OFFSET
0,2
COMMENTS
A quadrisection of A118968.
Convolved with A118969 (1, 2, 11, 80, 665, ...) = A002294: (1, 5, 35, 285, 2530, ...) - Gary W. Adamson, Nov 07 2011
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B33.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..924
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
Henri Muehle and Philippe Nadeau, A Poset Structure on the Alternating Group Generated by 3-Cycles, arXiv:1803.00540 [math.CO], 2018.
FORMULA
G.f.: F^3 where F is the g.f. of A002294. - Mark van Hoeij, Apr 23 2013
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
From Peter Bala, Oct 08 2015: (Start)
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.
(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)
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
MAPLE
ogf := series(RootOf(A = 1 + x * A^5, A)^3, x=0, 30); # Mark van Hoeij, Apr 22 2013
MATHEMATICA
Array[3 Binomial[5 # + 2, #]/(4 # + 3) &, 19, 0] (* Michael De Vlieger, May 30 2018 *)
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 *)
PROG
(PARI) a(n)=3*binomial(5*n+2, n)/(4*n+3); \\ Joerg Arndt, Apr 23 2013
CROSSREFS
KEYWORD
easy,nonn,changed
AUTHOR
Paul Barry, May 07 2006
STATUS
approved