A014432 -id:A014432 - cp's OEIS frontend

A025237 Expansion of (1 -x -sqrt(1-2x-11x^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

Clark Kimberling

nonn

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

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

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.

Cf. A217275.

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

{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 */

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

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

A374488 Expansion of 1/(1 - 2x - 11x^2)^(3/2).

Original entry on oeis.org

1, 3, 24, 100, 555, 2541, 12628, 59004, 281655, 1315765, 6171132, 28692456, 133315273, 616780815, 2848833120, 13124483344, 60364983987, 277142478921, 1270586298520, 5817063737100, 26600252408961, 121501917998263, 554429553154044, 2527595449990500

Offset: 0

Seiichi Manyama, Jul 09 2024

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A102839, A102840, A374487.

Cf. A014432, A025237, A084603.

Module[{x}, CoefficientList[Series[1/(1 - (11*x + 2)*x)^(3/2), {x, 0, 25}], x]] (* Paolo Xausa, Aug 25 2025 *)

a(n) = binomial(n+2, 2)*sum(k=0, n\2, 3^k*binomial(n, 2*k)*binomial(2*k, k)/(k+1));

a(0) = 1, a(1) = 3; a(n) = ((2*n+1)*a(n-1) + 11*(n+1)*a(n-2))/n.
a(n) = binomial(n+2,2) * A025237(n).
From Seiichi Manyama, Aug 20 2025: (Start)
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 3^k * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (1/2)^k * (11/2)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k). (End)
a(n) ~ sqrt(n) * (1 + 2*sqrt(3))^(n + 3/2) / (4 * 3^(3/4) * sqrt(Pi)). - Vaclav Kotesovec, Aug 21 2025

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A025236 Duplicate of A014432.

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

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A374488 Expansion of 1/(1 - 2x - 11x^2)^(3/2).

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cp's OEIS Frontend

A025236 Duplicate of A014432.

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

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A374488 Expansion of 1/(1 - 2*x - 11*x^2)^(3/2).

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

A374488 Expansion of 1/(1 - 2x - 11x^2)^(3/2).