Hermite Polynomials proof using generating functions

Question

I need to prove: $ \int H_n(x)H_m(x)xe^{-x^2}dx = \sqrt{\frac{n}{2}}\delta_{m,n-1}+\sqrt{\frac{n+1}{2}}\delta_{m,n+1} $ , but using the generating functions of the Hermite Polynomials and Gaussian integrals. So far: F(x,t) = $\sum_{n=0}^{\infty} \frac{H_n(x) t^n}{n!} = e^{x^2-(t-x)^2} $
$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{s^nt^m}{n!m!} \int H_n(x)H_m(x)xe^{-x^2}dx = \int e^{x^2-(t-x)^2}e^{x^2-(s-x)^2}xe^{-x^2}dx$
= $\int e^{-s^2+2sx-t^2+2tx-x^2}xdx$
= $\int e^{-(s+t-x)^2}e^{2st}xdx$
=$(s+t)e^{2st} \sqrt {\pi}$ = $\sqrt {\pi}\sum_{n=0}^{\infty}\frac{2^ns^{n+1}t^{n+1}}{n!}$
Now I know I need to compare coefficients of terms with equal powers of s and t but i cant get the right answer out