A261497 Number of necklaces with n white beads and 3*n black beads.

1, 1, 4, 19, 116, 776, 5620, 42288, 328756, 2615104, 21191904, 174303163, 1451430692, 12211799224, 103655949072, 886568153744, 7633233556276, 66105170315084, 575445692499952, 5032380942945322, 44191451788248416, 389514699013012242, 3444925385336301684

Offset: 0

Author

Alois P. Heinz, Aug 21 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• 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

Crossrefs

Column k=3 of A261494.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(4*n/d, n/d)
             *phi(d), d=divisors(n))/(4*n)):
seq(a(n), n=0..25);

Formula

a(n) = 1/(4*n) * Sum_{d|n} C(4*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ 2^(8*n-3/2) / (3^(3*n) * sqrt(3*Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015