This is inspired by an old Putnam problem from 2005, and a solution given by Professor Greg Martin (a Professor of Mathematics at the University of British Columbia, also a user on MO). The question is

Question (Putnam 2005): For non-negative integers $m,n$, let $f(m,n)$ denote the number of $n$-tuples $(x_1, \cdots, x_n)$ of integers such that $|x_1| + \cdots + |x_n| \leq m$. Show that $f(m,n) = f(n,m)$.

Greg's proof essentially boiled down to showing that the generating function $$\displaystyle G(x,y) = \sum_{m,n \geq 0} f(m,n) x^m y^n$$ is symmetric in $x,y$.

Update: Since it seems that MAA took down the Putnam directory and the old solutions are no longer easily accessible, I shall give the proof here. The credit goes entirely to Professor Martin.

We write

$$\displaystyle G(x,y) = \sum_{n \geq 0} \sum_{m \geq 0} f(m,n)x^m y^n$$ $$\displaystyle = \sum_{n \geq 0} \sum_{m \geq 0} x^m y^n \sum_{\substack{k_1, \cdots, k_n \in \mathbb{Z} \\ |k_1| + \cdots + |k_n| \leq m}} 1$$ $$\displaystyle = \sum_{n \geq 0} y^n \sum_{k_1, \cdots, k_n \in \mathbb{Z}} \sum_{m \geq |k_1| + \cdots + |k_n|} x^m$$ $$\displaystyle = \sum_{n \geq 0} y^n \sum_{k_1, \cdots, k_n \in \mathbb{Z}} \frac{x^{|k_1| + \cdots + |k_n|}}{1 - x}$$ $$\displaystyle = \frac{1}{1-x}\sum_{n \geq 0} y^n \left(\sum_{k \in \mathbb{Z}} x^{|k|}\right)^n$$ $$\displaystyle = \frac{1}{1-x} \sum_{n \geq 0} y^n \left(\frac{1+x}{1-x}\right)^n$$ $$\displaystyle = \frac{1}{1-x} \frac{1}{1 - y(1+x)/(1-x)}$$ $$\displaystyle = \frac{1}{1-x-y-xy}.$$

This seemed a fascinating approach to me back then (2005 was the first time I wrote the Putnam, and Greg was our Putnam coach at UBC), and even more so today when I looked back at it given that some of my work involves some clever generating function arguments (based on the answers given to me by Richard Stanley on a question I posted here). So the question I pose is:

Are there any other interesting quantities $f(n_1, \cdots, n_k)$ involving parameters $n_1, \cdots, n_k$ with $k \geq 2$ say that are symmetric in the parameters, and the proof comes from showing that the generating function

$$\displaystyle \sum_{n_1, \cdots, n_k} f(n_1, \cdots, n_k)x_1^{n_1} \cdots x_k^{n_k}$$ is symmetric?