I'm trying to show that
$$\dfrac{1}{2^n \ n! \ \sqrt{\pi}}\int_{-\infty}^{\infty}x^2 e^{-x^2}H_n(x)^2 = n+\dfrac{1}{2}.$$
It's the expectation value of $x^2$ in the quantum harmonic oscillator.
Does anyone have a hint here? Some propertie of the Hermitian polynomial, maybe, that is useful.