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
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Sep 20 2014
STATUS
approved