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A148790
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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)}.
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0
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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
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OFFSET
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0,3
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COMMENTS
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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)}.
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LINKS
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FORMULA
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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
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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, -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}]
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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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EXTENSIONS
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STATUS
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approved
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