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Search: id:A151307
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| A151307 |
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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, 1)} |
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+0
1
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| 1, 2, 9, 34, 151, 659, 2999, 13714, 63799, 298397, 1408415, 6678827, 31841195, 152374091, 731802083, 3524706626, 17021524103, 82383673241, 399539775647, 1941095088373, 9445526397891, 46028331970139, 224587864915595, 1097124938773915, 5365254892362091, 26263285466953979, 128675997398671299
(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/(x-x^2)*(x+Int(Int(2*x/(1-5*x)^(5/2)/(1+3*x)^(3/2)*(13+Int((1-5*x)^(3/2)*((1+3*x)/(1-4*x^2))^(1/2)*((24*x^4+32*x^3+x^2+12*x+1)*(1-4*x^2)^2*hypergeom([1/4, 3/4],[1],64*(1+x)*x^3/(1-4*x^2)^2)-3*x*(3-10*x-63*x^2-212*x^3-220*x^4-464*x^5-288*x^6+64*x^7)*hypergeom([5/4, 7/4],[2],64*(1+x)*x^3/(1-4*x^2)^2))/(1+x)/(4*x^2+4*x-1)/(1-4*x^2)^3/x^2,x)),x),x)) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Oct 14 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, -1 + 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
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Sequence in context: A091526 A150937 A150938 this_sequence A150939 A150940 A150941
Adjacent sequences: A151304 A151305 A151306 this_sequence A151308 A151309 A151310
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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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