Lets denote $\mathcal N_{k,n}$ the function that return the number of ways to write a given number as sum of $k$ numbers to the $n$-th power.
For example : $\mathcal N_{3,2}(1)=3$ because \begin{align*} 1 &=1^2+0^2+0^2 \\ &=0^2+1^2+0^2 \\ &=0^2+0^2+1^2 \\&=(-1)^2+0^2+0^2 \\ &=0^2+(-1)^2+0^2 \\ &=0^2+0^2+(-1)^2. \end{align*}
We already now that
$$\forall n\in\mathbb N \quad \mathcal N_{4,2}=\sum_{d\mid n \text{ and } 4\nmid d}d.$$
Is there a more general formula for most $k$ and $n$ ?
Is there at least other formulas like that for some non trivial $k$ and $n$ ?
Yes, there are more formulas for squares, see here. Two more examples for the sum of squares are given by $$ \mathcal{N}_{6}(n)=16\sum_{d\mid n}\chi (\frac{n}{d})d^2-4\sum_{d\mid n}\chi(d) d^2, $$
with $\chi$ a character modulo $4$ and
$$ \mathcal{N}_{8}(n)=6\sum_{d\mid n}(-1)^{n+d}d^3. $$ For higher degree, there are also results, in connection with Waring's problem, starting with an integral for $\mathcal{N}_{k,n}$ by Hardy and Littlewood.