A182881 Number of (1,1)-steps in all weighted lattice paths in L_n.

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

Offset: 0

Author

Emeric Deutsch, Dec 11 2010

Keywords

• nonn

Comments

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.

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, containing a total of 1+1+0+0+0=2 u steps.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• 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. A182880.

Programs

Maple

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);

Mathematica

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 *)

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

Formula

a(n) = Sum_{k>=0} k*A182880(n,k).

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