  The algebraic properties of formal power series, whose coefficients show
factorial growth and admit a certain well-behaved asymptotic expansion, are
discussed. It is shown that these series form a subring of $\mathbb{R}[[x]]$.
This subring is also closed under composition and inversion of power series. An
'asymptotic derivation' is defined which maps a power series to the asymptotic
expansion of its coefficients. Product and chain rules for this derivation are
deduced. With these rules asymptotic expansions of the coefficients of
implicitly defined power series can be obtained. The full asymptotic expansions
of the number of connected chord diagrams and the number of simple permutations
are given as examples.
