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A151312
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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, 1), (-1, 0), (1, -1), (1, 0), (1, 1)}.
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1
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1, 2, 10, 39, 210, 960, 5340, 26250, 148610, 761796, 4360356, 22971102, 132469260, 711426144, 4124979144, 22471772895, 130833575730, 720642479700, 4209268283220, 23389011317958, 136971640993044, 766537623135648, 4498668129173400, 25325478605719656, 148897823229533196, 842422760884567800
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OFFSET
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0,2
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REFERENCES
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Alin Bostan, Calcul Formel pour la Combinatoire des Marches [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d’Informatique de Paris Nord, Université Paris 13, December 2017; https://specfun.inria.fr/bostan/HDR.pdf
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LINKS
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M. Bousquet-Mélou 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.: Int(hypergeom([3/2,3/2],[2],16*x/((2*x+1)*(6*x+1)))*(1-2*x)/((2*x+1)*(6*x+1))^(3/2),x)/x. - Mark van Hoeij, Aug 14 2014
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MAPLE
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Gf:= Int(hypergeom([3/2, 3/2], [2], 16*x/((2*x+1)*(6*x+1)))*(1-2*x)/((2*x+1)*(6*x+1))^(3/2), x)/x:
S:= series(Gf, x, 101):
# The following programs a0 and a1 give the even and odd terms:
a0 := proc(n) options remember;
if n=0 then 1 elif n=1 then 10
else 2*(-18*(2*n-3)^2*a0(n-2)+(2*n-1)*(20*n^2-2*n-3)*a0(n-1)/(2*n+1))/n^2
fi
end:
a1 := proc(n) options remember;
if n=0 then 2 elif n=1 then 39 else
2*(-18*(2*n-1)^2*(1-1/n^2)*a1(n-2)+(20*n^2+18*n+1)*a1(n-1))/(n+1)^2
fi
end:
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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, j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[1 + 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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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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