Close form to polynomial times Gaussian sum $\displaystyle\sum_{m\in\mathbb{Z}} m^{2k} e^{-m^2}, \ k \in \mathbb{Z}_{+}$

Question

Is there a closed form to the sum:$\displaystyle\sum_{m\in\mathbb{Z}} m^{2k} e^{-m^2}, \ k \in \mathbb{Z}_{+}$?

Thanks!