A182895 Number of (1,0)-steps at level 0 in all weighted lattice paths in L_n.

0, 1, 3, 7, 19, 50, 130, 341, 893, 2337, 6119, 16020, 41940, 109801, 287463, 752587, 1970299, 5158310, 13504630, 35355581, 92562113, 242330757, 634430159, 1660959720, 4348449000, 11384387281, 29804712843, 78029751247, 204284540899

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

Table of n, a(n) for n=0..28.

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.

Index entries for linear recurrences with constant coefficients, signature (2,1,2,-1)

Formula

a(n) = Sum_{k>=0} k*A182893(n,k).

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

a(n) = (A000032(2n+1) - A010892(2n))/4. - John M. Campbell, Dec 30 2016

4*a(n) = -A057078(n) +A002878(n). - R. J. Mathar, Jul 26 2022

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; they contain 0+0+2+2+3=7 (1,0)-steps at level 0.

Maple

G:=z*(1+z)/(1+z+z^2)/(1-3*z+z^2): Gser:=series(G, z=0, 32): seq(coeff(Gser, z, n), n=0..28);

Mathematica

LinearRecurrence[{2, 1, 2, -1}, {0, 1, 3, 7}, 30] (* Harvey P. Dale, Jan 05 2022 *)

Crossrefs

Cf. A182893.

Sequence in context: A151266 A147234 A171854 * A087224 A308398 A341703

Adjacent sequences: A182892 A182893 A182894 * A182896 A182897 A182898

Keyword

nonn

Author

Emeric Deutsch, Dec 12 2010

Status

approved