If we want to find the integer solutions of $x_1+x_2+x_3=n$ such that $x_1$ is positive using a generating function.
We would first make the parentheses for each $x_i$, for $x_{2,3}$ it would be the standard: $(1+x+x^2...)$ but how can we represent the positive part of $x_1$?
You need to rule out the possibility $x_1=0$; that’s represented by an exponent of $0$, so you need to leave off the $x^0=1$ term. The factor corresponding to $x_1$ should therefore be
$$x+x^2+x^3+\ldots=x(1+x+x^2+\ldots)\;.$$