We write as (1 + x)(1 − x2)−½, and expand the latter term as a binomial series. We have
The coefficient of the general term, x2n, is given by .
The numerator of this expression is a double factorial: (2n − 1)!!.
The denominator may then be written in a similar form by "absorbing" the factors of 2n into n!, giving (2n)!!.
(Alternatively, we could multiply numerator and denominator by 2n n!, and simplify the expression, obtaining .)
Therefore, the power series expansion of = .
Source: To be announced