inversion of coefficients in taylor expansion

Question

There are two different functions, one has ArcSin x multiplied or divided by some function of x as for example the square of (1-x^2) or x itself. The other function is only a power of (1-x^2). x goes from 0 to 1 excluded. The fact is that when I expand in series around x=0, the first function has a development of the kind a0 + a2 x^2+ a4 x^4 +... with all the even powers. The other function has a development such as (1/a0)+ (1/a2) x^2+ (1/a4) x^4+ ... i.e. same structure of the power series but ALL the infinite coefficients are for one function the reciprocal of the other function. I think that's not possible such a coincidence for two functions that seems just arbitrarily different one from te other, there should be some relation between the two functions. Someone knows if there is any relation between two functions whose development has all the coefficients different from zero one reciprocal of the other?