A045738 Number of branches in all noncrossing rooted trees on n nodes on a circle.

1, 4, 24, 148, 925, 5838, 37128, 237576, 1527867, 9867000, 63946740, 415683216, 2709186844, 17697136408, 115833872400, 759517409424, 4987999112007, 32804320226580, 216018805979760, 1424151150922500, 9398957079664845, 62090203617715350, 410536632908307360

Offset

2,2

Links

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

Index entries for sequences related to rooted trees

Formula

a(n) = binomial(3n-3, n-2) - 2*binomial(3n-6, n-3).

G.f.: (2*g^3-4*g^2+2*g-1)/((1-3*g)*(g-1)^3) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011

D-finite with recurrence +2*(2*n-1)*(n-2)*a(n) +(-43*n^2+169*n-160)*a(n-1) +4*(31*n^2-196*n+292)*a(n-2) -12*(3*n-13)*(3*n-14)*a(n-3)=0. - R. J. Mathar, Jul 26 2022

Prog

(PARI) a(n) = binomial(3*n-3, n-2) - 2*binomial(3*n-6, n-3); \\ Andrew Howroyd, Nov 12 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->(2*g^3-4*g^2+2*g-1)/((1-3*g)*(g-1)^3))(b(n)) + O(x^n)}
Vec(s(25)) \\ Andrew Howroyd, Nov 12 2017

Crossrefs

Sequence in context: A072949 A104531 A225050 * A215708 A369327 A192806

Adjacent sequences: A045735 A045736 A045737 * A045739 A045740 A045741

Keyword

nonn

Author

Emeric Deutsch

Status

approved