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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, 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
1,8
COMMENTS
It is assumed that all beads lie on a circle and distance between any two adjacent is the same.
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).
EXAMPLE
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
...
KEYWORD
nonn,tabl,more
AUTHOR
Maxim Karimov and Vladislav Sulima, Feb 10 2024
STATUS
approved