Does anyone know of a closed formula for the function
$f\_k(x)=\sum\_{n=1}^{\infty}{n^k x^n}$ ? That is, the generating function of the sequence $1^k,2^k,3^k...$. 


It is not hard to see that $f\_k(x)=\frac{P(x)}{(1-x)^{k+1}}$, where $P(x)$ is a monic polynomial of degree $k$ (this follows from the identity $f\_k(x)=x\cdot f'\_{k-1}(x)$ ). A closed formula for the coefficients of $P(x)$ would be very helpful.



Closed formula for the generating function of the sequence of powers
Your polynomials are $x$ times the [Eulerian polynomials](http://oeis.org/wiki/Eulerian_polynomials). 


