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.

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.

Cf. A208544, A273891, A330618.

Sequence in context: A021307 A170852 A245320 * A152893 A335682 A335681

Adjacent sequences: A330338 A330339 A330340 * A330342 A330343 A330344

Keyword

nonn,tabl

Author

Andrew Howroyd, Dec 20 2019

Status

approved