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A261494 Number A(n,k) of necklaces with n white beads and k*n black beads; square array A(n,k), n>=0, k>=0, read by antidiagonals. 13
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 10, 10, 1, 1, 1, 5, 19, 43, 26, 1, 1, 1, 6, 31, 116, 201, 80, 1, 1, 1, 7, 46, 245, 776, 1038, 246, 1, 1, 1, 8, 64, 446, 2126, 5620, 5538, 810, 1, 1, 1, 9, 85, 735, 4751, 19811, 42288, 30667, 2704, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

For k>=1 is column k 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, Antidiagonals n = 0..140, flattened

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,k) = 1/((k+1)*n) * Sum_{d|n} C((k+1)*n/d,n/d) * A000010(d) for n>0, A(0,k) = 1.

EXAMPLE

A(2,2) = 3: 000011, 000101, 001001.

A(3,2) = 10: 000000111, 000001011, 000010011, 000100011, 001000011, 010000011, 000010101, 000100101, 001000101, 001001001.

Square array A(n,k) begins:

  1,  1,    1,    1,     1,     1,      1, ...

  1,  1,    1,    1,     1,     1,      1, ...

  1,  2,    3,    4,     5,     6,      7, ...

  1,  4,   10,   19,    31,    46,     64, ...

  1, 10,   43,  116,   245,   446,    735, ...

  1, 26,  201,  776,  2126,  4751,   9276, ...

  1, 80, 1038, 5620, 19811, 54132, 124936, ...

MAPLE

with(numtheory):

A:= (n, k)-> `if`(n=0, 1, add(binomial((k+1)*n/d, n/d)

                    *phi(d), d=divisors(n))/((k+1)*n)):

seq(seq(A(n, d-n), n=0..d), d=0..14);

MATHEMATICA

A[n_, k_] := If[n==0, 1, DivisorSum[n, Binomial[(k+1)*n/#, n/#]*EulerPhi[#] /((k+1)*n)&]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

PROG

(PARI) a(n, k) = if(n<1, 1, sumdiv(n, d, binomial((k + 1)*n/d, n/d) * eulerphi(d)) / ((k + 1)*n));

for(d=0, 14, for(n=0, d, print1(a(n, d - n), ", "); ); print(); ) \\ Indranil Ghosh, Mar 25 2017

CROSSREFS

Columns k=0-10 give: A000012, A003239, A082936, A261497, A261498, A261499, A261500, A261501, A261502, A261503, A261504.

Main diagonal gives A261495.

Lower diagonal gives A261496.

Cf. A000010.

Sequence in context: A084097 A293991 A288638 * A168377 A122867 A124775

Adjacent sequences:  A261491 A261492 A261493 * A261495 A261496 A261497

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 21 2015

STATUS

approved

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Last modified March 18 19:58 EDT 2019. Contains 321293 sequences. (Running on oeis4.)