A151090 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, -1), (-1, -1, 0), (0, 0, 1), (0, 1, 0), (1, 1, 1)}.

1, 3, 11, 43, 175, 731, 3111, 13427, 58591, 257947, 1143943, 5104419, 22896303, 103169899, 466725143, 2118787187, 9648585791, 44060516667, 201709358631, 925531659971, 4255568177615, 19604179972363, 90468636882231, 418164385032723, 1935725673812575, 8973094439246811, 41648668456569671

Offset: 0

Author

Manuel Kauers, Nov 18 2008

Keywords

• nonn
• walk

Links

• A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008-2009.
• Xiang-Ke Chang, X.-B. Hu, H. Lei, Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.

Crossrefs

Cf. A151292 (similar in N^2).

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, -1 + k, -1 + n] + aux[i, -1 + j, k, -1 + n] + aux[i, j, -1 + k, -1 + n] + aux[1 + i, 1 + j, k, -1 + n] + aux[1 + i, 1 + j, 1 + k, -1 + n]]; Table[Sum[aux[i, j, k, n], {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 10}]