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%I A151331 #35 Aug 06 2024 04:51:22
%S A151331 1,3,18,105,684,4550,31340,219555,1564080,11271876,82059768,602215614,
%T A151331 4450146624,33076800900,247096919784,1854031805769,13965171795432,
%U A151331 105550935041552,800212396412000,6083310009164388,46360755048406656,354109165968099048,2710276234371255888,20782807250217463750
%N A151331 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, 0), (1, 1)}.
%H A151331 Alois P. Heinz, <a href="/A151331/b151331.txt">Table of n, a(n) for n = 0..500</a>
%H A151331 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 A151331 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 A151331 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 A151331 A. Bostan and M. Kauers, <a href="http://arxiv.org/abs/0811.2899">Automatic Classification of Restricted Lattice Walks</a>, ArXiv 0811.2899, 2008.
%H A151331 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, 2008.
%F A151331 G.f.: (1/x)*Int(-(16*x^2+24*x-1)/(1+4*x)^5*hypergeom([5/4, 5/4],[2],-2*x/(x+1/4)^4*(x+1)*(x-1/8)),x). - _Mark van Hoeij_, Oct 13 2009
%F A151331 G.f.: Int(hypergeom([3/2,3/2],[2],16*x*(1+x)/(1+4*x)^2)/(1+4*x)^3,x)/x. - _Mark van Hoeij_, Aug 14 2014
%t A151331 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, -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}]
%t A151331 CoefficientList[Series[Integrate[HypergeometricPFQ[{3/2,3/2},{2},16*x*(1+x)/(1+4*x)^2]/(1+4*x)^3,x]/x,{x,0,20}],x] (* _Vaclav Kotesovec_, Aug 16 2014, after _Mark van Hoeij_ *)
%K A151331 nonn,walk
%O A151331 0,2
%A A151331 _Manuel Kauers_, Nov 18 2008

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