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

Author

Emeric Deutsch, Dec 15 2010

Keywords

• nonn
• changed

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.

Examples

			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).
		

References

• M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.

Crossrefs

A182900.

Programs

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);

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.

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