I want to calculate the inverse Fourier transform of $\exp\left[-e^{-k^2}\right]$. One of the ways I can imagine is to expand the exponential in a series $$ \exp\left[-e^{-k^2}\right] = \sum_{n=0}^\infty \frac{(-1)^n}{n!}e^{-nk^2} $$
The first of the series gives a inverse Fourier Transform of $\delta(x)$, the rest is a sum over Gaussian. How can I evaluate this series to get a solution? Can I predict what will be the leading behaviour in the case of $x\rightarrow 0$?