Partial Chu-Vandermonde summation $\sum\limits_{i=0}^{k/2-1} {n/4 \choose i}{3n/4 \choose k-i}$

Question

Could someone help me find a tight upper bound for the following: $\sum\limits_{i=0}^{k/2-1} {n/4 \choose i}{3n/4 \choose k-i}$

It's essentially a Chu-Vandermonde summation of only the first $k/2$ terms so it's clearly upper bounded by ${n \choose k}$, but I'd like to know how fast it approaches ${n \choose k}$ as a function of k.

Thanks.

Answer 1

mathematica says this here $$\frac{\Gamma (n+1)}{\Gamma (k+1) \Gamma (-k+n+1)}-\binom{\frac{n}{4}}{\frac{k}{2}} \binom{\frac{3 n}{4}}{\frac{k}{2}} \, _3F_2\left(1,-\frac{k}{2},\frac{k}{2}-\frac{n}{4};\frac{k}{2}+1,-\frac{k}{2}+ \frac{3 n}{4}+1;1\right)$$ which containes a hypergeometric function