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A045684 Number of 2n-bead balanced binary necklaces of fundamental period 2n which are inequivalent to their reverse, complement and reversed complement.

%I A045684 #18 Sep 23 2019 12:22:48
%S A045684 0,0,0,0,0,8,32,168,616,2380,8464,30760,109612,394816,1420616,5149940,
%T A045684 18736128,68553728,251899620,929814984,3445425136,12814382452,
%U A045684 47817520376,178982546512,671813585080,2528191984496,9536849432000
%N A045684 Number of 2n-bead balanced binary necklaces of fundamental period 2n which are inequivalent to their reverse, complement and reversed complement.
%C A045684 The number of length 2n balanced binary Lyndon words is A022553(n) and the number which are equivalent to their reverse, complement and reversed complement are respectively A045680(n), A000048(n) and A000740(n). - _Andrew Howroyd_, Sep 29 2017
%H A045684 Jean-François Alcover, <a href="/A045684/b045684.txt">Table of n, a(n) for n = 0..100</a>
%H A045684 <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>
%F A045684 From _Andrew Howroyd_, Sep 28 2017: (Start)
%F A045684 Moebius transform of A045675.
%F A045684 a(n) = A022553(n) - A045680(n) - A000048(n) - A000740(n) + 2*A045683(n).
%F A045684 (End)
%t A045684 a22553[n_] := If[n == 0, 1, Sum[MoebiusMu[n/d]*Binomial[2d, d], {d, Divisors[n]}]/(2n)];
%t A045684 a45680[n_] := If[n == 0, 1, DivisorSum[n, MoebiusMu[n/#] Binomial[# - Mod[#, 2], Quotient[#, 2]] &]];
%t A045684 a48[n_] := If[n == 0, 1, Total[MoebiusMu[#]*2^(n/#)& /@ Select[Divisors[n], OddQ]]/(2n)];
%t A045684 a740[n_] := Sum[MoebiusMu[n/d]*2^(d - 1), {d, Divisors[n]}];
%t A045684 b[n_] := Module[{t = 0, r = n}, If[n == 0, 1, While[EvenQ[r], r = Quotient[r, 2]; t += 2^(r - 1)]]; t + 2^Quotient[r, 2]];
%t A045684 a45683[n_] := If[n == 0, 1, DivisorSum[n, MoebiusMu[n/#]*b[#] &]];
%t A045684 a[n_] := If[n == 0, 0, a22553[n] - a45680[n] - a48[n] - a740[n] + 2 a45683[n]];
%t A045684 a /@ Range[0, 100] (* _Jean-François Alcover_, Sep 23 2019 *)
%Y A045684 Cf. A000048, A000740, A022553, A045675, A045680, A045683.
%K A045684 nonn
%O A045684 0,6
%A A045684 _David W. Wilson_