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A327693 Triangle read by rows: T(n,k) is the number of n-bead necklace structures which are not self-equivalent under a nonzero rotation using exactly k different colored beads.

%I A327693 #16 Oct 30 2019 20:01:49
%S A327693 1,0,0,0,1,0,0,1,1,0,0,3,5,2,0,0,4,13,9,2,0,0,9,43,50,20,3,0,0,14,116,
%T A327693 206,127,31,3,0,0,28,335,862,772,293,51,4,0,0,48,920,3384,4226,2263,
%U A327693 580,72,4,0,0,93,2591,13250,22430,16317,5817,1080,105,5,0
%N A327693 Triangle read by rows: T(n,k) is the number of n-bead necklace structures which are not self-equivalent under a nonzero rotation using exactly k different colored beads.
%C A327693 Permuting the colors does not change the structure.
%C A327693 The definition requires that a necklace must not be equivalent to itself by permutation of colors and rotation (except for identity rotation). For example the length 2 necklace AB is excluded because a rotation of 1 gives BA and permutation of colors brings back to AB.
%H A327693 Andrew Howroyd, <a href="/A327693/b327693.txt">Table of n, a(n) for n = 1..1275</a>
%e A327693 Triangle begins:
%e A327693   1;
%e A327693   0,  0;
%e A327693   0,  1,   0;
%e A327693   0,  1,   1,    0;
%e A327693   0,  3,   5,    2,    0;
%e A327693   0,  4,  13,    9,    2,    0;
%e A327693   0,  9,  43,   50,   20,    3,   0;
%e A327693   0, 14, 116,  206,  127,   31,   3,  0;
%e A327693   0, 28, 335,  862,  772,  293,  51,  4, 0;
%e A327693   0, 48, 920, 3384, 4226, 2263, 580, 72, 4, 0;
%e A327693   ...
%e A327693 T(6, 4) = 9: {aaabcd, aabacd, aabcad, aabbcd, aabcbd, aabcdb, aacbdb, ababcd, abacbd}. Compared with A107424 the patterns {abacad, aacbbd, abcabd, acabdb} are excluded.
%o A327693 (PARI)
%o A327693 R(n) = {Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, moebius(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
%o A327693 { my(A=R(12)); for(n=1, #A, print(A[n, 1..n])) }
%Y A327693 Columns k=2..4 are A051841, A328740, A328741.
%Y A327693 Row sums are A327696.
%Y A327693 Partial row sums include A328742, A328743.
%Y A327693 Cf. A324802 (not self-equivalent under reversal and rotations).
%Y A327693 Cf. A107424, A152175.
%K A327693 nonn,tabl
%O A327693 1,12
%A A327693 _Andrew Howroyd_, Sep 22 2019