Evaluating the sum $\sum\limits_k \ k\binom{n}{k}^2$ using generating functions

Question

I have to evaluate this expression $\sum\limits_k \ k\binom{n}{k}^2$ using generating function. Could you help me please? Also with some hints.

Answer 1

I suggest that you interpret the sum as the convolution of the generating function with coefficents $k\binom{n}{k}$ and $\binom{n}{k}$.