A303865 Number of noncrossing path sets on 3*n nodes up to rotation with each path having exactly 3 nodes.

1, 1, 6, 38, 384, 4425, 57976, 807318, 11828706, 179826245, 2816100678, 45170552490, 739103543356, 12297976924176, 207577047945312, 3547290764931730, 61277684496311364, 1068648890500799799, 18794421104465407618, 333037302131948734566, 5941487005826379359448

Offset

0,3

Links

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

Formula

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

Mathematica

seq[n_] := Module[{p, q}, p = 1 + InverseSeries[x/(3*(1 + x)^3) + O[x]^n]; q = x*D[p, x]/p; Integrate[((p - 1)/3 + Sum[EulerPhi[d]*(q /. x -> x^d + O[x]^n), {d, 2, n}])/x, x] + 1];
CoefficientList[seq[21], x] (* 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(q=x*deriv(p)/p);
Vec(intformal(((p-1)/3 + sum(d=2, n, eulerphi(d)*subst(q, x, x^d+O(x*x^n))))/x) + 1)}

Crossrefs

Column 3 of A303864.

Cf. A303330, A303844.

Sequence in context: A349844 A367838 A319194 * A319647 A239983 A096674

Adjacent sequences: A303862 A303863 A303864 * A303866 A303867 A303868

Keyword

nonn

Author

Andrew Howroyd, May 01 2018

Status

approved