I'm trying to find a closed form expression (that doesn't involve an indefinite summation) for the following combinatorial sum :
$$\sum\limits\_{k=0}^{n}\left[\frac{k4^k}{n+k}{n+k\choose n-k}\frac{\displaystyle{n\choose k}}{\displaystyle{2n\choose 2k}}\right]$$
I don't know wether or not there is one in the first place.
The context in which this sum arised isn't relevant and won't provide any help.

I don't have any idea as of where to even start. Any suggestion ?