A025237 Expansion of (1 -x -sqrt(1-2*x-11*x^2))/(6*x^2).

1, 1, 4, 10, 37, 121, 451, 1639, 6259, 23923, 93502, 367852, 1465003, 5874103, 23740276, 96503554, 394542379, 1620716251, 6687296308, 27700303510, 115152607831, 480244735171, 2008802728819, 8425318166635, 35425680021397, 149296062114181, 630526903497706, 2668194946794124, 11311786743536125

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

a(n) = (1/3)*s(n+2), where s = A014432.

Also, number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of n steps taken from {(-1, -1), (-1, 0), (-1, 1), (0, 1), (1, 1)}. - Manuel Kauers, Nov 18 2008

Also, number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, -1, 1), (-1, 0, 1), (-1, 1, 1), (0, 0, -1), (1, 1, 0)}. - Manuel Kauers, Nov 18 2008

Reversion of x/(1+x+3x^2). Hankel transform is 3^C(n+1,2) [A047656(n+1)]. - Paul Barry, Sep 07 2009

Examples

			G.f.: 1 + x + 4*x^2 + 10*x^3 + 37*x^4 + 121*x^5 + 451*x^6 + 1639*x^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008.
• Mireille Bousquet-Mélou and Marni Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008.
• Stefano Capparelli and Alberto Del Fra, Dyck Paths, Motzkin Paths, and the Binomial Transform, Journal of Integer Sequences, 18 (2015), #15.8.5.
• Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
• Serkan Demiriz, Adem Şahin, and Sezer Erdem, Some topological and geometric properties of novel generalized Motzkin sequence spaces, Rendiconti Circ. Mat. Palermo Ser. 2 (2025) Vol. 74, No. 136. See p. 4.
• Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

Crossrefs

Cf. A217275.

Programs

Mathematica

CoefficientList[Series[(1 - x - Sqrt[1 - 2*x - 11*x^2])/(6*x^2), {x, 0, 50}], x] (* G. C. Greubel, Feb 07 2017 *)

PARI

{a(n) = polcoeff((1 - x - sqrt(1 - 2*x - 11*x^2 + x^3*O(x^n))) / (6*x^2), n)}; /* Michael Somos, Sep 23 2003 */

Formula

From Paul Barry, Sep 07 2009: (Start)

G.f.: 1/(1-x-3x^2/(1-x-3x^2/(1-x-3x^2/(1-... (continued fraction);

a(n) = Sum_{k=0..floor(n/2)} binomial(n,2k)*3^k*A000108(k). (End)

D-finite with recurrence: (n+2)*a(n) - (2*n+1)*a(n-1) + 11*(1-n)*a(n-2) = 0. - R. J. Mathar, Nov 15 2011

a(n) ~ (1+2*sqrt(3))^(n+3/2)/(2*sqrt(Pi)*3^(3/4)*n^(3/2)). - Vaclav Kotesovec, Sep 29 2012

G.f. A(x) satisfies: A(x) = 1 + x * (1 + 3*x*A(x)^2) / (1 - x). - Ilya Gutkovskiy, Jun 30 2020

Extensions

Edited by N. J. A. Sloane, Nov 28 2008