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A148934
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), (0, 1, -1), (1, -1, 0), (1, 0, 1)}.
0
1, 1, 3, 9, 28, 100, 365, 1365, 5334, 21126, 85092, 349272, 1447776, 6067776, 25694097, 109597917, 470986802, 2037484306, 8862487634, 38753956670, 170265385368, 751193676168, 3327282964304, 14790351109976, 65960626714384, 295064784866320, 1323639286393920, 5953246253340960, 26840923720400640
OFFSET
0,3
COMMENTS
Also, number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of n steps taken from {(-1, -1), (-1, 1), (0, -1), (0, 1), (1, 0)}.
LINKS
A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, ArXiv 0811.2899, 2008.
M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, ArXiv 0810.4387, 2008.
FORMULA
G.f.: Int(-1/2+((1/2+2*x+6*x^2)*hypergeom([1/4, 3/4],[1],64*x^3*(1+x)/((2*x+1)^2*(2*x-1)^2))-6*x^2*(1+x)*hypergeom([1/4, 3/4],[2],64*x^3*(1+x)/((2*x+1)^2*(2*x-1)^2)))/(1-4*x^2)^(3/2),x)/x^2. - Mark van Hoeij, Aug 20 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, -1 + k, -1 + n] + aux[-1 + i, 1 + j, k, -1 + n] + aux[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: A148932 A151421 A148933 * A148935 A148936 A121656
KEYWORD
nonn,walk
AUTHOR
Manuel Kauers, Nov 18 2008
STATUS
approved