A151093 Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, -1, 0), (-1, 0, -1), (1, 0, 0), (1, 0, 1), (1, 1, 0)}.

1, 3, 11, 43, 177, 751, 3263, 14421, 64613, 292639, 1337295, 6156635, 28523245, 132861795, 621783139, 2921870183, 13780286589, 65201337703, 309392108399, 1471944790095, 7019341847821, 33545270688803, 160625969491971, 770511786145853, 3702199537395337, 17815681545802971, 85853558110277963

Offset: 0

Author

Manuel Kauers, Nov 18 2008

Keywords

• nonn
• walk

Links

• Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008-2009.
• Mélika Tebni, Fonctions de Bessel et cheminements en 3D, Dec 2024.

Programs

Mathematica

aux[i_Integer, j_Integer, k_Integer, n_Integer] := Which[Min[i, j, k, n] < 0 || Max[i, j, k] > n, 0, n == 0, KroneckerDelta[i, j, k, n], True, aux[i, j, k, n] = aux[-1 + i, -1 + j, k, -1 + n] + aux[-1 + i, j, -1 + k, -1 + n] + aux[-1 + i, j, k, -1 + n] + aux[1 + i, j, 1 + k, -1 + n] + aux[1 + i, 1 + j, k, -1 + n]]; Table[Sum[aux[i, j, k, n], {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 10}]

Formula

E.g.f.: (BesselI(0, 2*x) + BesselI(1, 2*x))^2*exp(x). - Mélika Tebni, Feb 07 2025