$0^2{{n}\choose{0}} + 1^2{{n}\choose{1}} + 2^2 {{n}\choose{2}} + ... +n^2{{n}\choose{n}} = n (1+n)* 2^{n-2}, n \in \mathbb{N} \setminus \{0\}$ proof.
Induction: $\sum_{k=0}^n \binom nk k^2 = n(1+n)2^{n-2}$