If I have an generating function (GF) --- ordinary or exponential --- defining a series with at least one coefficient equal to zero, is there a general method to find the "inverse GF", *i.e.*, the GF defining the series which **only** includes those zeros?

For example [a very simplified one!], if I have the "odd number exponential GF"

$EG(2n+1;x) = x^1 + x^3 + x^5 + x^7 + \dots,$

how can I use it to derive the "even number exponential GF"

$EG'(2n;x) = 1 + x^2 + x^4 + \dots$

?

Thanks! Kieren.

Addendum: For my trivial example, above, the inversion is (d'oh!) simply to subtract that series from the "unit series" (where all coefficients are 1), *i.e.*,

$(1 + x^1 + x^2 + x^3 + x^4 + x^5 + \dots) - (x^1 + x^3 + x^5 + \dots) = 1 + x^2 + x^4 + \dots.$

Ergo, this "inversion" is trivial to implement --- and the problem is reduced to the open question referenced below (*i.e.*, to determine whether either the original GF or its "inverse" has any zeros).

Now my question is more specific: For **arbitrary** coefficients, is there a reasonable algorithm to derive the "inverted GF"?

anyinversion would be quite useful, in my opinion. Do you have an idea for an inversion algorithm? $\endgroup$