# Closed-form expression for the infinite sum in Dobiński's formula

In combinatorial mathematics, I learned about Dobiński's formula for the $$n$$-th Bell number $$B_n$$, which states that:

Dobiński's formula gives the $$n$$-th Bell number $$B_n$$ (i.e., the number of partitions of a set of size $$n$$): $$B_n = \dfrac{1}{e}\displaystyle\sum_{k=0}^\infty \frac{k^n}{k!},$$ where $$e$$ denotes Euler's number.

The infinite sum $$S_n=\displaystyle\sum_{k=0}^\infty\frac{k^n}{k!}$$ seems to be an integer multiple of $$e$$. For $$n=1$$, it equals $$e$$. Similarly, $$S_2=2e,\ S_3=5e,\ S_4=15e,\ldots$$. I tried to find a closed-form expression for this infinite sum in general as: $$S_n=Ne=B_ne$$ by expanding the summand and re-arranging the terms. I could not express it in terms of a function involving finite terms and operations. I also tried WolframAlpha. It does not give any closed-form expressions for the general case. But for a finite positive integer $$n$$, it gives the integer multiple of $$e$$. Does a closed-form expression exist for the infinite sum in general? In other words, can we express the $$n$$-th Bell number $$B_n$$ in a closed form? or in terms of some special functions?

• If a finite sum is a closed form, then there is the Stirling series representation. You can also find a form via the hypergeometric function: $S_n=\sum\limits_{k=1}^\infty\left(\frac{(2)_k}{(1)_k}\right)^n\frac1{k!}=\,_nF_n(2,\dots,2;1,\dots,1;1)$ Commented May 9 at 14:57

## 1 Answer

It seems there is no formula in closed form known.