Counting the number of solutions to certain infinite Diophantine equations

Let $r,v,n$ be positive integers. This paper investigate the number of solutions $s_{r,v}(n)$ of the following infinite Diophantine equations $$n=1^r\cdot \mathrm{\mid }{k}_1{\mathrm{\mid }}^v+2^r\cdot \mathrm{\mid }{k}_2{\mathrm{\mid }}^v+3^r\cdot \mathrm{\mid }{k}_3{\mathrm{\mid }}^v+\dots ,$$ for $\mathbf{k}=(k_1,k_2,k_3,\dots )\in {\mathbb{Z}}^{\mathrm{\infty }}$. For each $(r,v)\in \mathbb{N}\times \{1,2\}$, a generating function and some asymptotic formulas of $s_{r,v}(n)$ are established.