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A276543
Triangle read by rows: T(n,k) = number of primitive (period n) n-bead bracelet structures using exactly k different colored beads.
15
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
OFFSET
1,8
COMMENTS
Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure.
REFERENCES
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]
LINKS
FORMULA
T(n, k) = Sum_{d|n} mu(n/d) * A152176(d, k).
EXAMPLE
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
...
PROG
(PARI) \\ Ach is A304972 and R is A152175 as square matrices.
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
CROSSREFS
Partial row sums include A000046, A056362, A056363, A056364, A056365.
Row sums are A276548.
Sequence in context: A063250 A348967 A285308 * A107424 A155161 A185937
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Apr 09 2017
STATUS
approved