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[itex]\int dx Hm(x)Hm(x)e^{-2x^2}[/itex]

Where Hm(x) is the hermite polynomial of m-th order. I know the hermite polynomials are a orthogonal set under the distribution exp(-x^2) but this is not the case here.

Using Hm(x)=[itex](-1)^m e^{x^2}[/itex][itex]\frac{d^m}{dx^m}e^{-x^2}[/itex] I was able to rewrite the integral as

[itex]\int \left(\frac{d^m}{dx^m}e^{-x^2}\right)^2 dx[/itex]

I have calculated this integral for m=0,1,2,3,4,5 with mathematica and the result seems to be [itex](2m-1)!!\sqrt{\frac{\pi}{2}}[/itex] but I need to prove it formally. Can you help me?

Thank you.