

A182901


Number of weighted lattice paths in B(n) having no valleys. The members of B(n) are paths of weight n that start at (0,0), end on but never go below 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 valley is a (1,1)step followed by a (1,1)step.


1



1, 1, 2, 4, 8, 17, 36, 78, 171, 379, 848, 1912, 4341, 9915, 22767, 52526, 121698, 283043, 660579, 1546556, 3631261, 8548643, 20174093, 47716388, 113095740, 268575321, 638954183, 1522668500, 3634346039, 8687404327, 20794957839, 49841956726, 119610395745
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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.


LINKS



FORMULA

G.f.: g=g(z) satisfies z^4*(1+z)g^2(1zz^2z^3)g+1=0.
Dfinite with recurrence (n+4)*a(n) +(n1)*a(n1) +3*(n2)*a(n2) +(n1)*a(n3) +(n+2)*a(n4) +3*(n+3)*a(n5) +(n+2)*a(n6) +(n5)*a(n7)=0.  R. J. Mathar, Jul 22 2022


EXAMPLE

a(3)=4. Indeed, denoting by h (H) the (1,0)step of weight 1 (2), and U=(1,1), D=(1,1), the four paths of weight 3 are hhh, hH, Hh, and UD; none of them has a valley.


MAPLE

eq := z^4*(1+z)*g^2(1zz^2z^3)*g+1 = 0: g := RootOf(eq, g): gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



