

A182904


Number of weighted lattice paths in L_n having no peaks. The members of L_n are paths of weight n that start at (0,0), end on the horizontal axis and whose steps are of the following four kinds: an (1,0)step with weight 1, an (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. A peak is a (1,1)step followed by a (1,1)step.


1



1, 1, 2, 4, 9, 21, 48, 112, 263, 623, 1484, 3550, 8525, 20537, 49612, 120136, 291519, 708699, 1725714, 4208364, 10276173, 25122829, 61486180, 150632012, 369361757, 906462529, 2226297008, 5471757126, 13457326605, 33117622245, 81547372396
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OFFSET

0,3


COMMENTS



REFERENCES

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.


LINKS



FORMULA

G.f.: g=1/sqrt(12zz^2z^42z^5+z^6).
Conjecture: n*a(n) +(n2)*a(n1) +(7*n+10)*a(n2) +3*(n+2)*a(n3) +(n+2)*a(n4) +(5*n+14)*a(n5) +(5*n+18)*a(n6) +3*(n4)*a(n7)=0.  R. J. Mathar, Jun 14 2016


EXAMPLE

a(3)=4. Indeed, denoting by h (H) the (1,0)step of weight 1 (2), and U=(1,1), D=(1,1), we have hhh, hH, Hh, and DU.


MAPLE

g := 1/sqrt(12*zz^2z^42*z^5+z^6): gser := series(g, z = 0, 35):seq(coeff(gser, z, n), n = 0 .. 30);


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



