|
%I #20 Dec 11 2022 14:29:37
%S 1,2,6,21,76,290,1148,4627,19038,79554,336112,1435522,6184704,
%T 26838474,117247440,515135847,2274656290,10090187786,44940868940,
%U 200897459804,901082056408,4053912011322,18289272082952,82724956638634,375064515961744,1704237546984170,7759645793395368,35398085705004882
%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), (1, -1), (1, 0)}.
%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 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(Int(x*(3*x+1)*(-4+Int(2*(1-2*x-15*x^2)^(3/2)*((4*x^2-1)*(92*x^4+76*x^3+43*x^2+6*x+1)*hypergeom([7/4, 9/4],[2],64*x^3*(1+x)/(1-4*x^2)^2)+14*x^3*(10*x+1)*(18*x^3+7*x^2+3*x-1)*hypergeom([9/4, 11/4],[3],64*x^3*(1+x)/(1-4*x^2)^2))/((3*x+1)*(1-4*x^2)^(9/2)*x^2),x))/(1-2*x-15*x^2)^(5/2),x),x)/x^2. - _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, j, -1 + n] + aux[-1 + 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
|
