Connected Chord Diagrams and the Combinatorics of Asymptotic Expansions

In this article we study an asymptotic expansion for $C_n$, the number of connected chord diagrams on $n$ chords. The expansion is obtained in earlier work by means of alien derivatives applied to the generating series of connected chord diagrams; we seek a combinatorial interpretation. The main outcome presented here is a new combinatorial interpretation for entry https://oeis.org/A088221 of the OEIS. We will show that https://oeis.org/A088221 counts pairs of connected chord diagrams (allowing empty diagrams). This gives a combinatorial interpretation for the part of the closed form of the asymptotic expansion of $C_n$.