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A153231
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a(n) = 2^n * binomial(3n,n)/(2n+1).
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8
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1, 2, 12, 96, 880, 8736, 91392, 992256, 11075328, 126297600, 1465052160, 17233182720, 205074874368, 2464404045824, 29864206663680, 364535993597952, 4477993284993024, 55316387638149120, 686720560048373760, 8563155161736806400, 107206525476085432320
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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Michael De Vlieger, Table of n, a(n) for n = 0..889
Hsien-Kuei Hwang, Mihyun Kang, 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.
Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.
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FORMULA
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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.: 2F2(1/3,2/3; 1,3/2; 27*x/2). - Ilya Gutkovskiy, Nov 23 2017
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MATHEMATICA
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Table[2^n Binomial[3 n, n]/(2 n + 1), {n, 0, 25}] (* Vincenzo Librandi, Sep 08 2015 *)
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PROG
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(MAGMA) [2^n*Binomial(3*n, n)/(2*n+1): n in [0..30]]; // Vincenzo Librandi, Sep 08 2015
(PARI) a(n) = 2^n*binomial(3*n, n)/(2*n+1); \\ Altug Alkan, Sep 24 2018
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CROSSREFS
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Sequence in context: A322543 A213422 A307103 * A052564 A014297 A193425
Adjacent sequences: A153228 A153229 A153230 * A153232 A153233 A153234
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KEYWORD
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nonn,easy
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AUTHOR
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Yidong Sun (sydmath(AT)yahoo.com.cn), Dec 21 2008
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EXTENSIONS
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More terms from N. J. A. Sloane, Dec 21 2008
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STATUS
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approved
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