I've come across an interesting matrix identity in my work. I'll define the NxN matrix as [tex] S_{ij} = 2^{-(2N - i - j + 1)} \frac{(2N - i - j)!}{(N-i)!(N-j)!}.[/tex] I find numerically that [tex] \sum_{i,j=1}^N S^{-1}_{ij} = 2N, [/tex] (the sum is over the elements of the matrix inverse). In fact, I expected to get 2N based on the problem I'm studying, but I don't know what this complicated matrix expression is doing or why it equals 2N. Does any of this look familiar to anyone here?

Thanks for your help!

P.S. If this is in the wrong subforum, please move it.