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?

If $f(z) = \sum\limits_{n=0}^\infty a_n z^n$ converges for $|z| < \rho$, then for any $m \ge 0$,
$$\frac{z^m}{m!} \frac{d^m}{dz^m} f(z) = \sum_{n=0}^\infty a_m\binom{n}{m} z^n
\quad\text{ for } |z| < \rho.$$
Apply this to $$f(z) = \frac{1}{1-z} = \sum_{n=0}^\infty z^n,$$ we get
$$\sum_{n=2}^\infty \binom{n}{2} z^n = \frac{z^2}{2!}\frac{d^2}{dz^2}\frac{1}{1-z} = \frac{z^2}{(1-z)^3}$$
Substitute $z = \frac14$ will give you
$$\sum_{n=2}^\infty \binom{n}{2} z^n = \frac{\frac14^2}{(1-\frac14)^3} = \frac{4}{27}$$
