Bell number

$$B_n$$ is the number of set partitions of an $n$ element set.

A000110 on the OEIS

Recurrence

• $$B_0 = 1$$
• $$B_{n+1}=\sum_{k=0}^{n} \binom{n}{k} B_k$$

Formulas

• $$B_n=\frac{1}{e}\sum_{k=0}^\infty \frac{k^n}{k!}$$
• $$B_{n}=\sum_{k=1}^{n}\frac{k^{n}}{k!}\sum_{j=0}^{n-k}\frac{(-1)^{j}}{j!}$$
• $$B_n=\sum_{k=0}^n \left\{{n\atop k}\right\}$$

Generating Function

$$B(x) = \sum_{n=0}^\infty \frac{B_n}{n!} x^n = e^{e^x-1}$$

Software

https://rosettacode.org/wiki/Bell_numbers

Articles

https://en.wikipedia.org/wiki/Bell_number

Blog Posts

Talks about computing bell numbers efficiently. https://fredrikj.net/blog/2015/08/computing-bell-numbers/