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A148548
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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, -1, -1), (-1, -1, 0), (-1, 1, 0), (-1, 1, 1), (1, 0, 0)}.
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0
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1, 1, 3, 5, 21, 44, 179, 405, 1835, 4490, 19779, 49829, 232721, 610882, 2788715, 7427785, 35253935, 96368290, 448865903, 1237270005, 5931202577, 16649102672, 78553909015, 221640074309, 1069716764575, 3059624598509, 14565629603329, 41799007929725, 202728660743415, 588026037157535, 2817795677054636
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OFFSET
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0,3
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LINKS
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MAPLE
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H := hypergeom([1/4, 7/12], [4/3], 64*x*(x-1)*(3*x+1)^2/(1-5*x)^4);
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;
Ord := 40; seriesInt := proc(A, x) global Ord; int(series(A, x, Ord), x) end:
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):
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);
# So the o.g.f. can be expressed in terms of 3F2 and 2F1 functions. - Mark van Hoeij, Apr 10 2012
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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STATUS
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approved
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