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A261497 Number of necklaces with n white beads and 3*n black beads. 2
1, 1, 4, 19, 116, 776, 5620, 42288, 328756, 2615104, 21191904, 174303163, 1451430692, 12211799224, 103655949072, 886568153744, 7633233556276, 66105170315084, 575445692499952, 5032380942945322, 44191451788248416, 389514699013012242, 3444925385336301684 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

F. Ruskey, Necklaces, unlabelled necklaces, Lyndon words, De Bruijn sequences

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/(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

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);

CROSSREFS

Column k=3 of A261494.

Sequence in context: A186359 A203268 A209673 * A217989 A281817 A060907

Adjacent sequences:  A261494 A261495 A261496 * A261498 A261499 A261500

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Aug 21 2015

STATUS

approved

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Last modified January 20 23:20 EST 2019. Contains 319343 sequences. (Running on oeis4.)