I am trying to compute $$\frac{d^{n-1}}{dz^{n-1}}\left(\frac{z}{\ln\left(z!\right)}\right)^{n}$$ The problem arises when dealing with inversion formulae. My question is, can this expression be evaluated in closed form?
My attempts:
I wrote $\dfrac{1}{\ln(z!)}$ as $\dfrac{1}{1+\ln(\frac{z!}{e})}$ and tried using the binomial theorem. After some manipulations we get $$\sum_{k=0}^{n}\frac{1}{k!}\left(\prod_{i=0}^{k-1}\left(-n-i\right)\right)\frac{d^{n-1}}{dx^{n-1}}x^{n}\ln\left(\frac{x!}{e}\right)^{k}$$
But to me this did not seem to be any closer to a solution. How would you solve this problem?