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

0,4

Comments

a(n)=A182893(n,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.

Links

Table of n, a(n) for n=0..32.

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

Example

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.

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);

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 *)

Crossrefs

Cf. A182893.

Sequence in context: A355566 A298477 A253677 * A286410 A285611 A288303

Adjacent sequences: A182891 A182892 A182893 * A182895 A182896 A182897

Keyword

nonn

Author

Emeric Deutsch, Dec 12 2010

Status

approved