

A182884


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


4



0, 1, 2, 5, 16, 44, 122, 341, 940, 2581, 7064, 19258, 52348, 141935, 383962, 1036633, 2793812, 7517698, 20200330, 54209775, 145309380, 389091111, 1040853492, 2781908250, 7429184976, 19824925429, 52866176702, 140883978971, 375216491080
(list;
graph;
refs;
listen;
history;
text;
internal format)



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), 291306.
E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163177.


FORMULA

a(n) = Sum_{k>=0} k*A182882(n,k).
G.f.: z*(1zz^2)/((13*z+z^2)*(1+z+z^2))^(3/2).
(n+3)*a(n)n*a(n+1)+(184*n)*a(n+2)+(6n)*a(n+3)+(14+3*n)*a(n+5)+(5n)*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*(1zz^2)/((13*z+z^2)*(1+z+z^2))^(3/2): Gser:=series(G, z=0, 35): seq(coeff(Gser, z, n), n=0..28);


CROSSREFS

Cf. A182882.
Sequence in context: A148373 A132734 A148374 * A152428 A317890 A138573
Adjacent sequences: A182881 A182882 A182883 * A182885 A182886 A182887


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Dec 11 2010


STATUS

approved



