A151261 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), (1, 0)}.

1, 1, 3, 5, 17, 34, 121, 265, 969, 2246, 8351, 20118, 75676, 187524, 711369, 1801161, 6876749, 17710990, 67962103, 177474154, 683797608, 1806245948, 6982719679, 18624035846, 72200420096, 194169343564, 754537816956, 2043762871460, 7958294450084, 21691396049160, 84615387463873, 231907336128537

Offset

0,3

Links

Table of n, a(n) for n=0..31.

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.

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.: (1-2*x)*(1+(-1+2*Int(((1-24*x^3)*hypergeom([1/2, 3/2],[2],16*x^2/(1+4*x^2))+18*x^2*(2*x-1)*hypergeom([1/2, 5/2],[3],16*x^2/(1+4*x^2)))/((1-4*x)^(1/2)*(1-2*x)^2*(1+4*x^2)^(1/2)),x))/(1-4*x)^(1/2))/(4*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[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: A056816 A156761 A371927 * A148504 A148505 A148506

Adjacent sequences: A151258 A151259 A151260 * A151262 A151263 A151264

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved