<s>[INST] Number of solutions via generating function How to find generating function for number of solutions
$x_1+x_2+x_3=n$ in set of positive integers such that $x_1 \ge x_2 \ge x_3$ and $x_1<x_2+x_3$. 
 [/INST] 
We   are looking  for a generating function $G(z)$ where the coefficient of $z^n$ gives the wanted number of solutions. We  want to find a proper representation for
  \begin{align*}
G(z)=\sum_{{{x_1,x_2,x_3\geq 1}\atop{x_1\geq   x_2\geq    x_3}}\atop{x_1<x_2+x_3}}z^{x_1+x_2+x_3}\tag{1}
\end{align*}

In order to simplify $x_1\geq x_2\geq x_3\geq 1$  (and also $x_1,x_2,x_3\geq 1$) we write
\begin{align*}
x_3&=y_3+1\qquad\qquad\  \, y_3\geq 0\\
x_2&=y_2+x_3\qquad\qquad y_2\geq 0\tag{2}\\
x_1&=y_1+x_2\qquad\qquad y_1\geq 0
\end{align*}
We  obtain  from (2):
\begin{align*}
x_1&<x_2+x_3\\
y_1+y_2+y_3+1&<(y_2+y_3+1)+(y_3+1)\\
y_1&<y_3+1\\
\color{blue}{y_1}&\color{blue}{\leq y_3}\tag{3}\\
\\
x_1+x_2+x_3&=(y_1+y_2+y_3+1)+(y_2+y_3+1)+(y_3+1)\\
&\,\,\color{blue}{=y_1+2y_2+3y_3+3}\tag{4}
\end{align*}

With the help of (2) and (3) we can write $G(z)$ as
  \begin{align*}
G(z)=\sum_{{y_1,y_2,y_3\geq 0}\atop{y_1\leq y_3}}z^{y_1+2y_2+3y_3+3}\tag{5}
\end{align*}

In order to simplify $y_1\leq y_3$ in (5) we write
\begin{align*}
y_3&=t_3+y_1\qquad\quad t_3\geq 0\\
y_2&=t_2\qquad\qquad\quad\, t_2\geq 0\tag{6}\\
y_1&=t_1\qquad\qquad\quad\, t_1\geq 0
\end{align*}
and we obtain from (6)
\begin{align*}
y_1+2y_2+3y_3+3&=t_1+2t_2+3(t_3+t_1)+3\\
&\,\,\color{blue}{=4t_1+2t_2+3t_3+3}\tag{7}
\end{align*}

We obtain from (5) - (7) 
  \begin{align*}
\color{blue}{G(z)}&=\sum_{t_1,t_2,t_3\geq 0}z^{4t_1+2t_2+3t_3+3}\\
&=z^3\sum_{t_2\geq 0}z^{2t_2}\sum_{t_3\geq 0}z^{3t_3}\sum_{t_1\geq 0}z^{4t_1}\\
&\,\,\color{blue}{=\frac{z^3}{(1-z^2)(1-z^3)(1-z^4)}}
\end{align*}

</s>