Math Help - Closed Formula for Generating Function

1. Closed Formula for Generating Function

Hi Guys,

Question: what is closed formula for generating function 1 /(1-2x^2)?

Thanks
Chicago Boy

2. The Taylor expansion of $f(x)$ around $x=0$ is...

$f(x) = \frac{1}{1- 2\cdot x^{2}} = 1 + 2\cdot x^{2} + 4\cdot x^{4} + \dots + (2\cdot x^{2})^{n} + \dots = \sum_{n=0}^{\infty} (2\cdot x^{2})^{n}$ (1)

... where the series converges for $|x|< \frac{\sqrt{2}}{2}$, so that $f(x) = \frac{1}{1-2\cdot x^{2}}$ is the generating function of the sequence...

$a_{n}= 2^{\frac{n}{2}}$ n even, $a_{n}=0$, n odd

Kind regards

$\chi$ $\sigma$