A074922 Number of ways of arranging n chords on a circle (handshakes between 2n people across a table) with exactly 2 simple intersections.

0, 0, 0, 3, 28, 180, 990, 5005, 24024, 111384, 503880, 2238390, 9806280, 42493880, 182530530, 778439025, 3300049200, 13919756400, 58462976880, 244639718730, 1020422356200, 4244365452600, 17610393500700, 72907029092898

Offset

0,4

Links

Table of n, a(n) for n=0..23.

Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.

Vincent Pilaud and Juanjo Rué, Analytic combinatorics of chord and hyperchord diagrams with k crossings, arXiv preprint arXiv:1307.6440, 2013

Henry Bottomley, Illustration for A000108, A001147, A002694, A067310 and A067311

Formula

a(n) = C(2n, n-2)*(n-2)/2 = A002694(n)*(n-2)/2 = A067310(n, 2) = Sum_{0<=j<n} (-1)^j*C((n-j)*(n-j+1)/2-1-2, n-1)*(C(2n, j)-C(2n, j-1)).

Example

a(3)=3 since the only possibility is to have one of the three chords intersected by the other two.

Mathematica

Table[Binomial[2n, n-2] (n-2)/2, {n, 0, 30}] (* Harvey P. Dale, Nov 04 2011 *)

Crossrefs

Cf. A067310, A002694, A232224, A274404.

Sequence in context: A012778 A302522 A303405 * A285365 A366689 A160872

Adjacent sequences: A074919 A074920 A074921 * A074923 A074924 A074925

Keyword

nonn,easy

Author

Henry Bottomley, Oct 06 2002

Status

approved