If $i\geq 1$ is an integer, we have the following integer valued functions (for any integer $n\geq 0$):
\begin{align} p_i(n)&=\textrm{the number of }i\textrm{-dimensional partitions of }n,\notag\\ \sigma_i(n)&=\sum_{d|n}d^i. \end{align}
If $i=1$, $p_1(n)=p(n)$ is the number of "usual" partitions of $n$ (corresponding to decompositions of $n$ as a sum of natural numbers; these also correspond to Young tableaux). If $i=2$, we get the so-called plane partitions. These generalize for higher $i$'s.
Now, for $i=1,2$ there is a formula relating $p_i$ and $\sigma_i$, namely $$np_i(n)=\sum_{k=1}^n\sigma_i(k)p_i(n-k).$$
My question is purely a reference request: is it known whether there is an analogous formula for $i>2$?
I would not be surprised if this was not known, because few is known about higher dimensional partitions. Up to the plane ones, we know the generating function (but no closed formula for $p_i(n)$, for any $i$). For $i>2$, we do not even know the generating function.
Thank you!
EDIT. References for the recursive formulas: $i=1$ (formula 13) and $i=2$ (formula 6). (The case $i=1$ is somehow more classical, and the result can be found also in textbooks.)