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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.

%I A330618 #12 Jan 09 2020 19:34:16
%S A330618 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 A330618 1440,720,0,1,33,696,4410,11340,12600,5040,0,0,58,1960,18760,73920,
%U A330618 137760,120960,40320,0,1,105,5508,76368,433944,1209600,1753920,1270080,362880
%N 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.
%C A330618 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 A330618 Andrew Howroyd, <a href="/A330618/b330618.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%F A330618 T(n,k) = Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*A208535(n,j) for n > 1.
%F A330618 T(n,n) = (n-1)! for n > 1.
%e A330618 Triangle begins:
%e A330618   0;
%e A330618   0, 1;
%e A330618   0, 0,  2;
%e A330618   0, 1,  3,    6;
%e A330618   0, 0,  6,   24,    24;
%e A330618   0, 1, 11,   80,   180,   120;
%e A330618   0, 0, 18,  240,   960,  1440,    720;
%e A330618   0, 1, 33,  696,  4410, 11340,  12600,   5040;
%e A330618   0, 0, 58, 1960, 18760, 73920, 137760, 120960, 40320;
%e A330618   ...
%o A330618 (PARI) \\ here U(n,k) is A208535(n,k) for n > 1.
%o A330618 U(n, k)={sumdiv(n, d, eulerphi(n/d)*(k-1)^d)/n - if(n%2, k-1)}
%o A330618 T(n,k)={sum(j=1, k, (-1)^(k-j)*binomial(k,j)*U(n,j))}
%Y A330618 Column 3 is A093367.
%Y A330618 Row sums are A330620.
%Y A330618 Cf. A087854, A208535, A327396, A330341.
%K A330618 nonn,tabl
%O A330618 1,6
%A A330618 _Andrew Howroyd_, Dec 20 2019