Bell Numbers

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

Basic Recurrence

B_0 = 1

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

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\}

where $\left\{{n\atop k}\right\}$ is a Stirling Number of the second kind

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