A151265 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, 0), (0, -1), (1, 1)}.

1, 1, 3, 7, 17, 47, 125, 333, 939, 2597, 7183, 20505, 57859, 163201, 469795, 1341775, 3830529, 11092823, 31940165, 91927379, 267406401, 774447755, 2242022721, 6544458687, 19036737381, 55354815639, 162028272261, 472921269031, 1379896701413, 4048204328607, 11848014062621, 34665885291485

Offset

0,3

References

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 997.

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; https://specfun.inria.fr/bostan/HDR.pdf

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

M. Bousquet-Mélou and M. Mishna, 2008. Walks with small steps in the quarter plane, ArXiv 0810.4387.

A. Bostan, Computer Algebra for Lattice Path Combinatorics, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.

A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, ArXiv 0811.2899.

Formula

a(n) == 1 (mod 2). - Alois P. Heinz, Jul 12 2021

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[i, 1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]

Crossrefs

Column k=2 of A335570.

Sequence in context: A113985 A309302 A320774 * A351437 A179944 A071985

Adjacent sequences: A151262 A151263 A151264 * A151266 A151267 A151268

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved