Example 5 Let $n \in \mathbf{Z}_{+}$, from the origin $O$ along the coordinate grid $(y=k, x=h, k, h \in \mathbf{Z})$ to $P(n, n)$, what is the sum of the number of intersections of all the shortest paths with the line $y=x$ (including $(0,0)$ and $(n, n)$)? ${ }^{[2]}$
Let $A_{k}(k=0,1, \cdots, n)$ be the number of these broken lines passing through the point $(k, k)$. Then, the number of broken lines from the origin to the point $(k, k)$ is $\mathrm{C}_{2 k}^{k}$, and the number of broken lines from the point $(k, k)$ to $(n, n)$ is $\mathrm{C}_{2 n-2 k}^{n-k}$.
Therefore, $A_{k}=\mathrm{C}_{2 k}^{k} \mathrm{C}_{2 n-2 k}^{n-k}$.
For any shortest broken line from the point $(0,0)$ to $(n, n)$, let it intersect the line $y=x$ at $m$ points
$$
\left(x_{1}, x_{1}\right),\left(x_{2}, x_{2}\right), \cdots,\left(x_{m}, x_{m}\right) \text {. }
$$

Then this broken line is counted $m$ times in $A_{x_{1}}, A_{x_{2}}, \cdots, A_{x_{m}}$. Thus, the total number of intersections of all different broken lines with $y=x$ is exactly $\sum_{k=0}^{n} \mathrm{C}_{2 k}^{k} \mathrm{C}_{2 n-2 k}^{n-k}$, denoted as $S$.
Next, we calculate the value of $S$.
Construct the generating function $f(x)=\frac{1}{\sqrt{1-4 x}}$.
By the Taylor series expansion, we have
$$
-\frac{1}{\sqrt{1-4 x}}=\sum_{k=0}^{+\infty} \mathrm{C}_{2 k}^{k} x^{k} \text {. }
$$

On one hand, $S$ is the coefficient of $x^{n}$ in $\left(\sum_{k=0}^{+\infty} \mathrm{C}_{2 k}^{k} x^{k}\right)\left(\sum_{k=0}^{+\infty} \mathrm{C}_{2 k}^{k} x^{k}\right)$.
On the other hand,
$$
\begin{array}{l}
\left(\sum_{k=0}^{+\infty} \mathrm{C}_{2 k}^{k} x^{k}\right)\left(\sum_{k=0}^{+\infty} \mathrm{C}_{2 k}^{k} x^{k}\right)=\left(\frac{1}{\sqrt{1-4 x}}\right)^{2} \\
=\frac{1}{1-4 x}=\sum_{k=0}^{+\infty} 4^{k} x^{k} .
\end{array}
$$

Thus, the coefficient of $x^{n}$ is $4^{n}$.
Therefore, $\sum_{k=0}^{n} \mathrm{C}_{2 k}^{k} \mathrm{C}_{2 n-2 k}^{n-k}=4^{n}$.
Hence, the total number of intersections of all shortest paths with the line $y=x$ is $4^{n}$.