The Bell number $B_n$ is the number of partitions of $[n]$. Unlike other basic combinatorial quantities, $B_n$ has no simple finite closed form. This seems surprising to me. Can anyone explain why this is the case, or lead me to references that discuss the matter further?
2026-03-26 14:40:39.1774536039
No simple closed form for Bell numbers
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With $S_{n,k}$ the Stirling number of the second kind:
$B_n=\sum_{k=0}^nS_{n,k}$
This is already a finite closed form expression.
If you mean finite closed form expressions without a sum sign:
If combinatorial numbers are more complex, e.g. more complex composed from other combinatorial numbers, than they have no finite closed form expression. Usual representations are generating functions (e.g. sums, products, continued fractions) and recurrence equations. Also some fundamental combinatorial quantities cannot be represented in closed form. Look e.g. for the partition function p(n).
There are some mathematical theories which classes of generating functions or recurrence equations have representations in finite closed form.