I am having trouble to compute the following sum: $ \sum_{k=0}^n(n-2k)^p \frac{{n \choose k}{2m-n \choose m-k}}{{2m \choose m}} $ Here $p\geq 2$.
To simplify the question, we can even assume that $n/2-C\sqrt n\leq k \leq n/2+C\sqrt n$. Any help or sources will be very helpful.
Thank you.