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A330341
Triangle read by rows: T(n,k) is the number of n-bead bracelets using exactly k colors with no adjacent beads having the same color.
4
0, 0, 1, 0, 0, 1, 0, 1, 3, 3, 0, 0, 3, 12, 12, 0, 1, 10, 46, 90, 60, 0, 0, 9, 120, 480, 720, 360, 0, 1, 27, 384, 2235, 5670, 6300, 2520, 0, 0, 29, 980, 9380, 36960, 68880, 60480, 20160, 0, 1, 75, 2904, 38484, 217152, 604800, 876960, 635040, 181440
OFFSET
1,9
COMMENTS
In the case of n = 1, the single bead is considered to be cyclically adjacent to itself giving T(1,1) = 0. If compatibility with A208544 is wanted then T(1,1) should be 1.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
FORMULA
T(n,k) = Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*A208544(n,j) for n > 1.
EXAMPLE
Triangle begins:
0;
0, 1;
0, 0, 1;
0, 1, 3, 3;
0, 0, 3, 12, 12;
0, 1, 10, 46, 90, 60;
0, 0, 9, 120, 480, 720, 360;
0, 1, 27, 384, 2235, 5670, 6300, 2520;
0, 0, 29, 980, 9380, 36960, 68880, 60480, 20160;
...
PROG
(PARI) \\ here U(n, k) is A208544(n, k) for n > 1.
U(n, k) = (sumdiv(n, d, eulerphi(n/d)*(k-1)^d)/n + if(n%2, 1-k, k*(k-1)^(n/2)/2))/2;
T(n, k)={sum(j=1, k, (-1)^(k-j)*binomial(k, j)*U(n, j))}
CROSSREFS
Column 3 is A330632.
Row sums are A330621.
Sequence in context: A021307 A170852 A245320 * A152893 A335682 A335681
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Dec 20 2019
STATUS
approved