A118644 -id:A118644 - cp's OEIS frontend

A208183 Number of distinct k-colored necklaces with n beads per color; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 6, 16, 4, 1, 1, 1, 24, 318, 188, 10, 1, 1, 1, 120, 11352, 30804, 2896, 26, 1, 1, 1, 720, 623760, 11211216, 3941598, 50452, 80, 1, 1, 1, 5040, 48648960, 7623616080, 15277017432, 586637256, 953056, 246, 1, 1

Offset: 0

Alois P. Heinz, Feb 24 2012

From Vaclav Kotesovec, Aug 23 2015: (Start)
Column k > 1 is asymptotic to k^(k*n-1/2) / ((2*Pi)^((k-1)/2) * n^((k+1)/2)).
Row r > 0 is asymptotic to (r*n)! / (r*n*(r!)^n). (End)

			A(1,4) =  6: {0123, 0132, 0213, 0231, 0312, 0321}.
A(3,2) =  4: {000111, 001011, 010011, 010101}.
A(4,2) = 10: {00001111, 00010111, 00100111, 01000111, 00011011, 00110011, 00101011, 01010011, 01001011, 01010101}.
Square array A(n,k) begins:
  1, 1,  1,     1,         1,              1, ...
  1, 1,  1,     2,         6,             24, ...
  1, 1,  2,    16,       318,          11352, ...
  1, 1,  4,   188,     30804,       11211216, ...
  1, 1, 10,  2896,   3941598,    15277017432, ...
  1, 1, 26, 50452, 586637256, 24934429725024, ...

Alois P. Heinz, Antidiagonals n = 0..35, flattened

Rows n=0-8 give: A000012, A104150, A137729, A208184, A208185, A208186, A208187, A208188, A208189.
Columns k=0+1, 2-8 give: A000012, A003239, A118644, A207816, A208190, A208191, A208192, A208193.
Main diagonal gives A252765.
Cf. A000010, A000142.

with(numtheory):
A:= (n, k)-> `if`(n=0 or k=0, 1,
              add(phi(n/d) *(k*d)!/(d!^k *k*n), d=divisors(n))):
seq(seq(A(n, d-n), n=0..d), d=0..10);

A[n_, k_] :=  If[n == 0 || k == 0, 1, Sum[EulerPhi[n/d]*(k*d)!/(d!^k*k*n), {d, Divisors[n]}]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

A(n,k) = Sum_{d|n} phi(n/d)*(k*d)!/(d!^k*k*n) if n,k>0; A(n,k) = 1 else.

A(n,k) = Sum_{i=1..n} (k*gcd(n,i))!/(gcd(n,i)!^k*k*n) = Sum_{i=1..n} (k*n/gcd(n,i))!/((n/gcd(n,i))!^k*k*n)*phi(gcd(n,i))/phi(n/gcd(n,i)) for n,k >= 1, where phi = A000010. - Richard L. Ollerton, May 19 2021

A207816 Number of distinct necklaces with n red, n green, n blue and n white beads.

Original entry on oeis.org

1, 6, 318, 30804, 3941598, 586637256, 96197661156, 16875655269948, 3111284141045598, 595909785174057204, 117634021777132574568, 23797087019979071174580, 4912693780461352534397604, 1031629572413246016139181544, 219809927417367534490107035244, 47426945434432859336092700072304

Offset: 0

Thotsaporn Thanatipanonda, Feb 20 2012

nonn

			For n=1, a(1)=6 since for four beads necklaces with each bead from each of the four colors say (R,G,B,W), we can arrange as following, [R,G,B,W], [R,G,W,B], [R,B,G,W], [R,B,W,G], [R,W,G,B] and [R,W,B,G].

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A003239, A118644, A137729.

Column k=4 of A208183. - Alois P. Heinz, Feb 24 2012

with(combinat): with(numtheory):
# This formula comes from Polya Counting Theorem:
# Z(C_n) = add(phi(d)*(a_d)^(n/d), d in divisors(n))/n;
PolyaBrace:= proc(S) option remember; local n, s, d;
               n:= add(s, s=S);
               add(phi(d) *PolyaCoeff(d, S), d=divisors(n))/n
             end:
# Find coeff of prod(a[i]^s[i], i=1..n) of a_d^(n/d) (symmetric function)
PolyaCoeff:= proc(d, S) option remember; local n, pow, s;
               n:= add(s, s=S);
               pow:= n/d;
               if {seq(s mod d, s = S)} = {0}
                  then multinomial(pow, seq(s/d, s = S))
                  else 0
               fi:
             end:
a:= n-> `if`(n=0, 1, PolyaBrace([n$4])):
seq(a(n), n=0..20);

a[n_] := DivisorSum[n, EulerPhi[n/#] (4#)!/(#!^4 * 4n)&]; a[0]=1;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 24 2017, after Alois P. Heinz *)

a(n) = Sum_{d|n} phi(n/d)*(4*d)!/(d!^4*4*n) if n>0 and a(0) = 1. - Alois P. Heinz, Feb 24 2012

a(n) ~ 2^(8*n-5/2) / (Pi^(3/2) * n^(5/2)). - Vaclav Kotesovec, Aug 23 2015

cp's OEIS Frontend

A208183 Number of distinct k-colored necklaces with n beads per color; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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