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A151326 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, 0), (0, -1), (0, 1), (1, -1), (1, 0), (1, 1)}. 0

%I

%S 1,3,15,74,392,2116,11652,64967,365759,2074574,11836868,67863126,

%T 390625864,2256008404,13066434500,75864388248,441412162944,

%U 2573133492918,15024422196084,87856077334712,514419919265976,3015635977208784,17697278566338720,103958103858046662,611220388506542904

%N Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, 0), (0, -1), (0, 1), (1, -1), (1, 0), (1, 1)}.

%H A. Bostan, <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.410.1160&amp;rep=rep1&amp;type=pdf">Computer Algebra for Lattice Path Combinatorics</a>, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.

%H Alin Bostan, <a href="https://specfun.inria.fr/bostan/HDR.pdf">Calcul Formel pour la Combinatoire des Marches</a> [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d’Informatique de Paris Nord, Université Paris 13, December 2017.

%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], 2008-2009.

%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], 2008-2009.

%F G.f.: Int(1+Int((2*x+1)*(6*x+1)*(6+Int(3*(1-4*x-12*x^2)^(3/2)*((8*x^2-1)*(96*x^3-248*x^2-150*x-13)*hypergeom([5/4, 7/4],[2],64*x^3*(2*x+1)/(8*x^2-1)^2)+5*(8*x^2+4*x+1)*(32*x^3-32*x^2-42*x-5)*hypergeom([7/4, 9/4],[2],64*x^3*(2*x+1)/(8*x^2-1)^2))/(2*(2*x+1)*(6*x+1)^2*(1-8*x^2)^(7/2)),x))/(1-4*x-12*x^2)^(5/2),x),x)/x. - _Mark van Hoeij_, Aug 16 2014

%F a(n) ~ [x^n]( (1-2*x)*(1+2*x)^(1/2)/(4*x*(1-6*x)^(1/2)) ), where [x^n](f) denotes the coefficient of x^n in the series expansion of f. - _Mark van Hoeij_, May 28 2020

%F a(n) ~ 2^(n+1) * 3^(n - 1/2) / sqrt(Pi*n) [Bostan and Kauers, p.13]. - _Vaclav Kotesovec_, May 29 2020

%t aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[-1 + i, j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]

%K nonn,walk

%O 0,2

%A _Manuel Kauers_, Nov 18 2008

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Last modified April 20 09:21 EDT 2021. Contains 343125 sequences. (Running on oeis4.)