Generating Function

Generating functions are useful in finding explicit formulas for sequences. The general idea is to

1. Take an unknown sequence $(a_n)$
2. Define $f(x):={\sum }_{n=0}^{\infty }{a}_n{x}^n$.
3. Find an equation for $f(x)$ describe by constraints on the coefficients $(a_n$).
4. Obtain the individual terms of $(a_n)$ as a Taylor series expansion of $f(x)$.