%I #18 Sep 23 2019 12:22:48

%S 0,0,0,0,0,8,32,168,616,2380,8464,30760,109612,394816,1420616,5149940,

%T 18736128,68553728,251899620,929814984,3445425136,12814382452,

%U 47817520376,178982546512,671813585080,2528191984496,9536849432000

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

%C 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 Jean-François Alcover, <a href="/A045684/b045684.txt">Table of n, a(n) for n = 0..100</a>

%H <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>

%F From _Andrew Howroyd_, Sep 28 2017: (Start)

%F Moebius transform of A045675.

%F a(n) = A022553(n) - A045680(n) - A000048(n) - A000740(n) + 2*A045683(n).

%F (End)

%t a22553[n_] := If[n == 0, 1, Sum[MoebiusMu[n/d]*Binomial[2d, d], {d, Divisors[n]}]/(2n)];

%t a45680[n_] := If[n == 0, 1, DivisorSum[n, MoebiusMu[n/#] Binomial[# - Mod[#, 2], Quotient[#, 2]] &]];

%t a48[n_] := If[n == 0, 1, Total[MoebiusMu[#]*2^(n/#)& /@ Select[Divisors[n], OddQ]]/(2n)];

%t a740[n_] := Sum[MoebiusMu[n/d]*2^(d - 1), {d, Divisors[n]}];

%t 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 a45683[n_] := If[n == 0, 1, DivisorSum[n, MoebiusMu[n/#]*b[#] &]];

%t a[n_] := If[n == 0, 0, a22553[n] - a45680[n] - a48[n] - a740[n] + 2 a45683[n]];

%t a /@ Range[0, 100] (* _Jean-François Alcover_, Sep 23 2019 *)

%Y Cf. A000048, A000740, A022553, A045675, A045680, A045683.

%K nonn

%O 0,6

%A _David W. Wilson_