weird Generating Function prove

[phoenixy]

Q: show that (1-4x)^(-1/2) generates the sequence 2n chooses n, n is defined as natural

All the formulas I have requires integer exponent. I am not sure how to deal with (-1/2).

Thanks for any input!!!

[Dr Transport]

expand the function in a Taylor series

f(x) = f(0) + x\frac{df(x)}{dx}|_0 +\frac{x}{2!}\frac{d^{2}f(x)}{dx^{2}}|_0 + ...

differentiate your function

\frac{1}{\sqrt{1-4x}}

as you normally would, for example the first derivative is \frac{2x}{(1-4x)^{3/2}} .