I try to prove several hard combinatorial identities. One of them is following \begin{align*} \sum_{s=0}^{\min\{k,n-1\}} \sum_{i=0}^{k-s} (-1)^{i} {2n+k-i-1 \choose k-s-i} {i-n \choose s} {n+i-1 \choose i} {n+k-s-1 \choose k} =\\ =\sum_{j=0}^{[\frac k2]} \sum_{i=0}^{min\{j, n-1\}}{ n-1 \choose i}^2 {{2n+j-i-2} \choose j-i} { n+k-2j-1 \choose n-1} .\text{ ($n,k$ are nonnegative integer)} \end{align*} Using the identity of Le-Jen Shoo , we have
\begin{align*} \sum_{s=0}^{\min\{k,n-1\}} (-1)^s {n+k-s-1 \choose k}\sum_{i=0}^{k-s} (-1)^{i} {2n+k-i-1 \choose k-s-i} {n+s-i-1 \choose s} {n+i-1 \choose i} =\\ =\sum_{j=0}^{[\frac k2]} {n+j-1 \choose j}^2 { n+k-2j-1 \choose n-1}. \end{align*} I tried to use many formulas from manuscripts of H.W. Gould, books of Riordan, of Egorichev and other. But I can't prove this. I want to use Zeilbergers algorithm in Maple. But I I have no idea how to start it.
