Derivatives of exponential and Hermite polynomials

Question

I interested in computing a closed expression in $m,n\in \mathbb N_0$ for the function \begin{equation*} f(n,m) =\left. \left(\frac{\mathrm d}{\mathrm d x}\right)^m \left[e^{-x^2/2} H_n(x)\right]\right|_{x=0} \end{equation*} Vi $H_n(x)$ denotes the $n-th$ Hermite polynomial, defined by \begin{equation*} H_n(x) = (-1)^n e^{x^2}\left(\frac{\mathrm d }{\mathrm d x}\right)^n e^{-x^2} \end{equation*} I tried to use the recursion relation for subsequent hermite functions and the closed form for the Hermite numbers, but I could not get to any satisfactory result. How shall I proceed?