Generating Functions To Deal With I've been working on producing a closed-form generating function for the
coefficients $a_n = \binom{n}{2}.$ I was wondering what might be a good
procedure to start on this. I get that
$$\sum_{n=0}^{\infty} x^n = x^0 + x^1 + x^2 + ... = \frac{1}{1-x}$$
by the convergence of geometric series. However, I'm wondering what might be
the proper steps to get LHS to look like $\sum_{n=0}^{\infty} \binom{n}{2}x^n.$
Unfortunately, I have a general lack of familiarity with discrete math, so any
help would be greatly appreciated.

Differentiate term-by-term twice:
$$
\begin{aligned}
\frac{d}{dx} \sum_{n=0}^\infty x^n
&= \sum_{n=1}^\infty n x^{n-1} \\
\frac{d^2}{dx^2} \sum_{n=0}^\infty x^n
&= \sum_{n=2}^\infty n(n-1) x^{n-2}.
\end{aligned}
$$
Next, since $a_n = \binom{n}{2} = \frac{n(n-1)}{2}$ and $a_0 = a_1 = 0$, we have
$$
\begin{aligned}
\sum_{n=0}^\infty a_n x^n
&= a_0 + a_1 x + \sum_{n=2}^\infty \frac{n (n - 1)}{2} x^n \\
&= \frac{x^2}{2} \sum_{n=2}^\infty n (n-1) x^{n-2} \\
&= \frac{x^2}{2} \frac{d^2}{dx^2} \sum_{n=0}^\infty x^n \\
&= \frac{x^2}{2} \frac{d^2}{dx^2} \frac{1}{1-x} \\
&= -\frac{x^2}{\left(1 - x\right)^3}.
\end{aligned}
$$
