To a symmetric sequence $V_\bullet$ of vector spaces, associate the generating function $F_V(z) = \sum_n \frac{\dim(V_n)}{n!} z^n$. Then

$$F_{Comm_\ast}(z) = \exp(z)-1 \qquad F_{Lie}(z) = \ln(1-z)$$

where $Comm_\ast$ is the reduced commutative operad and $Lie$ is the Lie operad. Notice that

On the one hand, these power series are inverse up to a sign.

On the other hand, these operads are Koszul dual (and perhaps the sign corresponds to the shift that appears in Koszul duality?).

Similarly,

$$F_{Ass_\ast}(z) = \frac{z}{1-z}$$

where $Ass_\ast$ is the reduced associative operad. On the one hand, this power series is its own inverse up to a sign. On the other hand, this operad is Koszul self-dual.

**Question:** Is this a coincidence? Or is there some deeper connection between (1) and (2)? More concretely, is it the case (under certain conditions, perhaps) that Koszul dual operads have inverse generating functions, up to some sign?