I want to prove for $2\le k<n$ (and $k$ is even if necessary)
$$\sum_{j=\frac{k}{2}+1}^{k-1}\binom{k-1}{j}\binom{n-(k-1)}{k-j}<2\binom{n-1}{k-1}.$$
MATLAB verifies it is true when $n=30$ and $k=10$. So I tend to believe in some range the inequality is true. Maybe some combinatorial proof will provide the range.