A385666 - OEIS

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

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OFFSET

1,4

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.

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

FORMULA

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

EXAMPLE

Triangle begins:

k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A003239(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.

CROSSREFS

Cf. A022553, A003239, A385665.

Sequence in context: A208758 A320161 A373637 * A291763 A161129 A011074

Adjacent sequences: A385663 A385664 A385665 * A385667 A385668 A385669

KEYWORD

nonn,tabl

AUTHOR

Tilman Piesk, Jul 16 2025

STATUS

approved