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
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
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