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%I #17 Aug 06 2024 04:57:56
%S 1,2,7,26,105,444,1944,8728,39999,186266,879108,4195762,20217136,
%T 98220992,480623748,2366735352,11720044199,58329063714,291607308864,
%U 1463803013168,7375252291592,37285645521840,189084169866584,961635234827346,4903573489214352,25065602340246704,128419743299166764
%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, 0), (0, -1), (0, 1), (1, -1), (1, 0)}.
%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 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.: (1/128)*(1-62*x)/x^2+(2*x+1)/x^2*(-1/128+Int(x/((2*x+1)*(1-6*x))^(3/2)*(-3-1/2*Int(((1-6*x)/(2*x+1)/(1-8*x^2)^3)^(1/2)*((24*x^3+4*x^2+4*x+1)/x^2*hypergeom( [1/4,3/4],[1],64/(8*x^2-1)^2*(2*x+1)*x^3)+12*x*(8*x^2+4*x+1)*(1+14*x+16*x^2)/(1-8*x^2)^2*hypergeom([5/4, 7/4],[2],64/(8*x^2-1)^2*(2*x+1)*x^3)),x)),x)). [_Mark van Hoeij_, Oct 13 2009]
%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, 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] + 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