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, 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

Offset

0,3

Comments

a(n)=A182900(n,0).

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

G.f.: g=g(z) satisfies z^4*(1+z)g^2-(1-z-z^2-z^3)g+1=0.

D-finite with recurrence (n+4)*a(n) +(-n-1)*a(n-1) +3*(-n-2)*a(n-2) +(-n-1)*a(n-3) +(-n+2)*a(n-4) +3*(-n+3)*a(n-5) +(-n+2)*a(n-6) +(n-5)*a(n-7)=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-(1-z-z^2-z^3)*g+1 = 0: g := RootOf(eq, g): gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);

Crossrefs

A182900.

Sequence in context: A275691 A251691 A157904 * A002845 A072925 A002955

Adjacent sequences: A182898 A182899 A182900 * A182902 A182903 A182904

Keyword

nonn

Author

Emeric Deutsch, Dec 15 2010

Status

approved