A151314 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, 1)}.

1, 2, 11, 49, 277, 1479, 8679, 49974, 301169, 1805861, 11097563, 68225081, 425527103, 2660509721, 16787151965, 106242633509, 676589551793, 4321724272449, 27729861453735, 178418992948065, 1151938855186131, 7455729526078989, 48388319643548481, 314727421388892459, 2051665321667625351

Offset

0,2

Links

Table of n, a(n) for n=0..24.

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.

Bostan, Alin ; Chyzak, Frédéric; van Hoeij, Mark; Kauers, Manuel; Pech, Lucien 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 [math.CO], 2008-2009.

M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.

Formula

G.f.: Int(-1+Int((2*x+1)*(5*x+1)*(-2+Int(6*(1-2*x-35*x^2)^(3/2)*((1+12*x^2)*(1808*x^5+1084*x^4+540*x^3+107*x^2-132*x-5)*hypergeom([7/4, 9/4],[2],64*(x^2+x+1)*x^2/(1+12*x^2)^2)-14*x*(636*x^6+2104*x^5+811*x^4-500*x^3-403*x^2-55*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)/((x-1)*x). - Mark van Hoeij, Aug 16 2014

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, 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}]

Crossrefs

Sequence in context: A265610 A036996 A262296 * A316263 A342906 A187000

Adjacent sequences: A151311 A151312 A151313 * A151315 A151316 A151317

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved