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A182887
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Number of (1,0)-steps in all weighted lattice paths in L_n. These are paths that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1; an (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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1
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0, 1, 3, 7, 21, 60, 166, 463, 1281, 3521, 9645, 26322, 71606, 194283, 525897, 1420595, 3830445, 10311510, 27718028, 74410105, 199519155, 534400491, 1429944603, 3822761742, 10211093226, 27254110405, 72691102131, 193750155673, 516100470051
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OFFSET
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0,3
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COMMENTS
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a(n)=A182884(n)+A182884(n-1).
a(n)=Sum(k*A182886(n,k),k>=0).
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REFERENCES
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M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.
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LINKS
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Table of n, a(n) for n=0..28.
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FORMULA
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G.f.: G=z(1+z)(1-z-z^2)/[(1-3z+z^2)(1+z+z^2)]^{3/2}.
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EXAMPLE
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a(3)=7. 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 (1,0) steps in them are 0+0+2+2+3=7.
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MAPLE
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G:=z*(1+z)*(1-z-z^2)/((1-3*z+z^2)*(1+z+z^2))^(3/2): Gser:=series(G, z=0, 32): seq(coeff(Gser, z, n), n=0..28);
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CROSSREFS
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Cf. A182884, A182886.
Sequence in context: A178718 A091650 A096240 * A035080 A091486 A056779
Adjacent sequences: A182884 A182885 A182886 * A182888 A182889 A182890
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch, Dec 11 2010
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STATUS
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approved
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