Complex Tetration, to base exp(1/e)

[sheldonison]

Hi,

I was reading the article[1] and i can't reproduce it in mathematica.
I need some help, and very much need some code.

Edison
[1]https://arxiv.org/abs/1105.4735

Equation 18 is the key, which is the asymptotic ecalle formal power series Abel function for iterating \( z\mapsto\exp(z)-1 \) which is congruent to iterating \( \eta=\exp(1/e);\;\;\;y\mapsto\eta^y;\;\;\;z=\frac{y}{e}-1; \)
The asymptotic series for the Abel equation for iterating z is given by equation 18.  I have used this equation to also get the value of Tetration or superfunction for base \( \eta=\exp(1/e) \), by using a good initial estimate, and then Newton's method.  If you use pari-gp or are interested in downloading pari-gp, I can post the pari-gp code here, including the logic to generate the formal asymptotic series equation for equation 18, and the logic to generate the two superfunctions and their inverses.  
\( \alpha(z)=-\frac{2}{z}+\frac{1}{3}\log(\pm z)-\frac{1}{36}z+\frac{1}{540}z^2+\frac{1}{7776}z^3-\frac{71}{435456}z^4+... \)
If the the asymptotic series is properly truncated then the Abel function approximation can be superbly accurate.  
\( \alpha(z)\approx\alpha(\exp(z)-1)+1 \)
To get arbitrarily accurate results, we iterate \( z\mapsto\exp(z)-1 \) enough times or for the repellilng flower, we can iterate \( z\mapsto\log(z+1) \) enough times so that z is small and the asymptotic series works well.