Near-origin probability in a maximal-angular-momentum hydrogen state ($P(r<r_p)\sim \frac{2^{2l+3}(r_p/a{)}^{2l+3}}{(2l+3)!(l+1{)}^{2l+3}}$)

For a normalized maximal-angular-momentum hydrogen state, the probability inside a small radius $r_p$ is the integral of $|R{|}^2{r}^2$. Since $R$ behaves as $r^l$ near zero, it equals $2^{2l+3}(r_p/a{)}^{2l+3}/[(2l+3)!(l+1{)}^{2l+3}]$ times $1+O(r_p/a)$. Increasing angular momentum strongly suppresses penetration toward the Coulomb centre.