I am trying to calculate $\sum_{i=0}^n i^K {n \choose i}$ for $K \in \mathbb{N}$. Clearly, the case $K=0$ is trivially $2^n$ by the binomial theorem. For higher $K$ I am stumped. I know I can use:
$(1+X)^n = \sum_i {n \choose i} X^i$
and differentiate with respect to $X$, and set $X=1$. This works for $K=1$ and $K=2$ but already by $K=3$ this becomes difficult to handle. If anyone knows of any references or knows how to calculate this I would be most grateful. Part of the problem is I don't work in combinatorics and don't know quite what terms to search for! Thanks.
