Define $f(u)=\sum_{k=0}^\infty \frac{k^u}{k!}$ for natural numbers $u$. Using maple I can find the sum for specific values of $u$, but there is no general answer (unsurprisingly). All values I found are integer multiples of $e$. Below is a short table (omitting $e$), which cannot be found in the OEIS
0 1
1 1
2 2
3 5
4 15
5 52
6 203
7 877
8 4140
9 21147
10 115975
What is known about this sequence, or the associated function we get by extending the argument $u$ to the nonnegative real line?
There should be no problems in differentiating term by term, and so we find the derivatives $$ f^{(n)}(u)=\sum_{k=n}^\infty \frac{k^u (\log k)^n}{k!} $$ so all the derivatives are positive.
