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   $x^y/f(x)$, for some $y=0,1,2...$ that does not require the use of Faà di Bruno's formula, or does not require multiple iterations of some recursive scheme to get the x^n coefficients . 

for example:

https://www.wolframalpha.com/input/?i=%281%2F%28x%5E3-2x%2Bz%29%29++about+x%3D0

The 6th coefficient is $((32 - 12 z^2))/z^6$
 
So how does one derive such functions of z without having to use a cumbersome recursive scheme ?  

It is possible to derive such a closed-form expression for the coefficients of  $\frac{3x^2-2}{x^3-2x+z}$ 

$\displaystyle -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 generalizations?