A182887 Number of (1,0)-steps in all weighted lattice paths in L_n.

0, 1, 3, 7, 21, 60, 166, 463, 1281, 3521, 9645, 26322, 71606, 194283, 525897, 1420595, 3830445, 10311510, 27718028, 74410105, 199519155, 534400491, 1429944603, 3822761742, 10211093226, 27254110405, 72691102131, 193750155673, 516100470051

Offset: 0

Author

Emeric Deutsch, Dec 11 2010

Keywords

• nonn

Comments

These are paths that start at (0,0) and end on the horizontal axis and whose steps are of the following four kinds: a (1,0)-step with weight 1; a (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)=7. 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; the total number of (1,0) steps in them are 0+0+2+2+3=7.
		

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. A182884, A182886.

Programs

Maple

G:=z*(1+z)*(1-z-z^2)/((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[x*(1+x)*(1-x-x^2)/((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], Vec(z*(1+z)*(1-z-z^2)/((1-3*z+z^2)*(1+z+z^2))^(3/2))) \\ G. C. Greubel, Mar 26 2017

Formula

a(n) = A182884(n) + A182884(n-1).

a(n) = Sum_{k>=0} k*A182886(n,k).

G.f.: z*(1+z)*(1-z-z^2)/((1-3*z+z^2)*(1+z+z^2))^(3/2).

a(n) ~ ((3 + sqrt(5))/2)^n * sqrt(n) / (2*5^(1/4)*sqrt(Pi)). - Vaclav Kotesovec, Mar 06 2016

Conjecture: (n-1)*(182*n-279)*a(n) + (-230*n^2+11*n+643)*a(n-1) + (-450*n^2+1603*n-315)*a(n-2) + (-498*n^2+971*n+57)*a(n-3) + (-86*n^2+959*n-529)*a(n-4) + (134*n-59)*(n-3)*a(n-5) = 0. - R. J. Mathar, Jun 14 2016