Tetration Power Series

Question

While reading through the Citizendium article on tetration, the first hyper-operation above exponentiation, I came across a power series approximate of tetration. The article said that it got the series coefficients from cauchy’s integral formula for the nth derivative of a function. There are a few questions I have. Firstly how did they get the power series coefficients given only the recursive definition for tetration? Secondly I’m familiar with the fact that you can determine the nth iterate of the exponential function through Carleman matrices and then plug 1 into the function and obtain tet(n) for all n, will this function converge to the same function represented through the power series expansion of tetration?