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A182884
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Number of (1,0)-steps of weight 1 in all weighted lattice paths in L_n.
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4
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0, 1, 2, 5, 16, 44, 122, 341, 940, 2581, 7064, 19258, 52348, 141935, 383962, 1036633, 2793812, 7517698, 20200330, 54209775, 145309380, 389091111, 1040853492, 2781908250, 7429184976, 19824925429, 52866176702, 140883978971, 375216491080
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OFFSET
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0,3
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COMMENTS
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L_n is the set of lattice paths of weight n that start at (0,0) and end on the horizontal axis and whose steps are of the following four kinds: a (1,0)-step with weight 1; a (1,0)-step with weight 2; a (1,1)-step with weight 2; a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.
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LINKS
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FORMULA
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G.f.: z*(1-z-z^2)/((1-3*z+z^2)*(1+z+z^2))^(3/2).
(n+3)*a(n)-n*a(n+1)+(-18-4*n)*a(n+2)+(6-n)*a(n+3)+(14+3*n)*a(n+5)+(-5-n)*a(n+6) = 0. - Robert Israel, Dec 30 2016
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EXAMPLE
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a(3)=5. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; the total number of h steps in them is 0+0+1+1+3=5.
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MAPLE
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G:=z*(1-z-z^2)/((1-3*z+z^2)*(1+z+z^2))^(3/2): Gser:=series(G, z=0, 35): seq(coeff(Gser, z, n), n=0..28);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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