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A182897
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Number of (1,-1)-returns to the horizontal axis in all weighted lattice paths in L_n. The members of L_n are paths of weight n 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, and 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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2
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0, 0, 0, 1, 3, 9, 27, 76, 211, 580, 1578, 4267, 11484, 30789, 82301, 219465, 584060, 1551770, 4117061, 10910049, 28881387, 76387179, 201875129, 533145603, 1407161007, 3711981168, 9787157469, 25793933410, 67952779665, 178954077522
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OFFSET
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0,5
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COMMENTS
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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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FORMULA
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G.f.: G(z)=z^3*c/[(1+z+z^2)(1-3z+z^2)], where c satisfies c = 1+zc+z^2*c+z^3*c^2.
D-finite with recurrence n*a(n) +(-4*n+3)*a(n-1) +(2*n-3)*a(n-2) +11*(n-3)*a(n-4) +(2*n-9)*a(n-6) +(-4*n+21)*a(n-7) +(n-6)*a(n-8)=0. - R. J. Mathar, Jul 24 2022
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EXAMPLE
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a(3)=1 because, 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; they contain 1+0+0+0+0=1 (1,-1)-return to the horizontal axis.
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MAPLE
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eq := c = 1+z*c+z^2*c+z^3*c^2: c := RootOf(eq, c): G := z^3*c/((1+z+z^2)*(1-3*z+z^2)): Gser := series(G, z = 0, 32): seq(coeff(Gser, z, n), n = 0 .. 29);
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MATHEMATICA
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CoefficientList[Series[x^3*(1 - x - x^2 - Sqrt[1+x^4-2*x^3-x^2-2*x]) / (2*x^3*(1+x+x^2)*(1-3*x+x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2016 *)
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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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