Explicite formula for Almansi decomposition

Question

In 1899 E. Almansi proved that any polyharmonic function of order $p$, i.e. in the kernel of the differential operator $\Delta^p$, defined on a shar shaped domain $\Omega\subset\mathbb{R}^n$, can be written uniquely as a sum of $p$ harmonic function, i.e. if $\Delta^p u=0$, then $\exists!$ $f_0,...,f_{p-1}$ such that $$ u(x)=f_0(x)+|x|^2f_1(x)+...+|x|^{2(p-1)}f_{p-1}(x)=\sum_{j=0}^{p-1}|x|^{2j}f_j(x), $$ with $\Delta f_j=0$, for $j=0,...,p-1$.
In that paper (Sull'integrazione dell'equazione differenziale $\Delta^{2n}=0$) there is no formula to compute the components. In the monograph "Polyharmonic functions" of Aronszajn et al. there is an inductive proof that states that if $g_0,...,g_{p-2}$ are the components for $p-1$, the components of the inductive step can be computed by $$ f_j(x)=\frac{1}{4j}\int_0^1\tau^{j-2-n/2}g_{j-1}(\tau x)d\tau. $$ Is there a formula that explicitely gives the components $f_j$ that permorm the decomposition, starting from $u$? At least in particular case such as $u$ is a polynomial function.
Thank you very much for your help.