Let $G \in \mathsf{N}(0, \sigma^2)$ be a D210: Gaussian random real number.
Let $n \in \mathbb{N}$ be a D996: Natural number.
Let $n \in \mathbb{N}$ be a D996: Natural number.
Then
\begin{equation}
\mathbb{E}(G^n)
=
\begin{cases}
\sigma^n (n - 1) !!, \quad & n \in 2 \mathbb{N} \\
0, \quad & n \in 2 \mathbb{N} + 1
\end{cases}
\end{equation}