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A182898 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k returns to the horizontal axis (both from above and below). 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.

%I A182898 #5 Mar 30 2012 17:36:25
%S A182898 1,1,2,3,2,5,6,8,18,13,46,4,21,112,20,34,262,80,55,600,268,8,89,1356,
%T A182898 816,56,144,3046,2324,280,233,6832,6320,1144,16,377,15354,16620,4136,
%U A182898 144,610,34658,42652,13728,864,987,78706,107520,42816,4144,32,1597,180000,267564,127392,17264,352
%N A182898 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k returns to the horizontal axis (both from above and below). 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.
%C A182898 Sum of entries in row n is A051286(n).
%C A182898 T(n,0)=A000045(n+1) (the Fibonacci numbers).
%C A182898 Sum(k*T(n,k), k=0..n)=A182899(n).
%D A182898 M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
%D A182898 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 A182898 G.f.:  G(t,z) =1/[1-z-z^2-2tz^3*c], where c satisfies c = 1+zc+z^2*c+z^3*c^2.
%F A182898 The trivariate g.f. H=H(t,s,z), where t (s) marks (1,-1)-returns ((1,1)-returns) to the horizontal axis, and z marks weight is given by H=1+zH+z^2*H+(t+s)z^3*cH, where c satisfies c = 1+zc+z^2*c+z^3*c^2.
%e A182898 T(3,1)=2. Indeed, 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; two of them, namely ud and du, have 1 return to the horizontal axis.
%e A182898 Triangle starts:
%e A182898 1;
%e A182898 1;
%e A182898 2;
%e A182898 3,2;
%e A182898 5,6;
%e A182898 8,18;
%e A182898 13,46,4;
%e A182898 21,112,20;
%p A182898 eq := c = 1+z*c+z^2*c+z^3*c^2: c := RootOf(eq, c): G := 1/(1-z-z^2-2*t*z^3*c): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form
%K A182898 nonn,tabf
%O A182898 0,3
%A A182898 _Emeric Deutsch_, Dec 13 2010