Radial moments of an isotropic Gaussian vector ($\mathbb{E}|\mathbf{X}{|}^{2n}=2^n\Gamma (n+d/2)v^n/\Gamma (d/2)$)

For $d$ independent centered normal random variables of common variance $v$, $|\mathbf{X}{|}^2/v$ has a chi-squared distribution with $d$ degrees of freedom. Its density gives the displayed moments by a Gamma integral. In three dimensions, $\mathbb{E}|\mathbf{X}{|}^{2n}=(2n+1)!!v^n$. Radial odd moments need not vanish, although any odd Cartesian product does.