Let $p^{(s)}(n)$ be the number of ways of writing the positive integer $n$ as a sum of perfect $s$-powers, where the order does not matter. For example, $p^{(2)}(9) = 4$ since $$9 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2$$ $$9 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2\phantom{1^2 +\;\,+ 1^2 + 1^2}$$ $$9 = 1^2 + 2^2 + 2^2\phantom{ + 1^2 + 1^2 + 1^2 + 2^2\;\,+ 1^2 + 1^2}$$ $$9 = 3^2 \phantom{+ 2^2 + 2^2 + 1^2 + 1^2 + 1^2 + 2^2\;\,+ 1^2 + 1^2}$$ and there are no other ways of writing $9$ as sum of squares.

It is known that
$$\log p^{(s)}(n) \sim (s+1)\left(\frac1{2}\Gamma\!\left(1+\frac1{s}\right)\zeta\!\left(1+\frac1{s}\right)\right)^{s/(s+1)} n^{1/(s+1)},$$
as $n \to +\infty$ (See *Hardy and Littlewood, Asymptotic formulæ in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVII, 1918, 75-115*).

My question is: If $p_k^{(s)}(n)$ is the number of ways of writing the positive integer $n$ as a sum of *exactly $k$* perfect $s$-powers, is there an asymptotic formula for $\log p_k^{(s)}(n)$ holding in a reasonable range of $n,k \to \infty$? I am particularly interested in the case of squares $s = 2$.

Thank you in advance for any suggestion.

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