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%I #13 Aug 30 2024 12:13:37
%S 1,2,1,2,3,6,9,18,30,56,98,180,311,546,915,1520,2440,3855,5916,8935,
%T 13178,19138,27264,38303,52950,72311,97419,129839,171066,223260,
%U 288498,369708,469708,592363,741433,921933,1138761,1398343,1706956,2072696,2503513,3009482,3600515,4289032,5087253,6010305,7073122,8293962
%N Number of aperiodic necklaces (Lyndon words) with k<=8 black beads and n-k white beads.
%H <a href="/index/Rec#order_28">Index entries for linear recurrences with constant coefficients</a>, signature (1, 3, -2, -4, 1, 4, -1, -5, 2, 7, -1, -6, 0, 4, 0, -6, -1, 7, 2, -5, -1, 4, 1, -4, -2, 3, 1, -1).
%F G.f.: 1 + x + x/(1-x) + 1/2*x^2*(1/(1-x)^2 - 1/(1-x^2)) + 1/3*x^3*(1/(1-x)^3 - 1/(1-x^3)) + 1/4*x^4*(1/(1-x)^4 - 1/(1-x^2)^2) + 1/5*x^5*(1/(1-x)^5 - 1/(1-x^5)) + 1/6*x^6*(1/(1-x)^6 - 1/(1-x^2)^3 - 1/(1-x^3)^2 + 1/(1-x^6)) + 1/7*x^7*(1/(1-x)^7 - 1/(1-x^7)) + 1/8*x^8*(1/(1-x)^8 - 1/(1-x^2)^4).
%t (* The g.f. for the number of aperiodic necklaces (Lyndon words) with k<=m black beads and n-k white beads. Here we have the case m=8 *)
%t gf[x_, m_]:=Sum[x^i/i Plus@@(MoebiusMu[#](1-x^#)^(-(i/#))&/@Divisors[i]), {i, 1, m}]+x+1
%Y Cf. A001037 (k arbitrary), A008747 (k<=3), A277619 (k<=4), A277629 (k<=5), A277631 (k<=6).
%Y The Mathematica section of A032168 gives the g.f. for k=m black beads and n-k white beads.
%K nonn,easy
%O 0,2
%A _Herbert Kociemba_, Oct 24 2016