A148005 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, 1, -1), (1, 0, 0)}.

1, 1, 2, 3, 8, 15, 44, 91, 286, 633, 2072, 4796, 16180, 38727, 133548, 327895, 1150226, 2881857, 10247072, 26099008, 93830568, 242264113, 878987980, 2295723288, 8393889812, 22139557300, 81484257064, 216757035756, 802324361696, 2150196732767, 7998562017700, 21576806128743, 80615634684738

Offset

0,3

Links

Table of n, a(n) for n=0..32.

A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008.

Formula

G.f.: (1-4*x)^(1/2)*(2*x-1)*Int(x*(1+((3*x-1)*(2*x+1)*hypergeom([1/2, 3/2],[3],16*x^2/(1+4*x^2))-2*x^2*(2*x-1)*hypergeom([3/2, 3/2],[4],16*x^2/(1+4*x^2)))/((1+4*x^2)^(1/2)*(2*x-1)^2))/(1-4*x)^(3/2),x)/x^3. - Mark van Hoeij, Aug 27 2014

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, j, k, -1 + n] + aux[1 + i, -1 + j, 1 + 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}]

Crossrefs

Sequence in context: A300443 A321231 A156802 * A151394 A148006 A148007

Adjacent sequences: A148002 A148003 A148004 * A148006 A148007 A148008

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved