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Search: id:A151287
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| A151287 |
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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, -1), (-1, 0), (0, 1), (1, -1), (1, 0)} |
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+0
1
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| 1, 2, 6, 21, 76, 290, 1148, 4627, 19038, 79554, 336112, 1435522, 6184704, 26838474, 117247440, 515135847, 2274656290, 10090187786, 44940868940, 200897459804, 901082056408, 4053912011322, 18289272082952, 82724956638634, 375064515961744, 1704237546984170, 7759645793395368, 35398085705004882
(list; graph; listen)
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OFFSET
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0,2
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LINKS
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M. Bouquet-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
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G.f. -1/(2x)+(1+x)/x^2*Int(x^2/(1+x)^2/(1+3*x)^(1/2)/(1-5*x)^(3/2)*(-13/2+Int((1+x)*((1-5*x)/(1+3*x)/(1-2*x)^3/(1+2*x)^3)^(1/2)*((4*x^3-2*x-1)/x^3*hypergeom([1/4, 3/4],[1],64*(1+x)*x^3/(1-2*x)^2/(1+2*x)^2)+6*(-3-2*x+4*x^2)*(12*x^2+4*x+1)/(1+2*x)^2/(1-2*x)^2*hypergeom([5/4, 7/4],[2],64*(1+x)*x^3/(1-2*x)^2/(1+2*x)^2)),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. Slaone, Oct 15 2009]
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MATHEMATICA
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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, j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]
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CROSSREFS
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Sequence in context: A116772 A131792 A144904 this_sequence A101265 A101879 A063023
Adjacent sequences: A151284 A151285 A151286 this_sequence A151288 A151289 A151290
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KEYWORD
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nonn,walk
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AUTHOR
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Manuel Kauers (manuel(AT)kauers.de), Nov 18 2008
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