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%I #14 Jul 22 2022 08:04:03
%S 0,0,0,1,1,2,5,11,26,62,148,356,860,2085,5073,12382,30309,74391,
%T 183042,451427,1115741,2763228,6856327,17042633,42433166,105816857,
%U 264268595,660908408,1655040445,4149700172,10416866219,26178412875,65858360172,165850637772
%N Number of paths in B(n) that start with a u step and end with a d step.
%C 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.
%H M. Bona and A. Knopfmacher, <a href="http://dx.doi.org/10.1007/s00026-010-0060-7">On the probability that certain compositions have the same number of parts</a>, Ann. Comb., 14 (2010), 291-306.
%F 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).
%F 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
%F 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
%e a(6) = 5 because we have uhhhd, uhHd, uHhd, uudd, and udud.
%p 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);
%K nonn
%O 0,6
%A _Emeric Deutsch_, Sep 20 2014
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