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%I #4 Sep 17 2014 20:53:15
%S 0,0,0,1,4,14,43,123,337,898,2349,6072,15577,39776,101304,257689,
%T 655279,1666772,4242354,10807191,27557720,70342486,179736541,
%U 459714008,1176937542,3015862454,7734617111,19852352861,50992757233,131071123062,337122433947,867624835207,2234205069696
%N Number of weak peaks in all weighted lattice paths in B(n).
%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.
%C A weak peak in a lattice path is a vertex on the top of a hump. A hump is an upstep followed by 0 or more flatsteps followed by a downstep. For example, the weighted lattice path Hu*duu*h*H*dd has 4 weak peaks (shown by the stars).
%C a(n) = Sum(k*A247469(n,k), k>=0).
%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 = z^3*g/((1 - z - z^2)^2*(1 - z - z^2 - 2*z^3*g)), where g = 1 + z*g + z^2*g + z^3*g^2.
%e a(4) = 4 because B(4) = {hhhh, hhH, hHh, Hhh, HH, hu*d, u*h*d, u*dh} (weak peaks shown by *).
%p G := z^3*g/((1-z-z^2)^2*(1-z-z^2-2*z^3*g)): eq := g = 1+z*g+z^2*g+z^3*g^2: g := RootOf(eq, g): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);
%Y Cf. A247469
%K nonn
%O 0,5
%A _Emeric Deutsch_, Sep 17 2014
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