

A182895


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


3



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
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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), 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.
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)(13z+z^2)].
a(n) = (A000032(2n+1)  A010892(2n))/4.  John M. Campbell, Dec 30 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; they contain 0+0+2+2+3=7 (1,0)steps at level 0.


MAPLE

G:=z*(1+z)/(1+z+z^2)/(13*z+z^2): Gser:=series(G, z=0, 32): seq(coeff(Gser, z, n), n=0..28);


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



