\[\int_{0}^{\infty}{\mathrm{e}}^{-x^{2}}\mathit{He}_{2n+1}\left(2x\right)\sin%
\left(xy\right)\,\mathrm{d}x=(-1)^{n}\tfrac{1}{2}\sqrt{\pi}{\mathrm{e}}^{-%
\frac{1}{4}y^{2}}\mathit{He}_{2n+1}\left(y\right).\]