Invert transform of a generating function

Question

A055887's generating function is given as a transform of another generating function $\frac{1}{1-P(x)}$. It's clear from the definition that this is $\sum_{n=0}^{\infty} \left [ \sum_{i+j+k+...=n} P(i)P(j)P(k)... \right ] x^n$ but I'm struggling to derive this.

I've seen $x$ in $\frac{1}{1-x} = 1+x+x^2+x^3$ be replaced $x$ with $a x$, $x^2$, $x^m$, $-x$ but I only proved these by working backwards. Starting with the result and showing $S - Sx = 1$ then concluding $S = \frac{1}{1-x}$.

This was called "invert transform" in A067687 but that term doesn't seem to be defined or commonly used outside of OEIS.

Answer 1

Hint: A proof of \begin{align*} \color{blue}{\left(1-A(z)\right)^{-1}=\sum_{n=0}^{\infty}A^n(z)} \end{align*} with constant term of $A(z)$ equal to zero is given here.