A182897 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.
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
Offset: 0
Keywords
Examples
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.
References
- 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.
Crossrefs
Programs
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Maple
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
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 *)
Formula
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.
a(n) ~ ((1 + sqrt(5))/2)^(2*n+1) / (4*sqrt(5)). - Vaclav Kotesovec, Mar 06 2016
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
Comments