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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
OFFSET
0,3
COMMENTS
Paths are constructed using noncrossing line segments between the vertices of a regular 3n-gon. Isolated vertices are not allowed.
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.
Sequence in context: A364624 A062494 A183274 * A280828 A103437 A215201
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