i found the binomial expansion when z=0 is $\displaystyle \frac{1}{z}-2-3z-4z^2$
That is not correct. Consider $\displaystyle \frac{1}{1-z}=\sum_{n=0}^{+\infty}z^n$ and take derivatives in both sides. Around $\displaystyle z=1$ , use the substitution $\displaystyle u=z-1$.