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A151326
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Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, 0), (0, -1), (0, 1), (1, -1), (1, 0), (1, 1)}
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0
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1, 3, 15, 74, 392, 2116, 11652, 64967, 365759, 2074574, 11836868, 67863126, 390625864, 2256008404, 13066434500, 75864388248, 441412162944, 2573133492918, 15024422196084, 87856077334712, 514419919265976, 3015635977208784, 17697278566338720, 103958103858046662, 611220388506542904
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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LINKS
| M. Bousquet-Melou and M. Mishna, 2008. Walks with small steps in the quarter plane, ArXiv 0810.4387.
A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, ArXiv 0811.2899.
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FORMULA
| G.f. 1/4-3/8/x-3/8*(2-1/x)*((2*x+1)/(1-6*x))^(1/2)-6/x*Int(Int((6*x+1)/(1-6*x)^(5/2)/(2*x+1)^(3/2)*Int(((1-6*x)/(1-8*x^2))^(3/2)*(2*x+1)^(1/2)*((32*x^3-32*x^2 -42*x-5)*hypergeom([1/4,3/4],[1],64/(8*x^2-1)^2*(2*x+1)*x^3)+(2+14*x-128*x^2-832*x^3-1936*x^4-1600*x^5+1152*x^6+2048*x^7)/(1-8*x^2)^2*hypergeom([5/4, 7/4],[2],64/(1-8*x^2)^2*(2*x+1)*x^3))/(-1+4*x+8*x^2)/(6*x+1)^2,x),x),x) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Oct 13 2009] [Needs to be written avoiding the a/b/c/d... notation! - N. J. A. Sloane, Oct 15 2009]
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MATHEMATICA
| aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[-1 + i, j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]
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CROSSREFS
| Sequence in context: A124543 A007142 A190010 * A063000 A002902 A005053
Adjacent sequences: A151323 A151324 A151325 * A151327 A151328 A151329
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KEYWORD
| nonn,walk
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AUTHOR
| Manuel Kauers (manuel(AT)kauers.de), Nov 18 2008
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