A045737 Number of nonroot branch nodes in all noncrossing rooted trees on n nodes on a circle.

0, 0, 3, 28, 210, 1470, 9996, 67032, 446292, 2960100, 19594575, 129585456, 856703848, 5663913528, 37454912040, 247778648880, 1639890119016, 10858731869160, 71939098633185, 476841658085100, 3162310375905450

Offset

2,3

Links

Andrew Howroyd, Table of n, a(n) for n = 2..200

Index entries for sequences related to rooted trees

Formula

a(n) = 7*(n-1)*binomial(3n-6, n-4)/(2n-1).

G.f.: g^2*(2*g-3)/((1-3*g)*(g-1)^3) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011

a(n) = Sum_{k=0..ceiling(n/2)-1} k*A101431(n, k). - Andrew Howroyd, Nov 17 2017

D-finite with recurrence +2*(2*n-1)*(n-4)*a(n) -3*(3*n-7)*(3*n-8)*a(n-1)=0. - R. J. Mathar, Jul 26 2022

Mathematica

Table[7(n-1) Binomial[3n-6, n-4]/(2n-1), {n, 2, 30}] (* Harvey P. Dale, Nov 01 2017 *)

Prog

(PARI) a(n) = 7*(n-1)*binomial(3*n-6, n-4)/(2*n-1); \\ Andrew Howroyd, Nov 11 2017
(PARI) \\ here b(n) is x^2 * g.f. of A006013.
b(n)=serreverse(x - 2*x^2 + x^3 + O(x^n));
s(n)={(g->g^2*(2*g-3)/((1-3*g)*(g-1)^3))(b(n)) + O(x^n)}
concat([0, 0], Vec(s(25))) \\ Andrew Howroyd, Nov 11 2017

Crossrefs

Cf. A006013, A101431.

Sequence in context: A356975 A278183 A091120 * A003466 A337590 A092637

Adjacent sequences: A045734 A045735 A045736 * A045738 A045739 A045740

Keyword

nonn

Author

Emeric Deutsch

Status

approved