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Preliminaries I encountered the following triangle of positive integers: $c_{n,k}$ $n=1$ $n=2$ $n=3$ $n=4$ $n=5$ $n=6$ $n=7$ $n=8$ $k=0$ $1$ $3$ $15$ $105$ $315$ $3465$ $45045$ $45045$ $k=1$ $5$ $...

I want to know what is following sum coefficient looks like. We sum over all integers $p$, $q$ also we put the condition that $q$ is even. Also, it should depend on the parity of $k$ $$\bar{S}(k)=\...