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A025237 Expansion of (1 -x -sqrt(1-2*x-11*x^2))/(6*x^2). 3

%I

%S 1,1,4,10,37,121,451,1639,6259,23923,93502,367852,1465003,5874103,

%T 23740276,96503554,394542379,1620716251,6687296308,27700303510,

%U 115152607831,480244735171,2008802728819,8425318166635,35425680021397,149296062114181,630526903497706,2668194946794124,11311786743536125

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

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

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

%H G. C. Greubel, <a href="/A025237/b025237.txt">Table of n, a(n) for n = 0..1000</a>

%H A. Bostan and M. Kauers, <a href="http://arxiv.org/abs/0811.2899">Automatic Classification of Restricted Lattice Walks</a>, arXiv:0811.2899 [math.CO], 2008.

%H M. Bousquet-Mélou and M. 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 S. Capparelli, A. 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 Xiang-Ke Chang, X.-B. Hu, H. Lei, Y.-N. Yeh, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p8">Combinatorial proofs of addition formulas</a>, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.

%H 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 From _Paul Barry_, Sep 07 2009: (Start)

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

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

%F Conjecture: (n+2)*a(n) - (2*n+1)*a(n-1) + 11*(1-n)*a(n-2) = 0. - _R. J. Mathar_, Nov 15 2011

%F a(n) ~ (1+2*sqrt(3))^(n+3/2)/(2*sqrt(Pi)*3^(3/4)*n^(3/2)). - _Vaclav Kotesovec_, Sep 29 2012

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

%t 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 (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 Cf. A217275.

%K nonn

%O 0,3

%A _Clark Kimberling_

%E Edited by _N. J. A. Sloane_, Nov 28 2008

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Last modified February 19 22:38 EST 2019. Contains 320328 sequences. (Running on oeis4.)