How to get the two highest order terms in $$ \sum_{j=0}^n j^k \left(\frac{n^2}{j(n-2)}\right)^j, $$ where $k\in \mathbb{N}$, leaving the rest in big Oh of something.
I think this sum equals $$ n^k \left( \frac{n^2}{n-2}\right)^n + \mathcal{O}\left(n^{k-1} \left( \frac{n^2}{n-2}\right)^n\right),$$ but I don't know how to show it, and I need the first two terms.