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A182881
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Number of (1,1)-steps in all weighted lattice paths in L_n.
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2
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0, 0, 0, 2, 6, 18, 56, 162, 462, 1306, 3648, 10116, 27892, 76524, 209112, 569506, 1546542, 4189314, 11323480, 30548190, 82272330, 221240070, 594131160, 1593553452, 4269391596, 11426761548, 30554523096, 81631135502, 217918012002
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OFFSET
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0,4
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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), 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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LINKS
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FORMULA
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G.f.: 2*z^3/[(1-3*z+z^2)*(1+z+z^2)]^(3/2).
a(n) ~ ((3 + sqrt(5))/2)^n * sqrt(n) / (2*sqrt(Pi)*5^(3/4)). - Vaclav Kotesovec, Mar 06 2016
Conjecture: (-n+3)*a(n) +(2*n-5)*a(n-1) +(n-2)*a(n-2) +(2*n-3)*a(n-3) +(-n+1)*a(n-4)=0. - R. J. Mathar, Jun 14 2016
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EXAMPLE
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a(3)=2. 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, containing a total of 1+1+0+0+0=2 u steps.
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MAPLE
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g:=2*z^3/((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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MATHEMATICA
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CoefficientList[Series[2*x^3/((1-3*x+x^2)*(1+x+x^2))^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2016 *)
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PROG
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(PARI) z='z+O('z^50); concat([0, 0, 0], Vec(2*z^3/((1-3*z+z^2)*(1+z+z^2))^(3/2))) \\ G. C. Greubel, Mar 25 2017
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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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