Half-iterate exp(z)-1: hypothese on growth of coefficients

[Gottfried]

Yeah Gotfried at first sight about the a_k terms I thought about Borel summation, and I was also interested into divergent sums
I once also stucked in the summing-up terms d_n=O(exp(exp(n))) which is very much faster growing than O(n!) and any O(n!^k), I studied O(exp(exp(n))) because of the infinity q-pochhammer function \((q;q)_\infty\) which has nice expansions but O(q^n^2) cannot sum and even analytically extended to \(norm(q)>1\).
(...)

 Hi Leo -
  I liked to be reminded of that function-examples q-pochammer -there has been a question in I think MO (perhaps MSE) where this was discussed... got many "reputations" for it if I remember correctly :-) .
But well, in general I don't really know what to say after reading your post, I should go into it next week again.
Perhaps I made myself not clear so far, I'll try again.    

I had two problems in the question:   

 - what is the growthrate of the coefficients     

 - if it is more than hypergeometric, then how to sum that divergent series.   

But after my guess/conjecture for the limiting function for the coefficients \( c_k \) being \( O( (k-3)!/(4 \pi) ^k) \)  (which is even smaller than that \( O(k!) \) which is requiring  Borel and thus we had no further question of summability of this), it is only open, how to show what is really its growth rate?        

I might -using some hours of cpu-time- find the next 1024 coefficients, but this would not remove the duty to find some analytical argument for some upper bound of coefficient \( c_k \) for any \( k \) given.
I could not yet find any argument (which surely must be derived from the computation scheme of the coefficients) for that growthrate so far...   

But thanks for taking some time for this (and plz excuse if I missed something in your post) ! -

Gottfried