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

Author

Manuel Kauers, Nov 18 2008

Keywords

• nonn
• walk

References

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

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.
• 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, 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.

Crossrefs

Column k=2 of A335570.

Programs

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}]

Formula

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