A261494 -id:A261494 - cp's OEIS frontend

A261503 Number of necklaces with n white beads and 9*n black beads.

1, 1, 10, 136, 2290, 42376, 834472, 17125354, 362345362, 7847250409, 173103115760, 3875837737520, 87857163416200, 2012268157890524, 46497242510288804, 1082614775186919136, 25374686712458407442, 598217593341171422090, 14176230568954212214234

Offset: 0

Alois P. Heinz, Aug 21 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..710
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

Column k=9 of A261494.

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(10*n/d, n/d)
             *phi(d), d=divisors(n))/(10*n)):
seq(a(n), n=0..25);

a(n) = 1/(10*n) * Sum_{d|n} C(10*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ 10^(10*n-1/2) / (sqrt(2*Pi) * 9^(9*n+1/2) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015

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

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Aug 21 2015

nonn

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

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

Column k=10 of A261494.

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);

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 *)

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

cp's OEIS Frontend

A261503 Number of necklaces with n white beads and 9*n black beads.

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