A118969 a(n) = 2*binomial(5*n+1,n)/(4*n+2).

1, 2, 11, 80, 665, 5980, 56637, 556512, 5620485, 57985070, 608462470, 6474009360, 69682358811, 757366074080, 8300675584120, 91634565938880, 1018002755977245, 11372548404732930, 127677890035721025, 1439777493407492640

Offset: 0

Author

Paul Barry, May 07 2006

Keywords

• nonn
• easy

Comments

A quadrisection of A118968.

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

Examples

			a(3) = 80 = sum of top row terms in M^n = (35 + 35 + 9 + 1).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• 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.
• Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices: Fuss-Catalan and Raney distribution, arXiv:1103.3453 [math-ph], 2011.
• Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices: Fuss-Catalan and Raney distribution, Phys. Rev. E 83, 061118 (2011).
• Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.
• Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)

Crossrefs

Cf. A000292, A118968, A006013, A233832, A234505, A234868, A002294.

Programs

Magma

[2*Binomial(5*n+1,n)/(4*n+2): n in [0..20]]; // Vincenzo Librandi, Aug 12 2011

Mathematica

Table[2*Binomial[5n+1,n]/(4n+2),{n,0,20}] (* Harvey P. Dale, Aug 21 2011 *)

PARI

a(n)=2*binomial(5*n+1,n)/(4*n+2); \\ Joerg Arndt, Apr 20 2013

Formula

From Gary W. Adamson, Aug 11 2011: (Start)

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:

1, 1, 0, 0, 0, 0, ...

4, 1, 1, 0, 0, 0, ...

10, 10, 4, 1, 0, 0, ...

20, 20, 10, 4, 1, 0, ...

35, 35, 20, 10, 4, 1, ...

... (End)

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

a(n) = 2*binomial(5n+1,n-1)/n for n>0, a(0)=1. - Bruno Berselli, Jan 19 2014

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

G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(x)^2)^2. - Ilya Gutkovskiy, Nov 13 2021