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%I #44 Mar 09 2023 08:59:24
%S 1,2,12,96,880,8736,91392,992256,11075328,126297600,1465052160,
%T 17233182720,205074874368,2464404045824,29864206663680,
%U 364535993597952,4477993284993024,55316387638149120,686720560048373760,8563155161736806400,107206525476085432320
%N a(n) = 2^n * binomial(3n,n)/(2n+1).
%C a(n) is also the number of rooted generalized noncrossing trees on n+1 vertices.
%C 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 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
%H Michael De Vlieger, <a href="/A153231/b153231.txt">Table of n, a(n) for n = 0..889</a>
%H 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 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 a(n) = 2^n*A001764(n). - _R. J. Mathar_, Oct 06 2012
%F 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 a(n) = (n+1)*A000309(n). - _Johannes W. Meijer_, Aug 22 2013
%F G.f.: sqrt(2)/sqrt(3*x)*sin(1/3*asin(sqrt(27*x/2))). - _Vladimir Kruchinin_, Sep 08 2015
%F E.g.f.: Hypergeometric2F2(1/3,2/3; 1,3/2; 27*x/2). - _Ilya Gutkovskiy_, Nov 23 2017
%t Table[2^n Binomial[3n, n]/(2n+1), {n, 0, 25}] (* _Vincenzo Librandi_, Sep 08 2015 *)
%o (Magma) [2^n*Binomial(3*n,n)/(2*n+1): n in [0..30]]; // _Vincenzo Librandi_, Sep 08 2015
%o (PARI) a(n) = 2^n*binomial(3*n,n)/(2*n+1); \\ _Altug Alkan_, Sep 24 2018
%o (SageMath) [2^n*binomial(3*n,n)/(2*n+1) for n in range(31)] # _G. C. Greubel_, Mar 08 2023
%Y Cf. A000309, A001764.
%K nonn,easy
%O 0,2
%A Yidong Sun (sydmath(AT)yahoo.com.cn), Dec 21 2008
%E More terms from _N. J. A. Sloane_, Dec 21 2008
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