A232224 Number of ways of arranging n chords on a circle (handshakes between 2n people across a table) with exactly 3 simple intersections.

0, 0, 0, 1, 20, 195, 1430, 9009, 51688, 278460, 1434120, 7141530, 34648856, 164663785, 769491450, 3546222225, 16152872400, 72846725160, 325722299760, 1445598337950, 6373942543800, 27942072562950, 121863923024844, 529043313674106, 2287209524819120

Offset

0,5

Links

Lars Blomberg, Table of n, a(n) for n = 0..100

V. Pilaud, J. Rué, Analytic combinatorics of chord and hyperchord diagrams with k crossings, arXiv preprint arXiv:1307.6440 [math.CO], 2013.

Formula

Pilaud-Rue give an explicit g.f.

a(n) = [x^(2n)] (1-sqrt(1-4*x^2))^6*((1-x^2)*sqrt(1-4*x^2)+7*x^2-26*x^4) / (64*x^6*sqrt(1-4*x^2)^5). - Michel Marcus, Sep 30 2015

Mathematica

CoefficientList[Series[(1 - Sqrt[1 - 4 x^2])^6 ((1 - x^2) Sqrt[1 - 4 x^2] + 7 x^2 - 26 x^4)/(64 x^6 Sqrt[1 - 4 x^2]^5), {x, 0, 48}], x^2] (* Michael De Vlieger, Sep 30 2015 *)

Prog

(PARI) lista(nn) = {np = 2*nn+2; default(seriesprecision, np); pol = (1-sqrt(1-4*x^2))^6*((1-x^2)*sqrt(1-4*x^2)+7*x^2-26*x^4)/(64*x^6*sqrt(1-4*x^2)^5) + O(x^(np)); forstep (n=0, 2*nn, 2, print1(polcoeff(pol, n), ", "); ); } \\ Michel Marcus, Sep 30 2015
(PARI) x='x+O('x^33); concat([0, 0, 0], Vec((1-sqrt(1-4*x))^6*((1-x)*sqrt(1-4*x)+7*x-26*x^2) / (64*x^3*sqrt(1-4*x)^5))) \\ Joerg Arndt, Sep 30 2015

Crossrefs

Cf. A002694, A074922, A274404.

Sequence in context: A355590 A219369 A245137 * A338778 A375803 A343242

Adjacent sequences: A232221 A232222 A232223 * A232225 A232226 A232227

Keyword

nonn

Author

N. J. A. Sloane, Nov 22 2013

Extensions

Corrected initial terms and more terms from Lars Blomberg, Sep 30 2015

Status

approved