I'm working with this sum: $$S(p,n)=\sum _{k=0}^{n+3} (-1)^k \binom{n+2 p+4-k}{2 p+1} \binom{2 n+2 p+7}{k} (n+p+3-k)^{2 n+4}$$ and I want to prove that it is eventually positive as a function of $n$, i.e., there exist $n$ such that $S(p,n)>0$ (and then it will be positive afterwards).
Using a C.A.S. I could see that $S(p,n)>0$ for $n\geq p-1$ but I cannot prove any inequalities with that alternating factor in the sum.
Is there any general techniques that help understand the behaviour of such alternating sums in the limit?