A148790 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, 0, 0), (1, -1, 0), (1, 0, 1), (1, 1, -1)}.

1, 1, 3, 8, 25, 77, 257, 853, 2946, 10178, 36005, 127635, 459307, 1657395, 6042087, 22089132, 81346477, 300383933, 1115251580, 4151032922, 15515158110, 58122268630, 218459019113, 822782085889, 3107215915981, 11755843600909, 44576827929909, 169308082883825, 644271153070229, 2455271638906533

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

Links

Table of n, a(n) for n=0..29.

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

M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, ArXiv 0810.4387 [math.CO], 2008.

Formula

G.f.: (Int((4*x^2+2*x-1)*(1/4+Int(x*((1-4*x)*(1-x)*hypergeom([1/2, 3/2],[2],16*x^2/(1+4*x^2))-x*(1-16*x+24*x^2+16*x^3)*hypergeom([1/2, 1/2],[2],16*x^2/(1+4*x^2)))/((1-4*x)^(1/2)*(1+4*x^2)^(1/2)*(4*x^2+2*x-1)^2),x))/((1-4*x)^(1/2)*x^2),x)-1/(4*x)-x)/((x-1)*x). - 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, -1 + j, 1 + k, -1 + n] + aux[-1 + i, j, -1 + k, -1 + n] + aux[-1 + i, 1 + j, k, -1 + n] + aux[1 + i, 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: A292884 A148789 A088327 * A148791 A148792 A007563

Adjacent sequences: A148787 A148788 A148789 * A148791 A148792 A148793

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Extensions

Edited by N. J. A. Sloane, Nov 28 2008 at the suggestion of R. J. Mathar

Status

approved