$$C_{i,j}=\{\begin{array}{ccc}\frac{1}{2}& if& i=j=0\\ 0& if& j>i\\ -1^{i-j}S& & otherwise\end{array};S=\sum _{k=0}^i\left(\begin{array}{c}2i\\ 2k\end{array}\right)\left(\begin{array}{c}k\\ k+j-i\end{array}\right)$$