A276543 Triangle read by rows: T(n,k) = number of primitive (period n) n-bead bracelet structures using exactly k different colored beads.

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

Author

Andrew Howroyd, Apr 09 2017

Keywords

• nonn
• tabl

Comments

Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure.

Examples

			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
  ...
		

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

• Andrew Howroyd, Table of n, a(n) for n = 1..1275

Crossrefs

Columns 1-6 are A063524, A056366, A056367, A056368, A056369, A056370.

Partial row sums include A000046, A056362, A056363, A056364, A056365.

Row sums are A276548.

Cf. A276550, A152175, A152176, A107424, A137651, A276544, A304972.

Programs

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

Formula

T(n, k) = Sum_{d|n} mu(n/d) * A152176(d, k).