A182902 Number of valleys in all weighted lattice paths in B(n).

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

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), 291-306.

Links

Table of n, a(n) for n=0..32.

Formula

a(n) = Sum(k*A182900(n,k), k>=0).

G.f.: G:=z^6*g^4/(1-z^3*g^2), where g=g(z) satisfies g=1+zg+z^2*g+z^3*g^2.

Conjecture D-finite with recurrence -3*(n+3)*(n-6)*a(n) +(n+1)*(7*n-34)*a(n-1) +2*(5*n+26)*a(n-2) +(7*n^2-39*n+16)*a(n-3) +4*(-n^2+5*n+2)*a(n-4) +(3*n^2-29*n+64)*a(n-5) -(n-4)*(n-7)*a(n-6)=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/(1-z^3*g^2), z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);

Crossrefs

A182900.

Sequence in context: A125068 A184138 A318019 * A108765 A304068 A005775

Adjacent sequences: A182899 A182900 A182901 * A182903 A182904 A182905

Keyword

nonn

Author

Emeric Deutsch, Dec 15 2010

Status

approved