A303865 - OEIS

%I #13 Jun 01 2022 06:24:35

%S 1,1,6,38,384,4425,57976,807318,11828706,179826245,2816100678,

%T 45170552490,739103543356,12297976924176,207577047945312,

%U 3547290764931730,61277684496311364,1068648890500799799,18794421104465407618,333037302131948734566,5941487005826379359448

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

%H Andrew Howroyd, <a href="/A303865/b303865.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) ~ 3^(4*n - 1/2) / (sqrt(Pi) * n^(5/2) * 2^(2*n + 2)). - _Vaclav Kotesovec_, Jun 01 2022

%t 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];

%t CoefficientList[seq[21], x] (* _Jean-François Alcover_, Jul 05 2018, after _Andrew Howroyd_ *)

%o (PARI)

%o seq(n)={ my(p=1 + serreverse( x/(3*(1 + x)^3) + O(x*x^n) )); my(q=x*deriv(p)/p);

%o Vec(intformal(((p-1)/3 + sum(d=2, n, eulerphi(d)*subst(q, x, x^d+O(x*x^n))))/x) + 1)}

%Y Column 3 of A303864.

%Y Cf. A303330, A303844.

%K nonn

%O 0,3

%A _Andrew Howroyd_, May 01 2018