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A276543
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Triangle read by rows: T(n,k) = number of primitive (period n) n-bead bracelet structures using exactly k different colored beads.
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15
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1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 5, 2, 1, 0, 5, 13, 11, 3, 1, 0, 8, 31, 33, 16, 3, 1, 0, 14, 80, 136, 85, 27, 4, 1, 0, 21, 201, 478, 434, 171, 37, 4, 1, 0, 39, 533, 1849, 2270, 1249, 338, 54, 5, 1, 0, 62, 1401, 6845, 11530, 8389, 3056, 590, 70, 5, 1
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OFFSET
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1,8
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COMMENTS
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Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure.
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REFERENCES
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M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
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LINKS
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FORMULA
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T(n, k) = Sum_{d|n} mu(n/d) * A152176(d, k).
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EXAMPLE
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Triangle starts:
1
0 1
0 1 1
0 2 2 1
0 3 5 2 1
0 5 13 11 3 1
0 8 31 33 16 3 1
0 14 80 136 85 27 4 1
0 21 201 478 434 171 37 4 1
0 39 533 1849 2270 1249 338 54 5 1
...
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PROG
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Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
R(n)={Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
T(n)={my(M=(R(n)+Ach(n))/2); Mat(vectorv(n, n, sumdiv(n, d, moebius(d)*M[n/d, ])))}
{ my(A=T(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Sep 20 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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