field of research [@dunne2012resurgence; @aniceto2011resurgence].

During a conversation with David Sauzin it became plausible that the presented methods can also be derived from resurgence. In fact, the formalism can be seen as a toy model of resurgence’s *calcul différentiel étranger* [@ecalle1981fonctions Vol. 1] also called *alien calculus* [@mitschi2016divergent II.6]. This toy model is unable to fully reconstruct functions from asymptotic expansions, but does not rely on analytic properties of Borel transformed functions and therefore offers itself for combinatorial applications. A detailed and illuminating account on resurgence theory is given in David Sauzin’s review [@mitschi2016divergent Part II] or [@sauzin2014introduction]. For a review focused on applications to problems from physics consult [@aniceto2011resurgence].

Statement of results
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Power series with well-behaved asymptotic expansions, as in eq. , form a subring of ${\mathbb{R}}[[x]]$, which will be denoted as ${{\mathbb{R}}[[x]]^{\alpha}_{\beta}}$. This subring is also closed under composition and inversion of power series. A linear map, ${\mathcal{A}}^\alpha_\beta:{{\mathbb{R}}[[x]]^{\alpha}_{\beta}}\rightarrow {\mathbb{R}}[[x]]$, can be defined which *maps a power series to the asymptotic expansion of its coefficients*. A natural way to define such a map is to associate the power series $\sum_{n=0} c_n x^n$ to the series $\sum_{n=0}a_n x^n$ both related as in eq. . This map turns out to 