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A074922
Number of ways of arranging n chords on a circle (handshakes between 2n people across a table) with exactly 2 simple intersections.
2
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
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
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
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, Oct 06 2002
STATUS
approved