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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.

%I A276543 #21 Sep 23 2019 14:32:12
%S A276543 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,
%T A276543 136,85,27,4,1,0,21,201,478,434,171,37,4,1,0,39,533,1849,2270,1249,
%U A276543 338,54,5,1,0,62,1401,6845,11530,8389,3056,590,70,5,1
%N A276543 Triangle read by rows: T(n,k) = number of primitive (period n) n-bead bracelet structures using exactly k different colored beads.
%C A276543 Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure.
%D A276543 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]
%H A276543 Andrew Howroyd, <a href="/A276543/b276543.txt">Table of n, a(n) for n = 1..1275</a>
%F A276543 T(n, k) = Sum_{d|n} mu(n/d) * A152176(d, k).
%e A276543 Triangle starts:
%e A276543   1
%e A276543   0  1
%e A276543   0  1   1
%e A276543   0  2   2    1
%e A276543   0  3   5    2    1
%e A276543   0  5  13   11    3    1
%e A276543   0  8  31   33   16    3   1
%e A276543   0 14  80  136   85   27   4  1
%e A276543   0 21 201  478  434  171  37  4 1
%e A276543   0 39 533 1849 2270 1249 338 54 5 1
%e A276543   ...
%o A276543 (PARI) \\ Ach is A304972 and R is A152175 as square matrices.
%o A276543 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}
%o A276543 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))]))}
%o A276543 T(n)={my(M=(R(n)+Ach(n))/2); Mat(vectorv(n,n,sumdiv(n, d, moebius(d)*M[n/d,])))}
%o A276543 { my(A=T(12)); for(n=1, #A, print(A[n, 1..n])) } \\ _Andrew Howroyd_, Sep 20 2019
%Y A276543 Columns 1-6 are A063524, A056366, A056367, A056368, A056369, A056370.
%Y A276543 Partial row sums include A000046, A056362, A056363, A056364, A056365.
%Y A276543 Row sums are A276548.
%Y A276543 Cf. A276550, A152175, A152176, A107424, A137651, A276544, A304972.
%K A276543 nonn,tabl
%O A276543 1,8
%A A276543 _Andrew Howroyd_, Apr 09 2017