Let $$ S_k=\sum_{j=0}^{\alpha k}(-1)^j\binom{k/2}{j}\binom{\alpha k^2-j k}{k} $$ where $\alpha\in(0,1/2)$ is a constant. I'm interested in understanding the asymptotic behaviour of $S_k$.

It would be sufficient for my purposes to determine $c_\alpha=\lim_{k\to\infty} ((\log S_k)/k-\log k)$, which I believe should be something like $1+\log\alpha-h(\alpha)$ for some positive function $h$. Even an upper bound of this form would be useful.

I've no idea if this is tractable. None of the (probably very naive) ideas I've tried so far have achieved anything useful. Any pointers would be appreciated.