Sum with Generating Functions Find the sum
$$\sum_{n=2}^{\infty} \frac{\binom n2}{4^n} ~~=~~ \frac{\binom 22}{16}+\frac{\binom 32}{64}+\frac{\binom 42}{256}+\cdots$$

How can I use generating functions to solve this?

For an alternative technique, let
$$ S = \sum_{n\geq 2}\frac{\binom{n}{2}}{4^n}.$$
We have:
$$\begin{eqnarray*} 3S = 4S-S &=& \sum_{n\geq 2}\frac{\binom{n}{2}}{4^{n-1}}-\sum_{n\geq 2}\frac{\binom{n}{2}}{4^{n}}=\frac{1}{4}+\sum_{n\geq 2}\frac{\binom{n+1}{2}-\binom{n}{2}}{4^n}\\&=&\frac{1}{4}+\sum_{n\geq 2}\frac{n}{4^n}.\end{eqnarray*}$$
If we set:
$$ T = \sum_{n\geq 2}\frac{n}{4^n}$$
we have:
$$ \begin{eqnarray*}3T = 4T-T = \sum_{n\geq 2}\frac{n}{4^{n-1}}-\sum_{n\geq 2}\frac{n}{4^n}=\frac{1}{2}+\sum_{n\geq 2}\frac{1}{4^n}=\frac{7}{12}\end{eqnarray*} $$
hence $T=\frac{7}{36}$ and $3S=\frac{4}{9}$, so $\color{red}{S=\frac{4}{27}}$.
