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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

%I #12 Jan 09 2020 19:34:16

%S 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,

%T 1440,720,0,1,33,696,4410,11340,12600,5040,0,0,58,1960,18760,73920,

%U 137760,120960,40320,0,1,105,5508,76368,433944,1209600,1753920,1270080,362880

%N 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.

%C 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.

%H Andrew Howroyd, <a href="/A330618/b330618.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)

%F T(n,k) = Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*A208535(n,j) for n > 1.

%F T(n,n) = (n-1)! for n > 1.

%e Triangle begins:

%e 0;

%e 0, 1;

%e 0, 0, 2;

%e 0, 1, 3, 6;

%e 0, 0, 6, 24, 24;

%e 0, 1, 11, 80, 180, 120;

%e 0, 0, 18, 240, 960, 1440, 720;

%e 0, 1, 33, 696, 4410, 11340, 12600, 5040;

%e 0, 0, 58, 1960, 18760, 73920, 137760, 120960, 40320;

%e ...

%o (PARI) \\ here U(n,k) is A208535(n,k) for n > 1.

%o U(n, k)={sumdiv(n, d, eulerphi(n/d)*(k-1)^d)/n - if(n%2, k-1)}

%o T(n,k)={sum(j=1, k, (-1)^(k-j)*binomial(k,j)*U(n,j))}

%Y Column 3 is A093367.

%Y Row sums are A330620.

%Y Cf. A087854, A208535, A327396, A330341.

%K nonn,tabl

%O 1,6

%A _Andrew Howroyd_, Dec 20 2019