Could someone help me manipulate this sum? I need to be able to extract the coefficient of $x^{n-1}$ in the following: $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\binom{n-1}{i}\binom{n+j-1}{j}x^{i+2j}$. Ideally, this would also be something that has a closed form, since extracting this coefficient as a sum wouldn't help the other calculations that I need to do. This manipulation is actually part of a larger problem, but this is the part that I am currently stuck on. Help is appreciated :).


  • $\begingroup$ Does anyone know an identity for this? Or could someone show me how to manipulate this sum> $\endgroup$ – Nizbel99 Nov 9 '12 at 1:56

By inspection, $$\sum_{j=0}^{\lfloor (n-1)/2\rfloor} \binom{n+j-1}{j}\binom{n-1}{n-1-2j}.$$ I'm not aware of any closed form for such a sum.

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