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A056290 Number of primitive (period n) n-bead necklaces with exactly five different colored beads.

%I A056290 #17 Jun 06 2018 14:32:37
%S A056290 0,0,0,0,24,300,2400,15750,92680,510288,2691600,13793850,69309240,
%T A056290 343499100,1686135352,8221421250,39901776360,193053923860,
%U A056290 932142850800,4495236287850,21664357532920,104388118174500,503044634004000,2425003910574000,11696087875731600
%N A056290 Number of primitive (period n) n-bead necklaces with exactly five different colored beads.
%C A056290 Turning over the necklace is not allowed.
%D A056290 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 A056290 Alois P. Heinz, <a href="/A056290/b056290.txt">Table of n, a(n) for n = 1..1000</a>
%F A056290 sum mu(d)*A056285(n/d) where d|n.
%p A056290 with(numtheory):
%p A056290 b:= proc(n, k) option remember; `if`(n=0, 1,
%p A056290       add(mobius(n/d)*k^d, d=divisors(n))/n)
%p A056290     end:
%p A056290 a:= n-> add(b(n, 5-j)*binomial(5, j)*(-1)^j, j=0..5):
%p A056290 seq(a(n), n=1..30);  # _Alois P. Heinz_, Jan 25 2015
%t A056290 b[n_, k_] := b[n, k] = If[n==0, 1, DivisorSum[n, MoebiusMu[n/#]*k^# &]/n];
%t A056290 a[n_] := Sum[b[n, 5 - j]*Binomial[5, j]*(-1)^j, {j, 0, 5}];
%t A056290 Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Jun 06 2018, after _Alois P. Heinz_ *)
%Y A056290 Cf. A001692.
%Y A056290 Column k=5 of A254040.
%K A056290 nonn
%O A056290 1,5
%A A056290 _Marks R. Nester_