Generalisation of the identity $\sum\limits_{k=1}^n {k^3} = \bigg(\sum\limits_{k=1}^n k\bigg)^2$

Are there any generalisations of the identity $\sum\limits_{k=1}^n {k^3} = \bigg(\sum\limits_{k=1}^n k\bigg)^2$ ?

For example can $\sum {k^m} = \left(\sum k\right)^n$ be valid for anything other than $m=3 , n=2$ ?

If not, is there a deeper reason for this identity to be true only for the case $m=3 , n=2$?

We can't have a relationship of the form $$\forall n\in\mathbb N^*, \sum_{k=1}^nk^a=\left(\sum_{k=1}^nk^b\right)^c$$ for $a,b,c\in\mathbb N$, except in the case $c=1$ and $a=b$ or $a=3$, $b=1$ and $c=2$. Indeed, we can write $$\sum_{k=1}^nk^a =n^{a+1}\frac 1n\sum_{k=1}^n\left(\dfrac kn\right)^a$$ hence $$\sum_{k=1}^nk^a\;\overset{\scriptsize +\infty}{\large \sim}\;n^{a+1}\int_0^1t^adt=\dfrac{n^{a+1}}{a+1}$$ and if we have the initial equality we should have $a+1 =(b+1)c$ and $a+1=(b+1)^c$. In particular, $(b+1)^{c-1}=c$. If $c>1$, then $c= (b+1)^{c-1}\geq 2^{c-1}\geq c$, and we should have $c=2$ and $b=1$, therefore $a=3$.

The Faulhaber polynomials are expressions of sums of odd powers as a polynomial of triangular numbers $T_n=\frac{n(n+1)}{2}$. Nicomachus's theorem, $\sum\limits_{k\leq n} k^3=T_n^2$, is a particular special case.

Other examples include

$$\begin{align*}\sum\limits_{k\leq n} k^5&=\frac{4T_n^3-T_n^2}{3}\\\sum\limits_{k\leq n} k^7&=\frac{6T_n^4-4T_n^3+T_n^2}{3}\end{align*}$$