# How can the following “funny identity” be generalised?

$$\left(\sum\limits_{k=1}^n k\right)^2=\sum\limits_{k=1}^nk^3 .$$
1. First of all, I am interested in higher powers higher than $2$. So if we consider $$\left(\sum\limits_{k=1}^n k\right)^n$$ for $n>2$, are there always ways turn this expression into other series without a raising them to some power? Only raising the individual terms to some power/factorial/function in general?
2. Second of all, I was wondering if similar identities exist for $$\left(\sum\limits_{k=1}^n k^m\right)^n$$ for $m>1$ and $n>0$.