OFFSET
0,3
COMMENTS
The members of L_n are 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, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.
LINKS
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 and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
Index entries for linear recurrences with constant coefficients, signature (2,1,2,-1).
FORMULA
G.f: x/((1+x+x^2)*(1-3*x+x^2)).
a(n) = Sum_{k>=0} k*A182888(n,k).
a(n) = Sum_{m=0..n} C(2*n-2*m,2*m+1)/2. - Vladimir Kruchinin, Jan 24 2022
EXAMPLE
a(3)=5. Indeed, denoting by h (resp. H) the (1,0)-step of weight 1 (resp. 2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; they contain 0+0+1+1+3=5 h-steps at level 0.
MAPLE
G:=z/(1+z+z^2)/(1-3*z+z^2): Gser:=series(G, z=0, 33): seq(coeff(Gser, z, n), n=0..30);
MATHEMATICA
Table[Sum[Binomial[2n+2-2k, 2k-1]/2, {k, 0, n+1}], {n, 0, 30}]; (* Rigoberto Florez, Apr 10 2023 *)
PROG
(Maxima) a(n):=1/2*sum(binomial(2*n-2*m, 2*m+1), m, 0, (2*n-1)/4); /* Vladimir Kruchinin, Jan 24 2022 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Dec 12 2010
STATUS
approved