

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.


4



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

0,3


COMMENTS

Paths are constructed using noncrossing line segments between the vertices of a regular 3ngon. 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 ngon


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] (* JeanFrançois 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(((p1)/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: A002728 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



