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Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, -1), (-1, 1), (-1, 0), (0, -1), (0, 1), (1, -1), (1, 1)}.
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%I #23 Aug 06 2024 04:58:53

%S 1,2,11,49,277,1479,8679,49974,301169,1805861,11097563,68225081,

%T 425527103,2660509721,16787151965,106242633509,676589551793,

%U 4321724272449,27729861453735,178418992948065,1151938855186131,7455729526078989,48388319643548481,314727421388892459,2051665321667625351

%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, -1), (-1, 1), (-1, 0), (0, -1), (0, 1), (1, -1), (1, 1)}.

%H A. Bostan, <a href="https://citeseerx.ist.psu.edu/pdf/749aef4c6f3668e652b5074e5268346ccecc88c9">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 Bostan, Alin ; Chyzak, Frédéric; van Hoeij, Mark; Kauers, Manuel; Pech, Lucien <a href="https://doi.org/10.1016/j.ejc.2016.10.010">Hypergeometric expressions for generating functions of walks with small steps in the quarter plane.</a> Eur. J. Comb. 61, 242-275 (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)*(5*x+1)*(-2+Int(6*(1-2*x-35*x^2)^(3/2)*((1+12*x^2)*(1808*x^5+1084*x^4+540*x^3+107*x^2-132*x-5)*hypergeom([7/4, 9/4],[2],64*(x^2+x+1)*x^2/(1+12*x^2)^2)-14*x*(636*x^6+2104*x^5+811*x^4-500*x^3-403*x^2-55*x-10)*hypergeom([9/4, 11/4],[3],64*(x^2+x+1)*x^2/(1+12*x^2)^2))/((5*x+1)*(1+12*x^2)^(9/2)*(2*x+1)^2),x))/(1-2*x-35*x^2)^(5/2),x),x)/((x-1)*x). - _Mark van Hoeij_, Aug 16 2014

%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, 1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + 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