OFFSET
0,3
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.
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.
FORMULA
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
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 11 2010
STATUS
approved