A045628 Number of 2n-bead black-white reversible necklaces with n black beads and fundamental period 2n.

1, 1, 1, 2, 6, 15, 46, 132, 432, 1384, 4735, 16158, 56766, 200473, 718012, 2587000, 9398080, 34324173, 126067124, 465093570, 1723171550, 6407924165, 23910560070, 89494164972, 335913861648, 1264107416450, 4768452532025

Offset

0,4

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Index entries for sequences related to bracelets

Formula

Moebius transform of A005648. - Christian G. Bower

a(n) = 1/(4*n) * Sum_{d|n} mu(n/d) * (2*n*binomial(2*floor(d/2), floor(d/2)) + binomial(2*d, d)) for n > 0. - Andrew Howroyd, Sep 28 2017

Mathematica

a[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)];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 28 2019, from PARI *)

Prog

(PARI) a(n) = if(n<1, n==0, sumdiv(n, d, moebius(n/d)*(2*n*binomial(2*(d\2), d\2) + binomial(2*d, d)))/(4*n)); \\ Andrew Howroyd, Sep 28 2017

Crossrefs

Cf. A005648.

Sequence in context: A293114 A322188 A317976 * A186738 A127383 A237262

Adjacent sequences: A045625 A045626 A045627 * A045629 A045630 A045631

Keyword

nonn

Author

David W. Wilson

Status

approved