Generating functions and central binomial coefficient

Here is a simple derivation of the generating function: start with $$ (1+x)^{-1/2} = \sum_k \binom{-1/2}{k}x^k.$$ Now, \begin{align*} \binom{-1/2}{k} &= \frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)\left(\cdots\right)\left(-\frac{1}{2}-k+1\right)}{k!} \\ &= (-1)^k \frac{1\cdot 3\cdot 5\cdots (2k-1)}{2^k k!} \\ &= (-1)^k \frac{(2k)!}{2^k2^kk!k!} \\ &= (-1)^k 2^{-2k}\binom{2k}{k}. \end{align*} Finally replace $x$ by $-4x$ in the binomial expansion, giving $$ (1-4x)^{-1/2} = \sum_k (-1)^k 2^{-2k}\binom{2k}{k}\cdot (-4x)^k = \sum_k \binom{2k}{k}x^k.$$

Use the binomial theorem. Then $(1-4x)^{-{1 \over 2}} = \sum_{k=0}^\infty \binom{-{1 \over 2}}{k}(-1)^k 4^k x^k$.

$\binom{-{1 \over 2}}{k}(-1)^k 4^k = {\prod_{j=0}^{k-1}((-4)(-{1 \over 2}-j)) \over k!} = {\prod_{j=0}^{k-1}(2(1+2i)) \over k!} = {2^k (2k)!\over 2^kk! k!} = \binom{2k}{k}$