I would like to know how to derive the value of the following sum in terms of $\ell$, $n$ and $p$ where each is a positive integer: $$\sum_{k=1}^{\ell}(-1)^k\binom{n}{k}k^p\enspace.$$
For small values of $p$, WolframAlpha can evaluate this sum to $$\frac{(-1)^{\ell}\ell(\ell+1)P_p(n,\ell)\binom{n}{\ell+1}}{\prod_{i=1}^p(n-i)}$$ for some multivariate polynomial $P_p$ but it gets very complicated very quickly as $p$ grows and unfortunately it does not give any hint as to how it computes the result.
How can I proceed in evaluating this sum?
I tried to apply some complex analysis and look at the function $n!\Gamma(-z)z^p/\Gamma(n+1-z)$ which has residue $(-1)^k\binom{n}{k}k^p$ at the simple pole $z=k$ for $k=1,2,\cdots,n$ but this doesn't correspond to the sum I want to evaluate unless $\ell=n$ but in that case the sum is zero. I'm interested in the case $0<\ell<n$ when the sum is non-zero.