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
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
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).
Comments