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A327396 Triangle read by rows: T(n,k) is the number of n-bead necklace structures with beads of exactly k colors and no adjacent beads having the same color.

%I A327396 #20 Oct 10 2019 22:20:00
%S A327396 0,0,1,0,0,1,0,1,1,1,0,0,1,1,1,0,1,3,5,2,1,0,0,3,10,8,2,1,0,1,7,33,40,
%T A327396 18,3,1,0,0,11,83,157,104,28,3,1,0,1,19,237,650,615,246,46,4,1,0,0,31,
%U A327396 640,2522,3318,1857,495,65,4,1,0,1,63,1817,9888,17594,13311,4911,944,97,5,1
%N A327396 Triangle read by rows: T(n,k) is the number of n-bead necklace structures with beads of exactly k colors and no adjacent beads having the same color.
%C A327396 Permuting the colors does not change the necklace structure.
%C A327396 Equivalently, the number of k-block partitions of an n-set up to rotations where no block contains cyclically adjacent elements of the n-set.
%H A327396 Andrew Howroyd, <a href="/A327396/b327396.txt">Table of n, a(n) for n = 1..1275</a>
%e A327396 Triangle begins:
%e A327396   0;
%e A327396   0, 1;
%e A327396   0, 0,  1;
%e A327396   0, 1,  1,    1;
%e A327396   0, 0,  1,    1,    1;
%e A327396   0, 1,  3,    5,    2,     1;
%e A327396   0, 0,  3,   10,    8,     2,     1;
%e A327396   0, 1,  7,   33,   40,    18,     3,    1;
%e A327396   0, 0, 11,   83,  157,   104,    28,    3,   1;
%e A327396   0, 1, 19,  237,  650,   615,   246,   46,   4,  1;
%e A327396   0, 0, 31,  640, 2522,  3318,  1857,  495,  65,  4, 1;
%e A327396   0, 1, 63, 1817, 9888, 17594, 13311, 4911, 944, 97, 5, 1;
%e A327396   ...
%o A327396 (PARI)
%o A327396 R(n) = {Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace((y-1)*exp(-x + O(x*x^(n\m))) - y + exp(-x + sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d)) ), x, x^m))/x), -n)]))}
%o A327396 { my(A=R(12)); for(n=1, #A, print(A[n, 1..n])) } \\ _Andrew Howroyd_, Oct 09 2019
%Y A327396 Columns k=3..4 are A327397, A328130.
%Y A327396 Partial row sums include A306888, A309673.
%Y A327396 Row sums are A328150.
%Y A327396 Cf. A152175, A261139, A208535.
%K A327396 nonn,tabl
%O A327396 1,18
%A A327396 _Andrew Howroyd_, Oct 04 2019