Consider a cubic of the form $f(x)=x^3-2x+z$.

Is it possible to derive a power series of coefficients for the function $x^k/f(x)$, for some $k=0,1,2\ldots$ without using Faà di Bruno's formula, or multiple iterations of some recursive scheme to get the $x^n$ coefficients?

For example for this series, the 6th coefficient is $((32 - 12 z^2))/z^6$

So how does one derive such functions of $z$ directly without having to use a cumbersome recursive iteration scheme?

It is possible to derive such a closed-form expression for the coefficients of the function $$ \frac{3x^2-2}{x^3-2x+z} + -z^{-m}2^{m+1} + \sum_{n=0} \frac{z^{2-m}z^{2n} (1+m)}{2^{3n-m+2}(n+1)} \binom{3n+1-m}{n} $$ for $3n+1-m<0$, so what about other rational functions?