OEIS knows about your sequence, which it calls Crossing matchings: linear chord diagrams with $2n$ nodes and $n$ arcs in which each arc crosses another arc.
Some highlights:
The sequence starts $ \begin{multline*} 1, 0, 1, 4, 31, 288, 3272, 43580, 666143, 11491696, \\ 220875237, 4681264432, 108475235444, 2728591657920, 74051386322580, \\ 2156865088819692, 67113404608820943, \\ 2221948578439255200, 77990056655776149179, \ldots \end{multline*} $
The generating function $F$ is the solution of F' = \frac{-x^2F^3 + F - 1}{2x^3F^2 + 2x^2F}.
There is a reference: M. Klazar, Non-P-recursiveness of numbers of matchings or linear chord diagrams with many crossings, Adv. in Appl. Math., 30 (2003), 126-136.
There is a link: Alexander Stoimenow, On enumeration of chord diagrams and asymptotics of Vassiliev invariants, Chapter 3.
If you're wondering how I found the link: I just calculated the first 4 numbers and asked OEIS. I could also have guessed the name, though to be extra sure you need to calculate some numbers anyway.