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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 (list; graph; refs; listen; history; text; internal format)
 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 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 (* 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: 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

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Last modified November 29 21:32 EST 2021. Contains 349416 sequences. (Running on oeis4.)