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)}.
1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11271876, 82059768, 602215614, 4450146624, 33076800900, 247096919784, 1854031805769, 13965171795432, 105550935041552, 800212396412000, 6083310009164388, 46360755048406656, 354109165968099048, 2710276234371255888, 20782807250217463750
Offset: 0
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..500
- A. Bostan, Computer Algebra for Lattice Path Combinatorics, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.
- Alin Bostan, Calcul Formel pour la Combinatoire des Marches [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d'Informatique de Paris Nord, Université Paris 13, December 2017.
- Alin Bostan, Frédéric Chyzak, Mark van Hoeij, Manuel Kauers, and Lucien Pech, Hypergeometric expressions for generating functions of walks with small steps in the quarter plane. Eur. J. Comb. 61, 242-275 (2017).
- A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, ArXiv 0811.2899, 2008.
- M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, ArXiv 0810.4387, 2008.
Programs
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Mathematica
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}] 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 *)
Formula
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
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