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

0,5

Comments

a(n)=Sum(k*A182896(n,k), k>=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..29.

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

Example

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.

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);

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 *)

Crossrefs

A182896

Sequence in context: A330079 A361423 A135415 * A228734 A048481 A269488

Adjacent sequences: A182894 A182895 A182896 * A182898 A182899 A182900

Keyword

nonn

Author

Emeric Deutsch, Dec 13 2010

Status

approved