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

%I A025237 #53 Jun 12 2025 14:12:13
%S A025237 1,1,4,10,37,121,451,1639,6259,23923,93502,367852,1465003,5874103,
%T A025237 23740276,96503554,394542379,1620716251,6687296308,27700303510,
%U A025237 115152607831,480244735171,2008802728819,8425318166635,35425680021397,149296062114181,630526903497706,2668194946794124,11311786743536125
%N A025237 Expansion of (1 -x -sqrt(1-2*x-11*x^2))/(6*x^2).
%C A025237 a(n) = (1/3)*s(n+2), where s = A014432.
%C A025237 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
%C A025237 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
%C A025237 Reversion of x/(1+x+3x^2). Hankel transform is 3^C(n+1,2) [A047656(n+1)]. - _Paul Barry_, Sep 07 2009
%H A025237 G. C. Greubel, <a href="/A025237/b025237.txt">Table of n, a(n) for n = 0..1000</a>
%H A025237 Alin Bostan and Manuel Kauers, <a href="http://arxiv.org/abs/0811.2899">Automatic Classification of Restricted Lattice Walks</a>, arXiv:0811.2899 [math.CO], 2008.
%H A025237 Mireille Bousquet-Mélou and Marni Mishna, <a href="http://arxiv.org/abs/0810.4387">Walks with small steps in the quarter plane</a>, arXiv:0810.4387 [math.CO], 2008.
%H A025237 Stefano Capparelli and Alberto Del Fra, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Capparelli/cap3.html">Dyck Paths, Motzkin Paths, and the Binomial Transform</a>, Journal of Integer Sequences, 18 (2015), #15.8.5.
%H A025237 Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, <a href="https://doi.org/10.37236/4793">Combinatorial proofs of addition formulas</a>, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
%H A025237 Serkan Demiriz, Adem Şahin, and Sezer Erdem, <a href="https://doi.org/10.1007/s12215-025-01243-3">Some topological and geometric properties of novel generalized Motzkin sequence spaces</a>, Rendiconti Circ. Mat. Palermo Ser. 2 (2025) Vol. 74, No. 136. See p. 4.
%H A025237 Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%F A025237 From _Paul Barry_, Sep 07 2009: (Start)
%F A025237 G.f.: 1/(1-x-3x^2/(1-x-3x^2/(1-x-3x^2/(1-... (continued fraction);
%F A025237 a(n) = Sum_{k=0..floor(n/2)} binomial(n,2k)*3^k*A000108(k). (End)
%F A025237 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
%F A025237 a(n) ~ (1+2*sqrt(3))^(n+3/2)/(2*sqrt(Pi)*3^(3/4)*n^(3/2)). - _Vaclav Kotesovec_, Sep 29 2012
%F A025237 G.f. A(x) satisfies: A(x) = 1 + x * (1 + 3*x*A(x)^2) / (1 - x). - _Ilya Gutkovskiy_, Jun 30 2020
%e A025237 G.f.: 1 + x + 4*x^2 + 10*x^3 + 37*x^4 + 121*x^5 + 451*x^6 + 1639*x^7 + ...
%t A025237 CoefficientList[Series[(1 - x - Sqrt[1 - 2*x - 11*x^2])/(6*x^2), {x, 0, 50}], x] (* _G. C. Greubel_, Feb 07 2017 *)
%o A025237 (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 */
%Y A025237 Cf. A217275.
%K A025237 nonn
%O A025237 0,3
%A A025237 _Clark Kimberling_
%E A025237 Edited by _N. J. A. Sloane_, Nov 28 2008