Does any one know how to calculate the following integration?


$$
\int\_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0.
$$


This post is related to my previous question [here](https://mathoverflow.net/questions/161068/has-anyone-seen-this-series) , based on which one can have a upper bound. Can any one give an exact solution for this integration?


Thanks a lot!
Anand



How to integrate an exponential function of an exponential function?
You can expand the integrand in powers of z. The coefficient of $z^k$ is
$$e^{-ky^2}\sum\_{j=1}^{k-1} {1\over j!(k-j)!}=e^{-ky^2}(2^k-2)/k!.$$
This yields the following series expression for the integral:
$$\sum\_{k=1}^\infty \sqrt{\pi\over k}(-2+2^k)z^k/k!.$$


