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A182888 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps at level 0. These 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 A182888 #18 Mar 29 2025 04:22:30
%S A182888 1,0,1,1,0,1,2,2,0,1,3,4,3,0,1,8,7,6,4,0,1,17,20,12,8,5,0,1,38,44,36,
%T A182888 18,10,6,0,1,89,104,82,56,25,12,7,0,1,206,253,204,132,80,33,14,8,0,1,
%U A182888 485,604,513,344,195,108,42,16,9,0,1,1152,1466,1262,891,530,272,140,52,18,10,0,1
%N A182888 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps at level 0. These 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 A182888 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 A182888 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 A182888 G.f.: G(t,z) = 1/( z-tz+sqrt((1+z+z^2)(1-3z+z^2)) ).
%F A182888 Sum_{k=0..n} k*T(n,k) = A182890(n).
%e A182888 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 hH and Hh, have exactly two (1,0)-steps at level 0.
%e A182888 Triangle starts:
%e A182888    1;
%e A182888    0,   1;
%e A182888    1,   0,  1;
%e A182888    2,   2,  0,  1;
%e A182888    3,   4,  3,  0,  1;
%e A182888    8,   7,  6,  4,  0,  1;
%e A182888   17,  20, 12,  8,  5,  0, 1;
%e A182888   38,  44, 36, 18, 10,  6, 0, 1;
%e A182888   89, 104, 82, 56, 25, 12, 7, 0, 1;
%e A182888   ...
%p A182888 G:=1/(z-t*z+sqrt((1+z+z^2)*(1-3*z+z^2))): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,k),k=0..n) od; # yields sequence in triangular form
%Y A182888 Row sums give A051286.
%Y A182888 Column k=0 gives A182889.
%Y A182888 Cf. A182890.
%K A182888 nonn,tabl
%O A182888 0,7
%A A182888 _Emeric Deutsch_, Dec 11 2010