Finding a value using generating functions

Question

In a certain exercise I have been asked to find the value of $\sum_{k\geq0}{n+k\choose2k}\cdot2^{n-k}$. I think that the approach written in the following paragraph seems one of the possible "paths" to reach the value, but I get struck at a certain point:
Define $f(x) = \sum_{k\geq0} {n+k\choose2k}\cdot x^{(n+k)-2k} = \sum_{k=0}^{\infty}{n+k\choose n-k}\cdot x^{n-k}$. I know I have to find the generating function and just substitute $x=2, f(2)$, but I cannot go further this point. I would be grateful if somebody could give me just a hint to continue.

Answer 1

Consider the bivariate generating function $F(x, y) = \sum_{n,k\ge 0}\binom{n+k}{2k} x^n y^k$. Then, you are looking for $2^n\cdot [x^n]F(x,\frac 12)$ (here, $[x^n]$ denotes the coefficient of $x^n$ in this series).

Try rewriting this generating function as $$F(x,y) = \sum_{k\ge 0}y^k \left(\sum_{n\ge 0}\binom{n+k}{2k}x^n\right)$$ and computing a closed form for the inner generating function first (it should have a fairly simple one).