I am trying to find the eigenvalues of a matrix, and the degeneracy of each eigenvalue is given by the following expression: \begin{equation} deg(2l)=4\sum_{\substack{\{0\leqslant 2i,2k \leqslant L^2:\\ 2k+2i=2l\}}}\binom{L^2}{2k}\binom{L^2}{2i}, \end{equation} where $L\geqslant2$, and where $2l=\{0,2,...,2L^2\}$ if L is even, or $2l=\{0,...,2L^2-2\}$ if L is odd.
Clearly, if $2l\leqslant L^2$, then the previous expression is equivalent to: \begin{equation} deg(2l)=\sum_{k=0}^l\binom{L^2}{2k}\binom{L^2}{2l-2k}, \end{equation} which is "easy" to compute and has solution \begin{equation} deg(2l)=\frac{1}{2}\left((-1)^l\binom{n}{l}+\binom{2n}{2l}\right). \end{equation} However, I fail to find any expression of the sum for $2l>L^2$. Any indication on how can I proceed? Thanks a lot.