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A247471 Number of paths in B(n) that start with a u step and end with a d step. 0

%I

%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

%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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Last modified August 12 05:33 EDT 2020. Contains 336438 sequences. (Running on oeis4.)