OFFSET
0,3
COMMENTS
L_n is the set of lattice paths of weight n 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
Robert Israel, Table of n, a(n) for n = 0..2388
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*A182882(n,k).
G.f.: z*(1-z-z^2)/((1-3*z+z^2)*(1+z+z^2))^(3/2).
(n+3)*a(n)-n*a(n+1)+(-18-4*n)*a(n+2)+(6-n)*a(n+3)+(14+3*n)*a(n+5)+(-5-n)*a(n+6) = 0. - Robert Israel, Dec 30 2016
EXAMPLE
a(3)=5. 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 h steps in them is 0+0+1+1+3=5.
MAPLE
G:=z*(1-z-z^2)/((1-3*z+z^2)*(1+z+z^2))^(3/2): Gser:=series(G, z=0, 35): seq(coeff(Gser, z, n), n=0..28);
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 11 2010
STATUS
approved