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%I #65 Jun 27 2023 10:25:51
%S 1,2,11,80,665,5980,56637,556512,5620485,57985070,608462470,
%T 6474009360,69682358811,757366074080,8300675584120,91634565938880,
%U 1018002755977245,11372548404732930,127677890035721025,1439777493407492640
%N a(n) = 2*binomial(5*n+1,n)/(4*n+2).
%C A quadrisection of A118968.
%C If y = x + 2*x^3 + x^5, the series reversion is x = y - 2*y^3 + 11*y^5 - 80*y^7 + 665*y^9 - ... - _R. J. Mathar_, Sep 29 2012
%H Vincenzo Librandi, <a href="/A118969/b118969.txt">Table of n, a(n) for n = 0..100</a>
%H Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, <a href="https://arxiv.org/abs/2204.14023">Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k</a>, arXiv:2204.14023 [math.CO], 2022.
%H Karol A. Penson and Karol Zyczkowski, <a href="http://arxiv.org/abs/1103.3453/">Product of Ginibre matrices: Fuss-Catalan and Raney distribution</a>, arXiv:1103.3453 [math-ph], 2011.
%H Karol A. Penson and Karol Zyczkowski, <a href="http://dx.doi.org/10.1103/PhysRevE.83.061118">Product of Ginibre matrices: Fuss-Catalan and Raney distribution</a>, Phys. Rev. E 83, 061118 (2011).
%H Sheng-liang Yang and Mei-yang Jiang, <a href="https://journal.lut.edu.cn/EN/abstract/abstract528.shtml">Pattern avoiding problems on the hybrid d-trees</a>, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)
%F From _Gary W. Adamson_, Aug 11 2011: (Start)
%F a(n) is sum of top row terms in M^n, where M is an infinite square production matrix with the tetrahedral series in each column (A000292), as follows:
%F 1, 1, 0, 0, 0, 0, ...
%F 4, 1, 1, 0, 0, 0, ...
%F 10, 10, 4, 1, 0, 0, ...
%F 20, 20, 10, 4, 1, 0, ...
%F 35, 35, 20, 10, 4, 1, ...
%F ... (End)
%F G.f.: hypergeom([1/5, 2/5, 3/5, 4/5],[1/2, 3/4, 5/4], 3125*x/256)^2. - _Mark van Hoeij_, Apr 19 2013
%F a(n) = 2*binomial(5n+1,n-1)/n for n>0, a(0)=1. - _Bruno Berselli_, Jan 19 2014
%F D-finite with recurrence 8*n*(4*n+1)*(2*n+1)*(4*n-1)*a(n) - 5*(5*n+1)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1) = 0. - _R. J. Mathar_, Oct 10 2014
%F G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(x)^2)^2. - _Ilya Gutkovskiy_, Nov 13 2021
%e a(3) = 80 = sum of top row terms in M^n = (35 + 35 + 9 + 1).
%t Table[2*Binomial[5n+1,n]/(4n+2),{n,0,20}] (* _Harvey P. Dale_, Aug 21 2011 *)
%o (Magma) [2*Binomial(5*n+1,n)/(4*n+2): n in [0..20]]; // _Vincenzo Librandi_, Aug 12 2011
%o (PARI) a(n)=2*binomial(5*n+1,n)/(4*n+2); \\ _Joerg Arndt_, Apr 20 2013
%Y Cf. A000292, A118968, A006013, A233832, A234505, A234868, A002294.
%K nonn,easy
%O 0,2
%A _Paul Barry_, May 07 2006
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