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