combinatorics - Is there a closed form expression for the following sum?

Is there a closed form expression for the following sum? $$\sum_{0\le i_1

I can understand that there are $\binom{n}{k}$ such terms and the values that $i_1+\cdots+i_k$ can take vary from $\frac{k(k-1)}{2}$ to $k(n-k)+\frac{k(k+1)}{2}$. So it remains to find how often the term $\sum_{j=1}^k i_j=l$ is found as an exponent in the above sum. Can anyone give some idea? Thanks.

Answer

This is a particular case of a more general formula that I posted in here Sum of power functions over a simplex . By denoting your sum as $S_k(n)$ we have:
\begin{equation}
S_k(n) = \sum\limits_{j=0}^{k}
\frac{(-1)^j r^{\frac{1}{2} (-j+k-1) (k-j)+j (n+1)}}{(r;r)_j (r;r)_{k-j}}
\end{equation}
where $(r;r)_j := \prod\limits_{l=0}^{j-1} (1-r^{l+1})$.