Sum of series $\sum\limits_{k=0}^\infty \frac{a^{k^2}}{k!}$

Question

I am looking at this sum:

$\displaystyle\sum_{k=0}^\infty \frac{a^{(k^2)}}{k!}$ for some $a>0$.

Answer 1

If I get to choose $a$ I choose $1$ and the sum is $e$.

For $a \gt 1$ the sum diverges.

For $0 \lt a \lt 1$ it will converge quickly but I suspect has no easy answer. Alpha says that for $a=\frac 12$ the sum is about $0.531576$ but does not give a closed form.