How to write the code for polynomial $\sum^{\infty}_{j=0} \sum_{x_1+x_2+x_3=j}$ [closed]

Question

Hellow, how to write the code of $\sum^{\infty}_{j=0} \sum_{x_1+x_2+x_3=j}$. Here is the complete equation. $$\sum^{\infty}_{j=0} \gamma^j \{ \sum_{x_1+x_2+x_3=j} \frac{(\frac{i}{2})^{x_2}(-\frac{3i}{2})^{x_3}}{x_1!x_2!x_3!}[t-(2x_1+3x_2+x_3)\eta]^j e^{-(\gamma+iJ)[t-(2x_1+3x_2+x_3)\eta]} \Theta[t-(2x_1+3x_2+x_3)\eta]\}$$

Thanks.

Answer 1

Clear["Global`*"]

f[x1_, x2_, x3_] :=
  (I/2)^x2 (-3 I/2)^x3/(x1! x2! x3!)*(t - (2 x1 + 3 x2 + x3) η)^j*
   Exp[-(γ + I*J) (t - (2 x1 + 3 x2 + x3) η)]*
   Θ[t - (2 x1 + 3 x2 + x3) η];

expr = Sum[γ^j*Sum[f[x1, x2, j - x1 - x2],
    {x1, 0, j}, {x2, 0, j - x1}], {j, 0, Infinity}]