Is there a known closed form for $\sum_{k=0}^n k\binom{n}{k}^2$?

Question

I know that there are closed forms for $\sum_{k=0}^n \binom{n}{k}^2 = \binom{2n}{n}$ and a similar one for $\sum_{k=0}^n k \binom{n}{k}$.

Is there a known closed form for $\sum_{k=0}^n k\binom{n}{k}^2$?

Answer 1

$$k\left(\binom nk\right)^2=\binom nk\cdot\binom{n-1}{k-1}$$

Now $$(1+x)^{2n-1}=(x+1)^n(1+x)^{n-1}$$

$$\sum_{r=0}^{2n-1}\binom{2n-1}rx^{2n+1-r}=\sum_{s=0}^m\binom ns x^{n-s}\cdot\sum_{t=0}^{n-1}\binom{n-1}tx^t$$

Compare the coefficients of $x^n$