

A182902


Number of valleys in all weighted lattice paths in B(n).


1



0, 0, 0, 0, 0, 0, 1, 4, 14, 45, 135, 391, 1105, 3067, 8404, 22806, 61428, 164495, 438459, 1164363, 3082717, 8141422, 21457255, 56455195, 148323305, 389213825, 1020283146, 2672225692, 6993600748, 18291536552, 47814575243, 124929304664, 326280023426
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OFFSET

0,8


COMMENTS

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.


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:=z^6*g^4/(1z^3*g^2), where g=g(z) satisfies g=1+zg+z^2*g+z^3*g^2.
Conjecture Dfinite with recurrence 3*(n+3)*(n6)*a(n) +(n+1)*(7*n34)*a(n1) +2*(5*n+26)*a(n2) +(7*n^239*n+16)*a(n3) +4*(n^2+5*n+2)*a(n4) +(3*n^229*n+64)*a(n5) (n4)*(n7)*a(n6)=0.  R. J. Mathar, Jul 22 2022


EXAMPLE

a(7) = 4. Indeed, denoting by h (H) the (1,0)step of weight 1 (2), and U = (1,1), D = (1,1), among the 82 paths in B(7) only hUDUD, UDUDh, UDUhD, and UhDUD have valleys (1 in each).


MAPLE

eq := g = 1+z*g+z^2*g+z^3*g^2: g := RootOf(eq, g): gser := series(z^6*g^4/(1z^3*g^2), z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



