I observed that 
$f^{(n)}(x)=
\begin{cases}
e^{-x^2} & \text{if $n=0$}\\
-2xe^{-x^2} & \text{if $n=1$}\\
f^{(n-1)}(x)-f^{(n-2)}(x) & \text{otherwise.}
\end{cases}$  

How to get the closed form?  

**Edit:** This recurrence does not hold.