Calculate $\sum_{n=1}^\infty\frac{n^x}{n!}$ I want to evaluate function defined by following sum: $$\sum_{n=1}^\infty\frac{n^x}{n!}$$ I was thinking about writing Taylor series expansion for it. However my try resulted in sum that looks even harder to calculate:
$$\sum_{n=1}^{\infty}\frac{\ln^k(n)}{n!}$$Thanks for all the help in solving this problem.

Proof of the Dobiniski's formula:
See, $$\sum_{n=0}^{\infty} \frac{n^k}{n!}=\underbrace{\frac{d}{dx}(x\frac{d}{dx}(x.....(\frac{d}{dx}e^x)))),}_\text{$k$ times}$$ at $x=1$.
$$=\underbrace{\frac{d}{dx}(x\frac{d}{dx}(......\frac{d}{dx}(x^2e^x+xe^x))),}_\text{$k-2$ times}$$ at $x=1$.....(1) 
Also , $eB_k=\frac{d^k}{dt^k}(e^{e^t})$ at $t=0$
$=\frac{d^{k-2}}{dt^{k-2}}((e^t)^2e^{e^t}+e^te^{e^t})$ at $t=0$ ....(2)
See, (1) and (2) have exact the same structure. Also $e^t=1, e^{e^t}=e$ at $t=0$. And, at $x=1, e^x=e$.
Hence, we get $$eB_k=\sum_{n=0}^{\infty} \frac{n^k}{n!}$$.
