2
$\begingroup$

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 :).

Thanks!

$\endgroup$
  • $\begingroup$ 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
1
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.