\[\int_{-\infty}^{\infty}H_{\ell}\left(x\right)H_{m}\left(x\right)H_{n}\left(x%
\right){\mathrm{e}}^{-x^{2}}\,\mathrm{d}x=\frac{2^{\frac{1}{2}(\ell+m+n)}\ell%
\,!\,m\,!\,n\,!\,\sqrt{\pi}}{(\tfrac{1}{2}\ell+\tfrac{1}{2}m-\tfrac{1}{2}n)\,!%
\,(\tfrac{1}{2}m+\tfrac{1}{2}n-\tfrac{1}{2}\ell\,)\,!\,(\tfrac{1}{2}n+\tfrac{1%
}{2}\ell-\tfrac{1}{2}m\,)\,!},\]