Prove that $\sum_{k=0}^n \binom{n}{k} k^2 = n(n+1) 2^{n-2}$ $$\sum_{k=0}^n \binom{n}{k} k^2 = n(n+1) 2^{n-2}$$ 

HINT: You know $$f(x):=\sum_{k=0}^n \binom{n}{k}x^k = (1+x)^n.$$
Compute $f'(x)$ and $f''(x)$ using both the sum and the power expressions. Asuitable linear combination fo $f(1)$, $f'(1)$, $f''(1)$ produces the sum in question.
