What is $g$ for which this is true!

Question

For what $g\equiv g(r,q)$ such that following is true: $$ \int_0^\infty r^{2n-1}g(r,q)dr=\frac{\Gamma(qn+1)}{n^2}. $$ The function $g$ should be independent of $n$. It can only have $r$ and $q$.