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A327396
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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.
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7
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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, 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, 640, 2522, 3318, 1857, 495, 65, 4, 1, 0, 1, 63, 1817, 9888, 17594, 13311, 4911, 944, 97, 5, 1
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OFFSET
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1,18
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COMMENTS
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Permuting the colors does not change the necklace structure.
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.
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LINKS
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EXAMPLE
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Triangle begins:
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, 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, 640, 2522, 3318, 1857, 495, 65, 4, 1;
0, 1, 63, 1817, 9888, 17594, 13311, 4911, 944, 97, 5, 1;
...
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PROG
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(PARI)
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)]))}
{ my(A=R(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Oct 09 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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