Here is an observation (thanks to OEIS):


$$\sum\_{i=0}^\infty \frac{i^k}{i!}= B\_k e,$$ where $B\_k$ is the $k$-th Bell number. I might be having reading comprehension issues, but I don't see this formula in the OEIS notes. I assume this is very well known - can someone point me at a reference or a simple proof?



Another formula for Bell numbers
That is essentially [Dobinski's formula](https://en.wikipedia.org/wiki/Dobinski%27s_formula).


