When asked for a "funny identity", Andrey Rekalo answered the following:

$$\left(\sum\limits_{k=1}^n k\right)^2=\sum\limits_{k=1}^nk^3 .$$

Not only do I think it's funny, I also think it's very interesting. Therefore, I would like to know whether or not there are generalizations of this identity.

  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 $n>0$.

I guess the multinomial theorem could be used, but I'm not entirely sure how to create such nice identities akin to the one mentioned by Andrey Rekalo.


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