Counting Self-intersecting n-gons

[N. J. A. Sloane]

There is a paper in the Amer. Math. Monthly about this, 30 to 40
years ago.  The sequences in it are in the OEIS.  The authors were 
some of the usual suspects.

ah yes, here it is:

MR0123487 (23 #A812)
Golomb, S. W.; Welch, L. R.
On the enumeration of polygons.
Amer. Math. Monthly 67 1960 349--353.
05.65
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There exist $n!$ polygonal paths joining $n$ equally spaced points on a circle. Two polygonal paths which differ only in starting point or orientation give rise to `identical' polygons. If, besides possible difference in starting point and orientation, tw

Reviewed by Seymour Schuster