rule for the asymptotic derivation, will be proven. In section \[sec:applications\], the apparatus will be applied to the calculation of the full asymptotic expansions of the number of *connected chord diagrams* and of the number of *simple permutations*.

Notation
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A (formal) power series $f \in {\mathbb{R}}[[x]]$ will be denoted in the usual ‘functional’ notation $f(x) = \sum_{n=0}^\infty f_n x^n$. The coefficients of a power series $f$ will be expressed by the same symbol with the index attached as a subscript $f_n$ or with the coefficient extraction operator $[x^n]f(x) = f_n$. Ordinary (formal) derivatives are expressed as $f'(x) = \sum_{n=0}^\infty n f_n x^{n-1}$. The ring of power series, restricted to expansions of functions which are analytic at the origin, or equivalently power series with non-vanishing radius of convergence, will be denoted as ${\mathbb{R}}\{x\}$. The ${\mathcal{O}}$-notation will be used: ${\mathcal{O}}(a_n)$ denotes the set of all sequences $b_n$ such that $\limsup _{n\rightarrow \infty} | \frac{ b_n }{a_n} | < \infty$ and ${o}(a_n)$ denotes all sequences $b_n$ such that $\lim _{n\rightarrow \infty} \frac{ b_n }{a_n} = 0$. Equations of the form $a_n = b_n + {\mathcal{O}}(c_n)$ are to be interpreted as statements $a_n-b_n \in {\mathcal{O}}(c_n)$ as usual. See [@bender1974asymptotic] for an introduction to 