A118970 a(n) = 3*binomial(5n+2,n)/(4n+3).

1, 3, 18, 136, 1155, 10530, 100688, 996336, 10116873, 104819165, 1103722620, 11777187240, 127067830773, 1383914371728, 15194457001440, 167996704221280, 1868870731122405, 20903064321375315, 234927317665726686

Offset: 0

Author

Paul Barry, May 07 2006

Keywords

• easy
• nonn

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.

Crossrefs

Cf. A118969, A002294, A000108, A163456.

Programs

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 *)

PARI

a(n)=3*binomial(5*n+2,n)/(4*n+3); \\ Joerg Arndt, Apr 23 2013

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