A303330 a(n) is the number of noncrossing path sets on 3*n nodes up to rotation and reflection with each path having exactly 3 nodes.

1, 1, 4, 22, 201, 2244, 29096, 404064, 5915838, 89918914, 1408072452, 22585364697, 369552118682, 6148989874890, 103788529623864, 1773645405777098, 30638842342771863, 534324445644633987, 9397210553851138484, 166518651072771792918, 2970743502941350443069

Offset

0,3

Comments

Paths are constructed using noncrossing line segments between the vertices of a regular 3n-gon. Isolated vertices are not allowed.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Math StackExchange, Question from user Matan at math.stackexchange.com: Number of ways to connect sets of k dots in a perfect n-gon

Formula

a(n) ~ 3^(4*n - 1/2) / (sqrt(Pi) * n^(5/2) * 2^(2*n + 3)). - Vaclav Kotesovec, Jun 01 2022

Mathematica

seq[n_] := Module[{p, h, q, c}, p = 1 + InverseSeries[x/(3*(1 + x)^3) + O[x]^n , x]; h = (p /. x -> x^2 + O[x]^n); q = x*D[p, x]/p; c = Integrate[((p - 1)/3 + Sum[EulerPhi[d]*(q /. x -> x^d + O[x]^n), {d, 2, n}])/x, x]; CoefficientList[1 + c + (1 + h + x^2*h^3 + x*2*h^2)/2, x]/2];
seq[30] (* Jean-François Alcover, Jul 05 2018, after Andrew Howroyd *)

Prog

(PARI)
seq(n)={
my(p=1 + serreverse( x/(3*(1 + x)^3) + O(x*x^n) ));
my(h=subst(p, x, x^2 + O(x*x^n)), q=x*deriv(p)/p);
my(c=intformal(((p-1)/3 + sum(d=2, n, eulerphi(d)*subst(q, x, x^d+O(x*x^n))))/x));
Vec(1 + c + (1 + h + x^2*h^3 + x*2*h^2)/2)/2} \\ Andrew Howroyd, Apr 29 2018

Crossrefs

Column k=3 of A302828.

Cf. A303839, A303865, A303867.

Sequence in context: A364624 A062494 A183274 * A280828 A103437 A215201

Adjacent sequences: A303327 A303328 A303329 * A303331 A303332 A303333

Keyword

nonn

Author

J. Stauduhar, Apr 21 2018

Extensions

Terms a(8) and beyond from Andrew Howroyd, Apr 29 2018

a(6) corrected by Andrew Howroyd, May 03 2018

Status

approved