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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)}.
0

%I #14 Aug 27 2014 15:09:23

%S 1,1,3,8,25,77,257,853,2946,10178,36005,127635,459307,1657395,6042087,

%T 22089132,81346477,300383933,1115251580,4151032922,15515158110,

%U 58122268630,218459019113,822782085889,3107215915981,11755843600909,44576827929909,169308082883825,644271153070229,2455271638906533

%N 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)}.

%C 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)}.

%H A. Bostan and M. Kauers, <a href="http://arxiv.org/abs/0811.2899">Automatic Classification of Restricted Lattice Walks</a>, arXiv:0811.2899 [math.CO], 2008.

%H M. Bousquet-Mélou and M. Mishna, <a href="http://arxiv.org/abs/0810.4387">Walks with small steps in the quarter plane</a>, ArXiv 0810.4387 [math.CO], 2008.

%F 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

%t 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}]

%K nonn,walk

%O 0,3

%A _Manuel Kauers_, Nov 18 2008

%E Edited by _N. J. A. Sloane_, Nov 28 2008 at the suggestion of _R. J. Mathar_