This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A153231 #66 May 24 2024 00:44:16
%S A153231 1,2,12,96,880,8736,91392,992256,11075328,126297600,1465052160,
%T A153231 17233182720,205074874368,2464404045824,29864206663680,
%U A153231 364535993597952,4477993284993024,55316387638149120,686720560048373760,8563155161736806400,107206525476085432320
%N A153231 a(n) = 2^n * binomial(3n,n)/(2n+1).
%C A153231 a(n) is also the number of rooted generalized noncrossing trees on n+1 vertices.
%C A153231 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
%C A153231 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
%C A153231 From _Torsten Muetze_, May 08 2024: (Start)
%C A153231 a(n) also counts ternary trees with n nodes that are colored red or blue.
%C A153231 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)
%D A153231 Bruce E. Sagan, Proper partitions of a polygon and k-Catalan numbers, Ars Combinatoria, 88 (2008), 109-124.
%H A153231 Michael De Vlieger, <a href="/A153231/b153231.txt">Table of n, a(n) for n = 0..889</a>
%H A153231 CombOS - Combinatorial Object Server, <a href="http://combos.org/kary">Generate k-ary trees and dissections</a>
%H A153231 Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, <a href="https://doi.org/10.4230/LIPIcs.AofA.2018.29">Asymptotic Expansions for Sub-Critical Lagrangean Forms</a>, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
%H A153231 Bruce E. Sagan, <a href="https://arxiv.org/abs/math/0407280">Proper partitions of a polygon and k-Catalan numbers</a>, arXiv:math/0407280 [math.CO], 2004.
%H A153231 Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, <a href="https://arxiv.org/abs/1706.03357">Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing</a>, arXiv:1706.03357 [cs.CL], 2017.
%F A153231 a(n) = 2^n*A001764(n). - _R. J. Mathar_, Oct 06 2012
%F A153231 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
%F A153231 a(n) = (n+1)*A000309(n). - _Johannes W. Meijer_, Aug 22 2013
%F A153231 G.f.: sqrt(2)/sqrt(3*x)*sin(1/3*asin(sqrt(27*x/2))). - _Vladimir Kruchinin_, Sep 08 2015
%F A153231 E.g.f.: Hypergeometric2F2(1/3,2/3; 1,3/2; 27*x/2). - _Ilya Gutkovskiy_, Nov 23 2017
%t A153231 Table[2^n Binomial[3n, n]/(2n+1), {n, 0, 25}] (* _Vincenzo Librandi_, Sep 08 2015 *)
%o A153231 (Magma) [2^n*Binomial(3*n,n)/(2*n+1): n in [0..30]]; // _Vincenzo Librandi_, Sep 08 2015
%o A153231 (PARI) a(n) = 2^n*binomial(3*n,n)/(2*n+1); \\ _Altug Alkan_, Sep 24 2018
%o A153231 (SageMath) [2^n*binomial(3*n,n)/(2*n+1) for n in range(31)] # _G. C. Greubel_, Mar 08 2023
%Y A153231 Cf. A000309, A001764.
%Y A153231 Cf. A369510 (colorful triangulations with an odd number of points).
%K A153231 nonn,easy
%O A153231 0,2
%A A153231 Yidong Sun (sydmath(AT)yahoo.com.cn), Dec 21 2008
%E A153231 More terms from _N. J. A. Sloane_, Dec 21 2008