It is well known the Laplacian of a radial function $f(r)$ on $\mathbb{R}^n$ is itself a radial function given by $ (\Delta f)(r) = f''(r) + \frac{n-1}{r}f'(r)$. One can iterate this to compute $\Delta^2f(r), \Delta^3 f(r)$ and so on. If we write $$ \Delta^m f(r) = \sum_{l=0}^{2m-1} c_{m,l,n}\frac{f^{(2m-l)}(r)}{r^l} $$
then the constants must satisfy the recurrence $ c_{m,l,n} = c_{m-1,l,n} + (n - 2l + 1)c_{m-1,l-1,n} + (l-2)(l-n)c_{m-1,l-2,n}$. Does anyone know if these constants are known to have a closed form, or if this recurrence is likely to be solvable?