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%I #21 Nov 17 2022 18:15:59
%S 1,1,3,5,21,44,179,405,1835,4490,19779,49829,232721,610882,2788715,
%T 7427785,35253935,96368290,448865903,1237270005,5931202577,
%U 16649102672,78553909015,221640074309,1069716764575,3059624598509,14565629603329,41799007929725,202728660743415,588026037157535,2817795677054636
%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, -1, -1), (-1, -1, 0), (-1, 1, 0), (-1, 1, 1), (1, 0, 0)}.
%H A. Bostan and M. Kauers, <a href="http://arxiv.org/abs/0811.2899">Automatic Classification of Restricted Lattice Walks</a>, arXiv:0811.2899v2 [math.CO], 2008.
%p H := hypergeom([1/4, 7/12], [4/3], 64*x*(x-1)*(3*x+1)^2/(1-5*x)^4);
%p U := ((1816*x^5 - 3192*x^4 + 382*x^3 + 140*x^2 - 13*x - 1)*hypergeom([1/4, 3/4, 7/6], [1, 4/3], 256*x^4) + 3*(2*x-1)*(3*x+1)*(16*x^2+1)*(212*x^3 - 82*x^2 + 4*x + 1)*hypergeom([3/4, 7/6, 5/4], [1, 4/3], 256*x^4) - 16*x^3*(564*x^4 + 142*x^3 - 89*x^2 + 8*x - 1)*hypergeom([3/4, 7/6, 5/4], [4/3, 2], 256*x^4))/((5*x-1)*(2*x-1))^3;
%p Ord := 40; seriesInt := proc(A,x) global Ord; int(series(A,x,Ord),x) end:
%p S := seriesInt((1-5*x)*x^(-4/3)*(1+3*x)^(-5/3)*(1-5*x)^(4/3)*(1-x)^(-4/3)*H^(-2)*(1-seriesInt(H*U,x)),x):
%p ogf := series((1-2*x)/(4*x^2) + (1/12) * (1-2*x)*x^(-5/3)*(1+3*x)^(2/3)*(1-5*x)^(-4/3)*(1-x)^(1/3)*H*S,x,Ord);
%p # So the o.g.f. can be expressed in terms of 3F2 and 2F1 functions. - _Mark van Hoeij_, Apr 10 2012
%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, j, k, -1 + n] + aux[1 + i, -1 + j, -1 + k, -1 + n] + aux[1 + i, -1 + j, 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}]
%K nonn,walk
%O 0,3
%A _Manuel Kauers_, Nov 18 2008
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