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A182899 Number of returns to the horizontal axis (both from above and below) in all weighted lattice paths in L_n.

%I A182899 #15 Jul 22 2022 12:09:22
%S A182899 0,0,0,2,6,18,54,152,422,1160,3156,8534,22968,61578,164602,438930,
%T A182899 1168120,3103540,8234122,21820098,57762774,152774358,403750258,
%U A182899 1066291206,2814322014,7423962336,19574314938,51587866820,135905559330,357908155044
%N A182899 Number of returns to the horizontal axis (both from above and below) in all weighted lattice paths in L_n.
%C A182899 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.
%H A182899 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.
%H A182899 E. Munarini and N. Zagaglia Salvi, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00378-3">On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns</a>, Discrete Mathematics 259 (2002), 163-177.
%F A182899 a(n) = Sum_{k>=0} k*A182898(n,k).
%F A182899 a(n) = 2*A182897(n).
%F A182899 G.f.:  2*z^3*c/((1+z+z^2)*(1-3*z+z^2)), where c satisfies c = 1+z*c+z^2*c+z^3*c^2.
%F A182899 Conjecture D-finite with recurrence n*a(n) +(-4*n+3)*a(n-1) +(2*n-3)*a(n-2) +11*(n-3)*a(n-4) +(2*n-9)*a(n-6) +(-4*n+21)*a(n-7) +(n-6)*a(n-8)=0. - _R. J. Mathar_, Jul 22 2022
%e A182899 a(3)=2 because, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; they contain 1+1+0+0+0=1  returns to the horizontal axis.
%p A182899 eq := c = 1+z*c+z^2*c+z^3*c^2: c := RootOf(eq, c): G := 2*z^3*c/((1+z+z^2)*(1-3*z+z^2)): Gser := series(G, z = 0, 32): seq(coeff(Gser, z, n), n = 0 .. 29);
%Y A182899 Cf. A182896, A182897, A182898.
%K A182899 nonn
%O A182899 0,4
%A A182899 _Emeric Deutsch_, Dec 13 2010