Symbolic Partial derivative

Hello, I am trying to work out some partial derivative parameterised. Let's define p=1...9, i=1...4 and x={x_1,...,x_i} the equation to calculate is:

H_p=((-1)^p)*exp(0.5*x^T*x)*D^p/(Dx_1Dx_2..D_x^i[exp(-0.5*x^T*x)]

Which for p=1 and i=2 (Dx_1)H_1=x_1 and (Dx_2)H_1=x_2

Which for p=2 and i=2 (D^2x_1)H_2=(x_1^2-1) and (D^2x_2)H_2=(x_2^2-1) and (Dx_1Dx_2)H_2=(x_1*x_2)

Which for p=3 and i=2 (D^3x_1)H_3=(x_1^3-3*x_1) and (D^2x_1Dx_2)H_3=(x_1^2*x_2-x_2) and (D^2x_2Dx_1)H_3=(x_1*x_2^2-x_1) and (D^3x_2)H_3=(x_2^3-3*x_2)

etc....which come from the hermite polynomials of order p.

Do you have any suggestion how to calculate those symbolic derivatives and make th code flexible so that it will calculate them at different i and P or it is better to hard code the derivatives in your opinion?

Thanks