binomial coefficients - Proof by induction of $sum_{k=2}^n (k-1)(k)binom{n}{k} = n(n-1)2^{n-2}$

I've been struggling with this sum for an while, pluging $n+1$ instead of $n$, knowing that $\binom{n+1}{k} = \binom{n}{k-1} + \binom{n}{k}$ and after some manipulation i've found this sum.
$$2\sum_{k=1}^{n} k^2\binom{n}{k}$$

I coudn't see any way I could get out of here and I don't know how to start this proof without the property of the sum of binomial coefficients.

Thanks in advance.