A182894 Number of weighted lattice paths in L_n having no (1,0)-steps at level 0. 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.
1, 0, 0, 2, 2, 4, 12, 24, 54, 130, 300, 706, 1686, 4028, 9686, 23426, 56866, 138584, 338940, 831508, 2045736, 5046240, 12477290, 30919122, 76774382, 190995224, 475979602, 1188125394, 2970282794, 7436232760, 18641883396, 46792219972, 117590713254
Offset: 0
Keywords
Examples
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; two of them, namely ud and du, have no (1,0)-steps at level 0.
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
Cf. A182893.
Programs
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Maple
G:=1/(z+z^2+sqrt((1+z+z^2)*(1-3*z+z^2))): Gser:=series(G,z=0,35): seq(coeff(Gser,z,n),n=0..32);
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Mathematica
CoefficientList[Series[1/(x+x^2+Sqrt[(1+x+x^2)*(1-3*x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2016 *)
Formula
G.f.: G(z) =1/( z+z^2+sqrt((1+z+z^2)(1-3z+z^2)) ).
a(n) ~ sqrt(105 + 47*sqrt(5)) * ((3 + sqrt(5))/2)^n / (5*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 06 2016
Conjecture: n*a(n) +(-4*n+3)*a(n-1) +(n-3)*a(n-2) -3*a(n-3) +3*(5*n-14)*a(n-4) +6*(n-3)*a(n-5) +6*(n-4)*a(n-6) +4*(-n+6)*a(n-7)=0. - R. J. Mathar, Jun 14 2016
a(n) ~ phi^(2*n + 4) / (5^(3/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 23 2017
Comments