Computing the n-th Bell number Bn

There are at least six ways of defining Bn:
  1. \(B_n\) is the number of partitions of a set with \(n\) elements;
  2. \(B_{n+1}=\displaystyle{\sum_{k=0}^n}\binom{n}{k}B_k\), with \(B_0=1\);
  3. \(B_n=\displaystyle\sum_{k=1}^n\frac{k^n}{k!}\sum_{j=0}^{n-k}\frac{(-1)^j}{j!}; \)
  4. \(B_n=\displaystyle\frac{1}{e}\sum_{k=0}^\infty\frac{k^n}{k!}; \)
  5. Via the Aitken table \(A_{i,j}\):

    \(\displaystyle A_{1,1}=1,\)
    \(\displaystyle A_{n,1}=A_{n-1,n-1}\), \(\quad n\gt 1,\)
    \(\displaystyle A_{n,k}=A_{n,k-1}+A_{n-1,k-1}, \quad 1\lt k\leq n\).

    Then \(B_n=A_{n,n},\quad 1\leq n\);

    \(A_{n,k}\) is the number of partitions of \(\{1,\ldots,n+1\}\) in which \(\{k+1\}\) is the singleton with the largest entry in the partition. (See references (2) and (3).)

  6. \(B_n=\displaystyle{\sum_{k=1}^n}S_{n,k}\),
    where \(S_{n,k}, 1\le n, 1\le k\le n\) is the Stirling number of the second kind:
    \(S_{n,1}=1, n\ge 1\) and \(S_{n+1,k}=k\times S_{n,k} + S_{n,k-1}, \quad 2\leq k\leq n\).
  1. OEIS: A000110
  2. The largest singleton of set partitions, Yidong Sun, Xiaojuan Wu ( reference supplied by David Guichard)
  3. Bell numbers (David Guichard)

Enter n (1 ≤ n ≤ 450):

Last modified 5th March 2023
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