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 A330618 Triangle read by rows: T(n,k) is the number of n-bead necklaces using exactly k colors with no adjacent beads having the same color. 4
 0, 0, 1, 0, 0, 2, 0, 1, 3, 6, 0, 0, 6, 24, 24, 0, 1, 11, 80, 180, 120, 0, 0, 18, 240, 960, 1440, 720, 0, 1, 33, 696, 4410, 11340, 12600, 5040, 0, 0, 58, 1960, 18760, 73920, 137760, 120960, 40320, 0, 1, 105, 5508, 76368, 433944, 1209600, 1753920, 1270080, 362880 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 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 A208535 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)*A208535(n,j) for n > 1. T(n,n) = (n-1)! for n > 1. EXAMPLE Triangle begins: 0; 0, 1; 0, 0, 2; 0, 1, 3, 6; 0, 0, 6, 24, 24; 0, 1, 11, 80, 180, 120; 0, 0, 18, 240, 960, 1440, 720; 0, 1, 33, 696, 4410, 11340, 12600, 5040; 0, 0, 58, 1960, 18760, 73920, 137760, 120960, 40320; ... PROG (PARI) \\ here U(n, k) is A208535(n, k) for n > 1. U(n, k)={sumdiv(n, d, eulerphi(n/d)*(k-1)^d)/n - if(n%2, k-1)} T(n, k)={sum(j=1, k, (-1)^(k-j)*binomial(k, j)*U(n, j))} CROSSREFS Column 3 is A093367. Row sums are A330620. Cf. A087854, A208535, A327396, A330341. Sequence in context: A254281 A295682 A195772 * A062104 A257783 A226874 Adjacent sequences: A330615 A330616 A330617 * A330619 A330620 A330621 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, Dec 20 2019 STATUS approved

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Last modified February 26 02:39 EST 2024. Contains 370335 sequences. (Running on oeis4.)