A370108 Array read by antidiagonals: T(n,k) is the number of length n necklaces using at most k colors in which the convex hull of a set of beads of any color A can be transformed by rotation into the convex hull of a set of beads of any other color B (n >= 1, k >= 1).

0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 0, 6, 2, 2, 0, 0, 10, 8, 6, 0, 0, 0, 15, 20, 18, 0, 2, 0, 0, 21, 40, 50, 0, 10, 0, 0, 0, 28, 70, 120, 24, 28, 0, 4, 0, 0, 36, 112, 252, 144, 60, 0, 12, 0, 0, 0, 45, 168, 476, 504, 230, 0, 54, 8, 4, 0

Offset: 1

Author

Maxim Karimov and Vladislav Sulima, Feb 10 2024

Keywords

• nonn
• tabl
• more

Comments

It is assumed that all beads lie on a circle and distance between any two adjacent is the same.

Examples

			n\k| 1 2  3  4   5   6    7    8     9 ...
---+----------------------------------
 1 | 0 0  0  0   0   0    0    0     0 ... A000007
 2 | 0 1  3  6  10  15   21   28    36 ... A000217
 3 | 0 0  2  8  20  40   70  112   168 ... A007290
 4 | 0 2  6 18  50 120  252  476   828 ... A062026
 5 | 0 0  0  0  24 144  504 1344  3024 ... A059593
 6 | 0 2 10 28  60 230 1022 3640 10488
 7 | 0 0  0  0   0   0  720 5760 25920 ... A153760
 8 | 0 4 12 54 190 510 1134 7252 49284
 9 | 0 0  8 32  80 160  280  448 40992
...
		

Crossrefs

Cf. A000007, A000013, A000217, A007290, A062026, A059593, A153760.

Formula

T(n,2) = A000013(ceiling(n/2)) * [n mod 2 == 0], where [] is the Iverson bracket.

For prime p, T(p,k) = (p-1)! * binomial(k,p).