A153231 a(n) = 2^n * binomial(3n,n)/(2n+1).

1, 2, 12, 96, 880, 8736, 91392, 992256, 11075328, 126297600, 1465052160, 17233182720, 205074874368, 2464404045824, 29864206663680, 364535993597952, 4477993284993024, 55316387638149120, 686720560048373760, 8563155161736806400, 107206525476085432320

Offset: 0

Author

Yidong Sun (sydmath(AT)yahoo.com.cn), Dec 21 2008

Keywords

• nonn
• easy

Comments

a(n) is also the number of rooted generalized noncrossing trees on n+1 vertices.

The series reversion of y = x +2*x^3 is x = y -2*y^3 +12*y^5 -96*y^7 +880*y^9 -8736*y^11 +... - R. J. Mathar, Sep 29 2012

Lattice paths in the 1st quadrant from (0,0) to (3n,0) using steps D(1,-1) and two types of U(1,2). - David Scambler, Jun 22 2013

From Torsten Muetze, May 08 2024: (Start)

a(n) also counts ternary trees with n nodes that are colored red or blue.

a(n) also counts triangulations of a convex (2n+2)-gon whose points are colored red and blue alternatingly, and that do not have monochromatic triangles (i.e., every triangle has at least one red point and at least one blue point). (End)

References

• Bruce E. Sagan, Proper partitions of a polygon and k-Catalan numbers, Ars Combinatoria, 88 (2008), 109-124.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..889
• CombOS - Combinatorial Object Server, Generate k-ary trees and dissections
• Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
• Bruce E. Sagan, Proper partitions of a polygon and k-Catalan numbers, arXiv:math/0407280 [math.CO], 2004.
• Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.

Crossrefs

Cf. A000309, A001764.

Cf. A369510 (colorful triangulations with an odd number of points).

Programs

Magma

[2^n*Binomial(3*n,n)/(2*n+1): n in [0..30]]; // Vincenzo Librandi, Sep 08 2015

Mathematica

Table[2^n Binomial[3n, n]/(2n+1), {n, 0, 25}] (* Vincenzo Librandi, Sep 08 2015 *)

PARI

a(n) = 2^n*binomial(3*n,n)/(2*n+1); \\ Altug Alkan, Sep 24 2018

SageMath

[2^n*binomial(3*n,n)/(2*n+1) for n in range(31)] # G. C. Greubel, Mar 08 2023

Formula

a(n) = 2^n*A001764(n). - R. J. Mathar, Oct 06 2012

D-finite with recurrence n*(2*n+1)*a(n) -3*(3*n-1)*(3*n-2)*a(n-1) = 0. - R. J. Mathar, Nov 16 2012

a(n) = (n+1)*A000309(n). - Johannes W. Meijer, Aug 22 2013

G.f.: sqrt(2)/sqrt(3*x)*sin(1/3*asin(sqrt(27*x/2))). - Vladimir Kruchinin, Sep 08 2015

E.g.f.: Hypergeometric2F2(1/3,2/3; 1,3/2; 27*x/2). - Ilya Gutkovskiy, Nov 23 2017

Extensions

More terms from N. J. A. Sloane, Dec 21 2008