I posed this question to another user, but we weren't able neither to make a precise guess, nor to give a precise proof. Is it a "not-so-well" well-known fact?

Consider a connected non-finite CW complex $X$; denote its Poincare' series as $p_X(T)=a_0+a_1T+\dots + a_nT^n+\dots$

Since $X$ is connected, the series $p_X(T)$ is an invertible element in $\mathbb Z[\![T]\!]$: what is the geometric meaning of its inverse series $q(T)$? Can it arise as the series of a space $Y$? Which is the relation between $X$ and $Y$?

Edit: As it was pointed out in the comments, the question as it is stated makes no sense: whatever the series $p_X(T)$, its inverse will have negative coefficients, so there is no way to interpret them as dimensions of homology objects associated to a wouldbe space $Y$.

Nevertheless, there are cases where utterly counterintuitive statements like "there is a groupoid whose measure is $\sum_k \frac{1}{k!}$" have suitable meaning. So my question sounds more like

is there a "groupoidal-trick" to turn the (multiplicative) inverse Poincare' series into something meaningful, by stretching a bit the meaning of the word "space"?