Originally Posted by

**galactus** This is an interesting topic.

The nth Bell number, $\displaystyle b_{n}$, is the number of ways to partition a set of n objects into subsets.

You already have $\displaystyle b_{3}=5$ as an example.

The sequence of Bell numbers is $\displaystyle 1,1,2,5,15,52,.....$

The exponential generating function for the Bell number is

$\displaystyle e^{e^{x}-1}$. This can be derived by using $\displaystyle b_{n+1}=\sum_{k=0}^{\infty}\frac{k^{n-1}}{(k-1)!}$

The general formula is $\displaystyle b_{n}=\frac{1}{e}\sum_{k=0}^{\infty}\frac{k^{n}}{k !}$

Note the formula looks a whole lot like the MacLaurin series for e, huh?.

With some effort, these can be derived.