# Combinatorial interpretation for coefficients of reciprocal of power series

I've seen a number of combinatorial interpretations for the coefficients of the compositional inverse (aka reversion) of a power series. Is there a known combinatorial interpretation for the coefficients of the reciprocal of a power series?

Specifically: I'm looking for a family of combinatorially defined sets $S_0, S_1, S_2, \ldots$, with $S_n$ consisting of objects of "size" $n$, such that for a power series $a_0+a_1x+a_2x^2+\ldots$ ($a_0 \neq 0$) with reciprocal $b_0+b_1x+b_2x^2 \ldots$, $b_n$ can be understood as a weighted sum over $S_n$, with the weighting depending in some reasonable way on the $a_i$'s.

Since $a_0+a_1x+a_2x^2+\cdots=a_0(1+(a_1/a_0)x+(a_2/a_0)x^2+\cdots)$, we can assume $a_0=1$. Then $$b_n = \sum (-1)^k a_{i_1}\cdots a_{i_k},$$ where the sum is over all $2^{n-1}$ compositions $(i_1,\dots,i_k)$ of $n$. Thus we can take $S_n$ to be the set of compositions of $n$, etc.
Let $P$ be a set and $S \subseteq P^*$ be a subset of the free monoid (we actually want $S$ be a linear system which is defined in the thesis). For any $V \subseteq S$ we define its conuting series to be $\Gamma(V) = \sum_{\alpha \in V} \alpha$ which is a formal sum. We also define the alternating counting series $\bar{\Gamma}(V) = \sum_{\alpha \in V} (-1)^{r(\alpha)} \alpha$ where $r(\alpha)$ is the number of prime factors in $\alpha$ (i.e. length of the word). For any $\alpha = a_1a_2 \cdots a_n$ the links of $\alpha$ are $a_i a_{i+1}$. Take some set a links $L \subseteq P^2$ and define $C$ to be the set of $\alpha \in S$ such that all links of $\alpha$ are in $L$. Also let $\bar{C}$ be the set of $\alpha \in S$ such that all links of $\alpha$ are not in $L$. On page 37 we have Theorem 4.1 (The Inversion Theorem) we states $\Gamma(C) \bar{\Gamma}(\bar{C}) = 1$. Now let's use this for some single variable generating functions.
For a trivial example let $S = P^*$ where $P$ is a set of size $k$. Then let $L = P^2$. In this case $\Gamma(C) = \sum_{w \in P^*} w$ is the sum of all $k$-ary words and $\bar{\Gamma}(\bar{C}) = 1 - \sum_{a \in P} a$. If we subsitute $a = x$ for all $a \in P$ we get the $\Gamma(C)$ becomes $\sum_{n \geq 0} k^n x^n$ and $\bar{\Gamma}({\bar{C}})$ becomes $1 - kx$.
For a less trivial example let $S = P^* = \{a,b\}^*$ and let $L = \{aa,ab,ba\}$. Then $\Gamma(C)$ is the sum of all binary words which have no consecutive $b$'s and $\bar{\Gamma}(\bar{C}) = 1 - a - b + b^2 - b^3 + \cdots$. Again substituting $a = b = x$ we have $$1 + 2x + 3x^2 + 5x^3 + \cdots = \frac{1}{1 - 2x + x^2 - x^3 + \cdots}$$ the inverse of an "almost" Fibonacci generating function.