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a(n) = 3*binomial(5n+2,n)/(4n+3).
13

%I #45 Oct 02 2024 01:56:28

%S 1,3,18,136,1155,10530,100688,996336,10116873,104819165,1103722620,

%T 11777187240,127067830773,1383914371728,15194457001440,

%U 167996704221280,1868870731122405,20903064321375315,234927317665726686

%N a(n) = 3*binomial(5n+2,n)/(4n+3).

%C A quadrisection of A118968.

%C Convolved with A118969 (1, 2, 11, 80, 665, ...) = A002294: (1, 5, 35, 285, 2530, ...) - _Gary W. Adamson_, Nov 07 2011

%D R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B33.

%H Michael De Vlieger, <a href="/A118970/b118970.txt">Table of n, a(n) for n = 0..924</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 Henri Muehle and Philippe Nadeau, <a href="https://arxiv.org/abs/1803.00540">A Poset Structure on the Alternating Group Generated by 3-Cycles</a>, arXiv:1803.00540 [math.CO], 2018.

%F G.f.: F^3 where F is the g.f. of A002294. - _Mark van Hoeij_, Apr 23 2013

%F 8*n*(4*n+1)*(2*n+1)*(4*n+3)*a(n) -5*(5*n+1)*(5*n+2)*(5*n-2)*(5*n-1)*a(n-1)=0. - _R. J. Mathar_, Dec 02 2014

%F From _Peter Bala_, Oct 08 2015: (Start)

%F O.g.f. A(x) = (1/x) * series reversion ( x/C(x)^3 ), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108.

%F (1/3)*x*A'(x)/A(x) = x + 9*x^2 + 91*x^3 + 969*x^4 + ... is the o.g.f. for A163456. (End)

%F E.g.f.: hypergeom([3/5, 4/5, 6/5, 7/5], [1, 5/4, 3/2, 7/4], (5^5/4^4)*x). - _Stefano Spezia_, Oct 01 2024

%p ogf := series(RootOf(A = 1 + x * A^5,A)^3, x=0, 30); # _Mark van Hoeij_, Apr 22 2013

%t Array[3 Binomial[5 # + 2, #]/(4 # + 3) &, 19, 0] (* _Michael De Vlieger_, May 30 2018 *)

%t CoefficientList[Series[HypergeometricPFQ[{3/5,4/5,6/5,7/5},{1,5/4,3/2,7/4},(5^5/4^4)x],{x,0,18}],x]Range[0,18]! (* _Stefano Spezia_, Oct 01 2024 *)

%o (PARI) a(n)=3*binomial(5*n+2,n)/(4*n+3); \\ _Joerg Arndt_, Apr 23 2013

%Y Cf. A118969, A002294, A000108, A163456.

%K easy,nonn

%O 0,2

%A _Paul Barry_, May 07 2006