Series expansion of the power Reciprocal gamma function

Question

Based on the post Non-recursive closed-form of the coefficients of Taylor series of the reciprocal gamma function, is it possible to obtain some Taylor expansion series for $$ f(z)=\frac{1}{[\Gamma(z)]^n}, $$ around zero? Here $n$ is some positive integer and $z>0$.

Answer 1

$$f_n(z)=\frac{1}{\big(\Gamma (z)\big)^n}=\frac{z^n}{\big(z\,\Gamma (z)\big)^n}$$ By Taylor $$\frac{1}{z\,\Gamma (z)}=1+\sum_{k=1}^p a_k\, z^k + O(z^{p+1})$$ where the $a_k$'s are known.

$$f_n(z)=z^n \Bigg(1+\sum_{k=1}^p a_k\, z^k + O(z^{p+1}) \Bigg)^n=z^n\Bigg(1+n\sum_{k=1}^p \frac{b_k}{k!}\, z^k \Bigg)+ O(z^{p+1})$$

Now, using the binomial expansion,we have the $b_k$'s $$b_1=a_1$$ $$b_2= (n-1)a_1^2+2a_2$$ $$b_3=(n-1)(n-2)a_1^3+6(n-1)a_1a_2+6a_3$$ The next one is too long to type here but this is the idea.