A261504 Number of necklaces with n white beads and 10*n black beads.

1, 1, 11, 166, 3091, 63251, 1376738, 31231278, 730421043, 17485161178, 426342182761, 10551652603526, 264383070217234, 6693356751004275, 170957058318902538, 4399826606699783666, 113989308154754290611, 2970465160237643057033, 77808659386282667142686

Offset: 0

Author

Alois P. Heinz, Aug 21 2015

Keywords

• nonn

Comments

In general, for k>=1 is column k of A261494 asymptotic to (k+1)^((k+1)*n-1/2) / (sqrt(2*Pi) * k^(k*n+1/2) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..690
• 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=10 of A261494.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(11*n/d, n/d)
             *phi(d), d=divisors(n))/(11*n)):
seq(a(n), n=0..25);

Mathematica

a[n_] := If[n == 0, 1, DivisorSum[n, Binomial[11*n/#, n/#]*EulerPhi[#]&]/ (11*n)];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 28 2017, translated from Maple *)

Formula

a(n) = 1/(11*n) * Sum_{d|n} C(11*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ 11^(11*n-1/2) / (sqrt(2*Pi) * 10^(10*n+1/2) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015