A115120 - OEIS

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

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OFFSET

0,5

COMMENTS

a(p)=1 for prime p.

LINKS

Table of n, a(n) for n=0..50.

FORMULA

a(n) = A005648(n) - A045628(n).

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)];