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?
