A151287 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)}.

1, 2, 6, 21, 76, 290, 1148, 4627, 19038, 79554, 336112, 1435522, 6184704, 26838474, 117247440, 515135847, 2274656290, 10090187786, 44940868940, 200897459804, 901082056408, 4053912011322, 18289272082952, 82724956638634, 375064515961744, 1704237546984170, 7759645793395368, 35398085705004882

Offset

0,2

Links

Table of n, a(n) for n=0..27.

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(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

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, 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}]

Crossrefs

Sequence in context: A116772 A131792 A144904 * A294822 A294823 A294824

Adjacent sequences: A151284 A151285 A151286 * A151288 A151289 A151290

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved