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A209240 Triangular array read by rows. T(n,k) is the number of ternary length-n words in which the longest run of consecutive 0's is exactly k; n>=0, 0<=k<=n.

%I A209240 #22 Feb 27 2013 04:56:20
%S A209240 1,2,1,4,4,1,8,14,4,1,16,44,16,4,1,32,132,58,16,4,1,64,384,200,60,16,
%T A209240 4,1,128,1096,668,214,60,16,4,1,256,3088,2180,740,216,60,16,4,1,512,
%U A209240 8624,6992,2504,754,216,60,16,4,1,1024,23936,22128,8332,2576,756,216,60,16,4,1
%N A209240 Triangular array read by rows. T(n,k) is the number of ternary length-n words in which the longest run of consecutive 0's is exactly k; n>=0, 0<=k<=n.
%C A209240 Row sums are 3^n.
%C A209240 Column k=0 is A000079.
%C A209240 Column k=1 is A094309.
%C A209240 Limit of reversed rows gives A120926.
%H A209240 Alois P. Heinz, <a href="/A209240/b209240.txt">Rows n = 0..140, flattened</a>
%F A209240 O.g.f. for column k: (1-x)^2*x^k/(1-3*x+2*x^(k+1))/(1-3*x+2*x^(k+2)).
%e A209240 1;
%e A209240 2,   1;
%e A209240 4,   4,    1;
%e A209240 8,   14,   4,    1;
%e A209240 16,  44,   16,   4,   1;
%e A209240 32,  132,  58,   16,  4,   1;
%e A209240 64,  384,  200,  60,  16,  4,  1;
%e A209240 128, 1096, 668,  214, 60,  16, 4,  1;
%e A209240 256, 3088, 2180, 740, 216, 60, 16, 4,  1;
%t A209240 nn=10;f[list_]:=Select[list,#>0&];Map[f,Transpose[Table[CoefficientList[ Series[(1-x^k)/(1-3x+2x^(k+1))-(1-x^(k-1))/(1-3x+2x^k),{x,0,nn}],x],{k,1,nn+1}]]]//Grid
%Y A209240 Cf. A048004.
%K A209240 nonn,tabl
%O A209240 0,2
%A A209240 _Geoffrey Critzer_, Jan 13 2013