Chap4 q2) Of which sequence is $U(s)=(1-4pqs^2)^{\frac{-1}{2}}$ the generating function(where $0<p=1-q<1$)?
Solution: So we need to find out a sequence $u_0,u_1,...$ such that $u_0+u_1s+u_2s^2+...=(1-4pqs^2)^{\frac{-1}{2}}$. Given that $\sum_i ar^i=\frac{a}{1-r}$, I feel like I need to make the exponent of $(1-4pqs^2)$ be $-1$
By the binomial series:
$\begin{align*} (1 + x)^{-1/2} &= \sum_{n \ge 0} \binom{-1/2}{n} x^n \\ &= \sum_{n \ge 0} \frac{(-1)^k}{2^{2 n}} \binom{2 n}{n} x^n \end{align*}$