A385666 Triangle read by rows: T(n,k) is the number of 2n-bead balanced binary necklaces with period length 2n/k.

1, 1, 1, 3, 0, 1, 8, 1, 0, 1, 25, 0, 0, 0, 1, 75, 3, 1, 0, 0, 1, 245, 0, 0, 0, 0, 0, 1, 800, 8, 0, 1, 0, 0, 0, 1, 2700, 0, 3, 0, 0, 0, 0, 0, 1, 9225, 25, 0, 0, 1, 0, 0, 0, 0, 1, 32065, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 112632, 75, 8, 3, 0, 1, 0, 0, 0, 0, 0, 1, 400023

Offset: 1

Author

Tilman Piesk, Jul 16 2025

Keywords

• nonn
• tabl
• changed

Comments

There are A003239(n) balanced binary necklaces of length 2n. (Central numbers of A047996.)

T(n,k) is the number of those that can be rotated into themselves in k different ways (at least 1 for the trivial rotation).

A022553(n) necklaces (corresponding to Lyndon words) have only the trivial rotation.

All columns have the same positive entries, each preceded by k-1 zeros.

Compare triangle A385665, which counts only self-complementary balanced binary necklaces.

Examples

			Triangle begins:
         k     1   2  3  4  5  6  7  8  9 10 11 12 13 14 15 16    A003239(n)
  n
  1            1   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .            1
  2            1   1  .  .  .  .  .  .  .  .  .  .  .  .  .  .            2
  3            3   .  1  .  .  .  .  .  .  .  .  .  .  .  .  .            4
  4            8   1  .  1  .  .  .  .  .  .  .  .  .  .  .  .           10
  5           25   .  .  .  1  .  .  .  .  .  .  .  .  .  .  .           26
  6           75   3  1  .  .  1  .  .  .  .  .  .  .  .  .  .           80
  7          245   .  .  .  .  .  1  .  .  .  .  .  .  .  .  .          246
  8          800   8  .  1  .  .  .  1  .  .  .  .  .  .  .  .          810
  9         2700   .  3  .  .  .  .  .  1  .  .  .  .  .  .  .         2704
 10         9225  25  .  .  1  .  .  .  .  1  .  .  .  .  .  .         9252
 11        32065   .  .  .  .  .  .  .  .  .  1  .  .  .  .  .        32066
 12       112632  75  8  3  .  1  .  .  .  .  .  1  .  .  .  .       112720
 13       400023   .  .  .  .  .  .  .  .  .  .  .  1  .  .  .       400024
 14      1432613 245  .  .  .  .  1  .  .  .  .  .  .  1  .  .      1432860
 15      5170575   . 25  .  3  .  .  .  .  .  .  .  .  .  1  .      5170604
 16     18783360 800  .  8  .  .  .  1  .  .  .  .  .  .  .  1     18784170
Examples for n=4 with necklaces of length 8:
T(4, 1) = 8 necklaces have k=1 rotation, i.e. rotating 0 places:
 00001111, 00010111, 00011011, 00011101, 00100111, 00101011, 00101101, 00110101
T(4, 2) = 1 necklace has k=2 rotations:
 00110011 can be rotated onto itself by rotating 0 or 4 places.
T(4, 4) = 1 necklace has k=4 rotations:
 01010101 can be rotated onto itself by rotating 0, 2, 4 or 6 places.
		

Links

• Tilman Piesk, Rows 1..32, flattened
• Tilman Piesk, Triangle T(n,k)*2*n/k with row sums 2n choose n
• Tilman Piesk, List of imprimitive necklaces for n=1...15
• Tilman Piesk, List of primitive necklaces for n=1...8

Crossrefs

Cf. A022553, A003239, A385665.

Formula

T(n,k) = A022553(n/k) iff n divisible by k, otherwise 0.