During a discussion, I was asking to find a function $f$ which is holomorphic on $$\mathbb{C}\setminus \{(2n)! : n \in \mathbb{N}\}.$$ Naturally, I wanted to set $$f(z) = \prod_{n=0}^{+\infty} \frac{1}{z-(2n)!}$$ but this product does not converge. (Remember that $\prod_n z_n$ is convergent if and only if $\sum_n |z_n-1|$ is convergent.) So finally, I set $$f(z) = \prod_{n=0}^{+\infty} \frac{(2n)!}{(2n)!-z}$$ which is holomorphic on the required domain.
First, I'd like to know if someone has a better idea to find a holomorphic function on $\mathbb{C}\setminus \{(2n)! : n \in \mathbb{N}\}.$ (If possible with "usual" functions).
I would also like to know if one can get interesting informations about the function $f$ I sat. For example, can we obtain its Taylor expansion on $D(0,1)$ ?