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%I #19 May 11 2020 09:43:23
%S 1,3,16,86,509,3065,19088,120401,771758,4991255,32580974,214030883,
%T 1414515215,9392000723,62625868280,419056150250,2812961273801,
%U 18933686864261,127752378798614,863851325189957,5852685174009545,39721858765234379,270018313988209400,1838154015475958213
%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), (1, -1), (1, 0), (1, 1)}.
%H A. Bostan, <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.410.1160&rep=rep1&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)*(5*x+1)*(10+Int(12*(1-2*x-35*x^2)^(3/2)*((1+12*x^2)*(736*x^5+2208*x^4+1096*x^3-44*x^2+44*x+1)*hypergeom([7/4, 9/4],[2],64*(x^2+x+1)*x^2/(1+12*x^2)^2)-7*x*(1824*x^6+2496*x^5+1288*x^4+452*x^3+420*x^2+53*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)/((2*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, j, -1 + n] + aux[-1 + 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
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