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.

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, 4, 1, 128, 1096, 668, 214, 60, 16, 4, 1, 256, 3088, 2180, 740, 216, 60, 16, 4, 1, 512, 8624, 6992, 2504, 754, 216, 60, 16, 4, 1, 1024, 23936, 22128, 8332, 2576, 756, 216, 60, 16, 4, 1

Offset

0,2

Comments

Row sums are 3^n.

Column k=0 is A000079.

Column k=1 is A094309.

Limit of reversed rows gives A120926.

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

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)).

Example

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,  4,  1;
128, 1096, 668,  214, 60,  16, 4,  1;
256, 3088, 2180, 740, 216, 60, 16, 4,  1;

Mathematica

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

Crossrefs

Cf. A048004.

Sequence in context: A065109 A038207 A113988 * A263989 A202710 A200965

Adjacent sequences: A209237 A209238 A209239 * A209241 A209242 A209243

Keyword

nonn,tabl

Author

Geoffrey Critzer, Jan 13 2013

Status

approved