A115120 Number of imprimitive (periodic) 2n-bead black-white reversible necklaces with n black beads.
0, 0, 1, 1, 2, 1, 4, 1, 8, 3, 17, 1, 56, 1, 134, 18, 440, 1, 1434, 1, 4758, 135, 16160, 1, 57254, 16, 200475, 1387, 718152, 1, 2591800, 1, 9398520, 16161, 34324175, 148, 126125330, 1, 465093572, 200476, 1723176740, 1, 6408642359, 1, 23910576236, 2588402, 89494164974, 1, 335923316982, 133, 1264107421202
Offset: 0
Programs
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Mathematica
A005648[0] = 1; A005648[n_] := (1/2)(Binomial[2 Quotient[n, 2], Quotient[n, 2]] + DivisorSum[n, EulerPhi[#] Binomial[2n/#, n/#] &]/(2n)); A045628[n_] := If[n == 0, 1, Sum[MoebiusMu[n/d] (2n Binomial[2 Quotient[d, 2], Quotient[d, 2]] + Binomial[2d, d]), {d, Divisors[n]}]/(4n)]; a[n_] := A005648[n] - A045628[n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 28 2019 *)
Extensions
More terms from Jean-François Alcover, Aug 28 2019
Comments