A261496 Number of necklaces with n white beads and n^2-n black beads.

1, 1, 2, 10, 116, 2126, 54132, 1753074, 69159400, 3220837534, 173103115760, 10551652603526, 719578430426044, 54297978110913252, 4492502634679508204, 404469190271900056316, 39370123445405248353744, 4120204305690280446004838, 461365717080849798202175772

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..335

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]

Eric Weisstein's World of Mathematics, Necklace

Wikipedia, Necklace (combinatorics)

Index entries for sequences related to necklaces

Formula

a(n) = 1/(n^2) * Sum_{d|n} C(n^2/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ exp(n-1/2) * n^(n-5/2) / sqrt(2*Pi). - Vaclav Kotesovec, Aug 22 2015

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(n^2/d, n/d)
            *phi(d), d=divisors(n))/n^2):
seq(a(n), n=0..20);

Crossrefs

Lower diagonal of A261494.

Cf. A000010.

Sequence in context: A132522 A187653 A131811 * A347014 A356514 A006121

Adjacent sequences: A261493 A261494 A261495 * A261497 A261498 A261499

Keyword

nonn

Author

Alois P. Heinz, Aug 21 2015

Status

approved