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. 



Combinatorial interpretation for coefficients of reciprocal of power series
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.


