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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.

%I A330341 #30 Jan 09 2020 19:23:34
%S A330341 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,
%T A330341 360,0,1,27,384,2235,5670,6300,2520,0,0,29,980,9380,36960,68880,60480,
%U A330341 20160,0,1,75,2904,38484,217152,604800,876960,635040,181440
%N 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.
%C A330341 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.
%H A330341 Andrew Howroyd, <a href="/A330341/b330341.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%F A330341 T(n,k) = Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*A208544(n,j) for n > 1.
%e A330341 Triangle begins:
%e A330341   0;
%e A330341   0, 1;
%e A330341   0, 0,  1;
%e A330341   0, 1,  3,   3;
%e A330341   0, 0,  3,  12,   12;
%e A330341   0, 1, 10,  46,   90,    60;
%e A330341   0, 0,  9, 120,  480,   720,   360;
%e A330341   0, 1, 27, 384, 2235,  5670,  6300,  2520;
%e A330341   0, 0, 29, 980, 9380, 36960, 68880, 60480, 20160;
%e A330341   ...
%o A330341 (PARI) \\ here U(n, k) is A208544(n, k) for n > 1.
%o A330341 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;
%o A330341 T(n, k)={sum(j=1, k, (-1)^(k-j)*binomial(k, j)*U(n, j))}
%Y A330341 Column 3 is A330632.
%Y A330341 Row sums are A330621.
%Y A330341 Cf. A208544, A273891, A330618.
%K A330341 nonn,tabl
%O A330341 1,9
%A A330341 _Andrew Howroyd_, Dec 20 2019