%I #7 Jun 14 2016 12:27:48
%S 1,1,2,4,9,21,48,112,263,623,1484,3550,8525,20537,49612,120136,291519,
%T 708699,1725714,4208364,10276173,25122829,61486180,150632012,
%U 369361757,906462529,2226297008,5471757126,13457326605,33117622245,81547372396
%N Number of weighted lattice paths in L_n having no peaks. The members of L_n are paths of weight n that start at (0,0), end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps. A peak is a (1,1)-step followed by a (1,-1)-step.
%C a(n)=A182903(n,0).
%D M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
%D E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.
%F G.f.: g=1/sqrt(1-2z-z^2-z^4-2z^5+z^6).
%F Conjecture: n*a(n) +(n-2)*a(n-1) +(-7*n+10)*a(n-2) +3*(-n+2)*a(n-3) +(-n+2)*a(n-4) +(-5*n+14)*a(n-5) +(-5*n+18)*a(n-6) +3*(n-4)*a(n-7)=0. - _R. J. Mathar_, Jun 14 2016
%e a(3)=4. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and U=(1,1), D=(1,-1), we have hhh, hH, Hh, and DU.
%p g := 1/sqrt(1-2*z-z^2-z^4-2*z^5+z^6): gser := series(g, z = 0, 35):seq(coeff(gser, z, n), n = 0 .. 30);
%Y Cf. A182903.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Dec 16 2010