Radius of convergence after powering coefficients
= Radius of convergence after powering coefficients
If $L=\limsup a_n^{1/n}$ for positive coefficients, then the coefficients $a_n^2$ have root limsup $L^2$. Coefficients $a_n^{a_n}$ instead have root terms
$$
a_n^{a_n/n}=\exp\left(\frac{a_n\log a_n}{n}\right),
$$
so their radius depends on the growth of $a_n\log a_n$ relative to $n$.