So the exercise in my book on probability was to find the sequence that the function $U(s)=\sqrt{1-4pqs^2}$ generates given that $0<p=1-q<1$.
Immediately I thought to just write out some terms of the Taylor Expansion around 0. After about 3 terms or so I gave up because there didn't seem to be much order in it. So I went online to Wolfram Alpha and it gave me the following:
The main thing that I'm wondering now is how to describe this sequence and if this is a sequence I should know about? It doesn't seem like it is a generating function of something my book discusses.
The binomial series is $$ (1+x)^\alpha=\sum_{n=0}^\infty\binom{\alpha}{n}x^n,\quad|x|<1. $$ Then $$ \sqrt{1-4\,p\,q\,s^2}=(1-4\,p\,(1-p)\,s^2)^{1/2}=\sum_{n=0}^\infty\binom{1/2}{n}(-1)^n4^np^n(1-p)^n\,s^{2n}. $$