There is a relationship here, in the case of Gauß-Hermite Integration, where the weight function is exactly e^(-x^2) the weights have to add up sqrt(pi), becaus

There is a relationship here, in the case of Gauß-Hermite Integration, where the weight function is exactly e^(-x^2) the weights have to add up sqrt(pi), because the integral is exact for the constant 1 polynomial.