Binomial-coefficient equation problem

Question

Hey guys can't find to figure this one out

$\sum_k^n k^{2}\left(\begin{array}{c}n\\ k\end{array}\right) = n(n+1)2^{n-2} ,k\geq0$

Maybe one of you can help.

Answer 1

$$\sum k^{2}\binom{n}{k}=\sum k\left(k-1\right)\binom{n}{k}+\sum k\binom{n}{k}=n\left(n-1\right)\sum\binom{n-2}{k-2}+n\sum\binom{n-1}{k-1}=$$$$n\left(n-1\right)2^{n-2}+n2^{n-1}=n\left(n+1\right)2^{n-2}$$

Using the convention that $\binom{n}{k}=0$ if $k\notin\{0,\dots,n\}$ just let $k$ range over $\mathbb Z$.