Problem Statement

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}.

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.

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.

Let's start with the solution:

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]