Can we get closed form for $$\sum_{k=0}^m \left(-\frac12\right)^k \binom{2m}{m-k}k^p,\quad p\in\mathbb{N}\,?$$ In Concrete Mathematics Knuth describes Gosper's algorithm and its Zeilberger's extension, but they both fails on this partial hypergeometric sum. $-1/2$ can be replaced to $x$, but it doesn't help (I tried to differentiate it for any advances).
Hypergeometric function is not a closed form, however. Because this sum is a hypergeometric function yet (multiplied by binomial coeff.). It will be very cool to find closed form for $p=0$ or $p=1$. But if I understood correctly Knuth, there is no closed form.
Any ideas?