I want to find a closed form expression for $$ \sum_{t=0}^N \sum_{s=0}^{N-t} \sqrt{t+1} \sqrt{s} \binom{N-t-s+k-3}{k-3}. $$ If the coefficients in front of the binomial would be a polynomial in $t$ and $s$, one could rewrite it in terms of binomial coefficients and use an upper-index Vandermonde identity to find a closed form expression. I do not know what to do for non-polynomial functions.
The problem comes from condensed matter quantum mechanics, more specifically, from the weighted counting problem: $$ \sum_{n_1+...+n_k=N} \sqrt{n_l(n_m+1)}, $$ where $N,k,n_i$ are integers and $l,m$ are arbitrary with $1\le l,m \le k$.
All help is appreciated!