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?