What is the asymptotic expansion of the following expression as $\theta \to 0^+$? Alternatively, what is the asymptotic expansion of the MGF $M(\theta)$ of the squared Poisson(1) as $\theta \to 0^+$?
$$\sum_{n=0}^\infty \frac{\theta^n}{n!} \left\{\sum_{k=0}^\infty \frac{k^{2n}}{k!}\right\}$$
I was able to use properties of the Stirling numbers of the second kind to write it as $$e \sum_{n=0}^\infty \frac{\theta^n}{n!} \left\{\sum_{k=0}^{2n} S(2n, k)\right\}$$
but haven't been able to figure out how to proceed. Has this been studied before? Thank you.