A123223 Triangle read by rows: T(n,k) = number of ternary Lyndon words of length n with exactly k 1's

1, 2, 1, 1, 2, 0, 2, 4, 2, 0, 3, 8, 5, 2, 0, 6, 16, 16, 8, 2, 0, 9, 32, 38, 26, 9, 2, 0, 18, 64, 96, 80, 40, 12, 2, 0, 30, 128, 220, 224, 137, 56, 13, 2, 0, 56, 256, 512, 596, 448, 224, 74, 16, 2, 0, 99, 512, 1144, 1536, 1336, 806, 332, 96, 17, 2, 0, 186, 1024, 2560, 3840, 3840

Offset

0,2

Comments

Sum of rows equal to number of ternary Lyndon words A027376 first column (k=0) is equal to the number of binary Lyndon words A001037 third through sixth column (k=2,3,4,5) equal to A124720, A124721, A124722, A124723 T(n+1,n-1) entry equal to A042948

Links

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

Formula

G.f. for columns (except for k=0) given by 1/k*sum_{d|k} mu(d) x^k/(1-2*x^d)^(k/d) T(0,0) = 1 and T(n,0) = 1/n*sum_{d|n} mu(d)*2^(n/d) T(n,n) = 0 if n>1, T(n,n-1) = 2

Example

Triangle begins:
1,
2,1,
1,2,0,
2,4,2,0,
3,8,5,2,0,
6,16,16,8,2,0,
9,32,38,26,9,2,0,
18,64,96,80,40,12,2,0,
T(n,1) = 2^(n-1) because all words beginning with a 1 and consisting of the rest 2's or 3's are ternary Lyndon words with exactly one 1.

Crossrefs

Cf. A027376, A001037, A124720, A124721, A124722, A124723, A051168, A042948.

Sequence in context: A155997 A326171 A359393 * A088226 A344309 A358338

Adjacent sequences: A123220 A123221 A123222 * A123224 A123225 A123226

Keyword

nonn,tabl

Author

Mike Zabrocki, Nov 05 2006

Status

approved