extracting Catalan numbers from its generating function (binomial theorem)

Question

I am reading the Combinatorics by Cameron, and one sections buggs me:

$C_n = \sum_{i=1}^{n-1} C_iC_{n-i}$

We take $F(t) = \sum_{n \geq1}C_nt^n$

From which we conclude $F(t) = t + F(t)^2$ then $F(t) = \frac{1}{2}(1-(1-4t)^{1/2})$

And this is part I dont get: how we extract the coefficients of $t^n$ to be :$\frac{-1}{2} {1/2 \choose n} (-4)^n$

Thank you!

Answer 1

You use Newton's binomial theorem: $$ (1+z)^a = \sum_{n = 0}^\infty \binom{a}{n} z^n $$ for $a \in \mathbf{C}$. Here $\binom{a}{n} := \frac{a(a-1) \dots (a-n+1)}{n!}$. This is valid as an equality of complex analytic functions for $|z|<1$ and is shown by computing the Taylor expansion of $(1+z)^a$ (which helps explain the appearance of the binomial coefficients).