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A056289 Number of primitive (period n) n-bead necklaces with exactly four different colored beads.

%I A056289 #18 Aug 22 2017 20:53:07
%S A056289 0,0,0,6,48,260,1200,5100,20720,81828,318000,1222870,4675440,17813820,
%T A056289 67769504,257695800,980240880,3731732200,14222737200,54278498154,
%U A056289 207438936800,793940157900,3043140078000,11681056021300,44900438149248,172824327151140,666070256468960
%N A056289 Number of primitive (period n) n-bead necklaces with exactly four different colored beads.
%C A056289 Turning over the necklace is not allowed.
%D A056289 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 A056289 Alois P. Heinz, <a href="/A056289/b056289.txt">Table of n, a(n) for n = 1..1000</a>
%F A056289 a(n) = Sum_{d|n} mu(d)*A056284(n/d).
%p A056289 with(numtheory):
%p A056289 b:= proc(n, k) option remember; `if`(n=0, 1,
%p A056289       add(mobius(n/d)*k^d, d=divisors(n))/n)
%p A056289     end:
%p A056289 a:= n-> add(b(n, 4-j)*binomial(4, j)*(-1)^j, j=0..4):
%p A056289 seq(a(n), n=1..30);  # _Alois P. Heinz_, Jan 25 2015
%t A056289 b[n_, k_] := b[n, k] = If[n==0, 1, DivisorSum[n, MoebiusMu[n/#]*k^#&]/n]; a[n_] := Sum[b[n, 4-j]*Binomial[4, j]*(-1)^j, {j, 0, 4}]; Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Feb 20 2017, after _Alois P. Heinz_ *)
%Y A056289 Cf. A027377.
%Y A056289 Column k=4 of A254040.
%K A056289 nonn
%O A056289 1,4
%A A056289 _Marks R. Nester_