 To i - 1
tri(i)(j) = tri(i)(j - 1) + tri(i - 1)(j - 1)
Next
Next
Return tri
End Function
 
Sub Main()
Dim bt = BellTriangle(51)
Console.WriteLine("First fifteen Bell numbers:")
For i = 1 To 15
Console.WriteLine("{0,2}: {1}", i, bt(i)(0))
Next
Console.WriteLine("50: {0}", bt(50)(0))
Console.WriteLine()
Console.WriteLine("The first ten rows of Bell's triangle:")
For i = 1 To 10
Dim it = bt(i).GetEnumerator()
Console.Write("[")
If it.MoveNext() Then
Console.Write(it.Current)
End If
While it.MoveNext()
Console.Write(", ")
Console.Write(it.Current)
End While
Console.WriteLine("]")
Next
End Sub
 
End Module
Output:
First fifteen 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

The 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]

Wren[edit]

Translation of: Go
Library: Wren-fmt

Unable to calculate 50th Bell number accurately due to lack of 'big integer' support.

import "/fmt" for Fmt
 
var bellTriangle = Fn.new { |n|
var tri = List.filled(n, [])
for (i in 0...n) tri[i] = List.filled(i, 0)
tri[1][0] = 1
for (i in 2...n) {
tri[i][0] = tri[i-1][i-2]
for (j in 1...i) {
tri[i][j] = tri[i][j-1] + tri[i-1][j-1]
}
}
return tri
}
 
var bt = bellTriangle.call(16)
System.print("First fifteen Bell numbers:")
for (i in 1..15) System.print("%(Fmt.d(2, i)): %(bt[i][0])")
System.print("\nThe first ten rows of Bell's triangle:")
for (i in 1..10) System.print(bt[i])
Output:
First fifteen 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

The 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]

zkl[edit]

fcn bellTriangleW(start=1,wantRow=False){	// --> iterator
Walker.zero().tweak('wrap(row){
row.insert(0,row[-1]);
foreach i in ([1..row.len()-1]){ row[i]+=row[i-1] }
wantRow and row or row[-1]
}.fp(List(start))).push(start,start);
}
println("First fifteen Bell numbers:");
bellTriangleW().walk(15).println();
Output:
First fifteen Bell numbers:
L(1,1,2,5,15,52,203,877,4140,21147,115975,678570,4213597,27644437,190899322)
println("Rows of the Bell Triangle:");
bt:=bellTriangleW(1,True); do(11){ println(bt.next()) }
Output:
Rows of the Bell Triangle:
1
1
L(1,2)
L(2,3,5)
L(5,7,10,15)
L(15,20,27,37,52)
L(52,67,87,114,151,203)
L(203,255,322,409,523,674,877)
L(877,1080,1335,1657,2066,2589,3263,4140)
L(4140,5017,6097,7432,9089,11155,13744,17007,21147)
L(21147,25287,30304,36401,43833,52922,64077,77821,94828,115975)
Library: GMP
GNU Multiple Precision Arithmetic Library
print("The fiftieth Bell number: ");
var [const] BI=Import("zklBigNum"); // libGMP
bellTriangleW(BI(1)).drop(50).value.println();
Output:
The fiftieth Bell number: 10726137154573358400342215518590002633917247281