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

0,5

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)];
a[n_] := A005648[n] - A045628[n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 28 2019 *)

Crossrefs

Sequence in context: A072345 A200583 A366842 * A147373 A147441 A146269

Adjacent sequences: A115117 A115118 A115119 * A115121 A115122 A115123

Keyword

easy,nonn

Author

Valery A. Liskovets, Jan 17 2006

Extensions

More terms from Jean-François Alcover, Aug 28 2019

Status

approved