After playing a bit with some probabilities, I ended up with this:
$P(X=n) = \frac{1}{\Gamma(s)} \sum_{i=1}^n (-1)^{k-1} \binom{n}{k} \gamma(s, \frac{k}{\theta})$
Where $\gamma(s,x) = \int_{0}^{x} t^{s-1}e^{-t} dt$ (incomplete gamma function). I have the feeling that this converges for $n \to \inf$ as I plotted it for $k=0.1, \theta=21$ up to $n=30$ but I could not figure out a method yet.
In short, how to solve $\lim \limits_{n \to \inf} \sum_{i=1}^n (-1)^{k-1} \binom{n}{k} \gamma(s, \frac{k}{\theta})$ ?