

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



FORMULA

G.f.: z(1+z)/[(1+z+z^2)(13z+z^2)].


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);


MATHEMATICA

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


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



