A247471 Number of paths in B(n) that start with a u step and end with a d step.

0, 0, 0, 1, 1, 2, 5, 11, 26, 62, 148, 356, 860, 2085, 5073, 12382, 30309, 74391, 183042, 451427, 1115741, 2763228, 6856327, 17042633, 42433166, 105816857, 264268595, 660908408, 1655040445, 4149700172, 10416866219, 26178412875, 65858360172, 165850637772

Offset

0,6

Comments

B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps.

Links

Table of n, a(n) for n=0..33.

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

Formula

G.f.: G = (g - f)/f^2, where g = 1 + z*g + z^2*g + z^3*g^2 and f = 1/(1 - z - z^2).

G.f.: G = z^3*(1 - z - z^2)*g^2, where g = 1 + z*g + z^2*g + z^3*g^2. - Emeric Deutsch, Oct 12 2014

Conjecture D-finite with recurrence (n+3)*a(n) +3*(-n-1)*a(n-1) +(n-9)*a(n-3) +2*(2*n-9)*a(n-4) +(n-9)*a(n-5) +(-n+9)*a(n-6)=0. - R. J. Mathar, Jul 22 2022

Example

a(6) = 5 because we have uhhhd, uhHd, uHhd, uudd, and udud.

Maple

eqg := g = 1+z*g+z^2*g+z^3*g^2: g := RootOf(eqg, g): f := 1/(1-z-z^2): G := (g-f)/f^2: Gser := series(G, z = 0, 43): seq(coeff(Gser, z, n), n = 0 .. 40);

Crossrefs

Sequence in context: A182015 A124217 A095981 * A082397 A051286 A192475

Adjacent sequences: A247468 A247469 A247470 * A247472 A247473 A247474

Keyword

nonn

Author

Emeric Deutsch, Sep 20 2014

Status

approved