## Generating function for a sequence [closed]

Tackling problems on combinatorics course, I faced such a sequence of binomial coefficients: ${n \choose 0} + {n + 1 \choose 2} + ... + {n + k \choose 2k} + ...$ for which I need to build a generating function. Could anyone give a hint how to solve this problem?

-
Not the right website for this (try math.stackexchange or artofproblemsolving). But wherever you post a homework problem, you should tell us about the work you already have done on it; otherwise it looks like you are trying to get a solution without having thought about it yourself. – darij grinberg Dec 27 2011 at 18:57
Initially, I have such a problem: find $\sum_{k=0}^{n+1}(−1)^{n−k}4^k{n+k+1 \choose 2k}$.I thought,if I found such $g_n(x) = \sum_{k=0}^{n}{n+k \choose 2k}x^k$, the answer would be $(-1)^ng_{n+1}(-4)$.I tried to factorize ${n + k \choose 2k}to{n+k \choose k}{n \choose k}/{2k \choose k}$.It is known that $(1-x)^{-n-1} = \sum_{k=0}^{\infty}{n + k \choose k}x^k, (1-4x)^{-1/2} = \sum_{k=0}^{\infty}{2k \choose k}x^k, (1+x)^n = \sum_{k=0}^n{n \choose k}x^k$. I have tried to combine these formulae, but nothing interesting has been found yet. – feather Dec 27 2011 at 19:31