<s>[INST] Find a generating function for which $A(n)={n \choose 2}$ In the book I'm using, $A(x)$ denotes the formal power series (generating function), $A(x) = \sum a_ix^i$.
I'm really stuck on this problem. Thanks for any help.
My attempt after the given hint:
$$\begin{align}
A(x)&=\sum_{n\geq 0} \binom{n}{2}x^n\\
&=\sum_{n\geq 0} \frac{x^2}{2}\frac{d^2}{dx^2}(x^{n})\\
&=\frac{x^2}{2} \frac{d^2}{dx^2}\sum_{n\geq 0} x^{n}\\
&=\frac{x^2}{2} \frac{d^2}{dx^2}\frac{1}{1-x}\\
&=\frac{x^2}{2} \frac{2}{(1-x)^3}
\end{align}$$
Sorry, I'm new to $\rm \LaTeX$.
 [/INST] Do you mean $a_n=\binom{n}{2}$ and $A(x) =\sum_{n\geq 0}a_nx^n= \sum_{n\geq 0} \binom{n}{2}x^n$?
Then note that 
$$\binom{n}{2}x^n=\frac{x^2}{2}\cdot n(n-1) x^{n-2}=\frac{x^2}{2}\cdot\frac{d^2}{dx^2}(x^{n})$$
and recall the basic generating function $\sum_{n\geq 0}x^n=\frac{1}{1-x}$.
What is $A(x)$? 
</s>