I'm trying to use generating functions to get the value of some coefficients, namely
$\displaystyle\sum_{m\geq 0} f_{2m}x^m = 1 - \Big(\displaystyle\sum_{m\geq 0} u_{2m}x^m\Big)^{-1}\\$ and
$u_{2m} = \binom{2m}{m}4^{-2m}$ so that
$\displaystyle\sum_{m\geq 0} u_{2m}x^m = 2F1(\frac{1}{2},\frac{1}{2};1;x^2)$
(according to Wolfram Alpha).
For the first equation to work out and for me to find $f_{2m}$ I need to take $\displaystyle\sum_{m\geq 0} u_{2m}x^m$ out of the denominator. Any ideas or identities?