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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.
2

%I #22 Feb 27 2013 04:56:20

%S 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 4,1,128,1096,668,214,60,16,4,1,256,3088,2180,740,216,60,16,4,1,512,

%U 8624,6992,2504,754,216,60,16,4,1,1024,23936,22128,8332,2576,756,216,60,16,4,1

%N 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 Row sums are 3^n.

%C Column k=0 is A000079.

%C Column k=1 is A094309.

%C Limit of reversed rows gives A120926.

%H Alois P. Heinz, <a href="/A209240/b209240.txt">Rows n = 0..140, flattened</a>

%F 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 1;

%e 2, 1;

%e 4, 4, 1;

%e 8, 14, 4, 1;

%e 16, 44, 16, 4, 1;

%e 32, 132, 58, 16, 4, 1;

%e 64, 384, 200, 60, 16, 4, 1;

%e 128, 1096, 668, 214, 60, 16, 4, 1;

%e 256, 3088, 2180, 740, 216, 60, 16, 4, 1;

%t 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 Cf. A048004.

%K nonn,tabl

%O 0,2

%A _Geoffrey Critzer_, Jan 13 2013