Multiplicative inverses of formal series with non-negative coefficients
What are the formal series $f$ with non-negative integer coefficients and constant term equal to $1$ whose multiplicative inverse $1/f$ has all coefficients, apart from a finite subset, all non-positive?
In fact, assume the series converges in some disk if you want...
For example, $f(x) = (1 - a x)/(1 - b x)$ with $b > a > 0$ has all coefficients nonnegative while $1/f(x)$ has all coefficients nonpositive after the constant term.