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A208183
Number of distinct k-colored necklaces with n beads per color; square array A(n,k), n>=0, k>=0, read by antidiagonals.
16
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
0,12
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)
FORMULA
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
EXAMPLE
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, ...
MAPLE
with(numtheory):
A:= (n, k)-> `if`(n=0 or k=0, 1,
seq(seq(A(n, d-n), n=0..d), d=0..10);
MATHEMATICA
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 *)
CROSSREFS
Columns k=0+1, 2-8 give: A000012, A003239, A118644, A207816, A208190, A208191, A208192, A208193.
Main diagonal gives A252765.
Sequence in context: A253586 A347621 A318191 * A214810 A257248 A090737
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Feb 24 2012
STATUS
approved

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Last modified September 20 08:53 EDT 2024. Contains 376067 sequences. (Running on oeis4.)