A065065 Number of noncrossing connected graphs with nodes on a circle having n edges.

1, 3, 13, 64, 341, 1913, 11132, 66573, 406653, 2526351, 15913347, 101396034, 652378120, 4232439734, 27657380019, 181872596607, 1202641671293, 7991878198287, 53343146808137, 357464739709920, 2404073823950915

Offset: 1

Author

Emeric Deutsch, Nov 06 2001

Keywords

• nonn

Examples

			a(3)=13 because we have 1 triangle on 3 nodes and 12 non-crossing trees on 4 nodes.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.

Crossrefs

Cf. A007297, A127537.

Programs

Maple

A065065 := n-> sum(binomial(3*k-3,n+k)*binomial(n-1,n-k+1)/(k-1),k=ceil((n+3)/2)..n+1);

Mathematica

terms = 21;
A[_] = 0;
Do[A[x_] = x (1 + 3 A[x] + 4 A[x]^2 + A[x]^3) + O[x]^(terms+1), {terms+1}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Jul 29 2018, after Vladimir Kruchinin *)

PARI

a(n)=sum(k=ceil((n+3)/2), n+1, binomial(3*k-3,n+k)*binomial(n-1,n-k+1)/(k-1)); \\ Andrew Howroyd, Nov 12 2017

PARI

Vec(serreverse(x/(1+3*x+4*x^2+x^3) + O(x^20))) \\ Andrew Howroyd, Nov 12 2017

Formula

a(n) = Sum_{k=ceiling((n+3)/2)..n+1} binomial(3*k-3,n+k)*binomial(n-1,n-k+1)/(k-1).

G.f. satisfies: A(x) = x*(1+3*A(x)+4*A(x)^2+A(x)^3). - Vladimir Kruchinin, Nov 12 2014

a(n) = Sum_{m=n..2*n-2} A127537(m,n). - Andrew Howroyd, Nov 12 2017

D-finite with recurrence 8*n*(2*n+1)*a(n) +2*(-46*n^2+55*n-18)*a(n-1) +6*(-30*n^2+60*n-7)*a(n-2) +2*(n-3)*(28*n-163)*a(n-3) +93*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 26 2022