A143021 Number of vertices of degree 1 in all non-crossing connected graphs on n points on a circle.

2, 6, 36, 270, 2244, 19740, 179880, 1678446, 15927780, 153055188, 1485010488, 14518525164, 142821228648, 1412109087480, 14021321053392, 139725123309486, 1396698760714788, 13998927825197220, 140638610864578200

Offset: 2

Author

Emeric Deutsch, Jul 30 2008

Keywords

• nonn

Examples

			a(3)=6 because in the graphs (AB,BC,CA), (AB,AC), (AB,BC) and (AC,BC) the vertices of degree 1 are (none), {B,C}, {A,C} and {A,B}.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..200
• P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.

Crossrefs

Cf. A007297, A089436.

Programs

Maple

g:=-1/3+(2/3)*sqrt(1+9*z)*sin((1/3)*arcsin(((2+27*z+54*z^2)*1/2)/(1+9*z)^(3/2))): ser:=series(z*(diff(g^2,z)),z=0,25): seq(coeff(ser,z,n), n=2..21);

Mathematica

terms = 19;
g[x_] = 0; Do[g[x_] = g[x]^2 + x (1+g[x])^3 + O[x]^(terms+2), {terms+2}];
Drop[CoefficientList[(x+x g[x])^2+O[x]^(terms+2), x], 2] Range[2, terms+1] (* Jean-François Alcover, Jul 29 2018, after A089436 and Andrew Howroyd *)

PARI

{ my(n=30); Vec(deriv((x+x*serreverse((x-x^2)/(1+x)^3 + O(x^n)))^2)) } \\ Andrew Howroyd, Dec 22 2017

Formula

a(n) = n*A089436(n).

G.f.: z*(d/dz)g^2, where g=g(z), the g.f. for the number of non-crossing connected graphs on n nodes on a circle, satisfies g^3 + g^2 - 3zg + 2z^2 = 0 (A007297).

D-finite with recurrence (n-1)*(n-2)*a(n) -34*(n-2)*(n-4)*a(n-1) +4*(29*n^2-396*n+937)*a(n-2) +24*(153*n^2-1071*n+1810)*a(n-3) -2688*(3*n-14)*(3*n-16)*a(n-4)=0. - R. J. Mathar, Jul 22 2022