A208183 - OEIS

%I #35 May 20 2021 10:55:54

%S 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,

%T 11352,30804,2896,26,1,1,1,720,623760,11211216,3941598,50452,80,1,1,1,

%U 5040,48648960,7623616080,15277017432,586637256,953056,246,1,1

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

%C From _Vaclav Kotesovec_, Aug 23 2015: (Start)

%C Column k > 1 is asymptotic to k^(k*n-1/2) / ((2*Pi)^((k-1)/2) * n^((k+1)/2)).

%C Row r > 0 is asymptotic to (r*n)! / (r*n*(r!)^n). (End)

%H Alois P. Heinz, <a href="/A208183/b208183.txt">Antidiagonals n = 0..35, flattened</a>

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

%F 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

%e A(1,4) =  6: {0123, 0132, 0213, 0231, 0312, 0321}.

%e A(3,2) =  4: {000111, 001011, 010011, 010101}.

%e A(4,2) = 10: {00001111, 00010111, 00100111, 01000111, 00011011, 00110011, 00101011, 01010011, 01001011, 01010101}.

%e Square array A(n,k) begins:

%e   1, 1,  1,     1,         1,              1, ...

%e   1, 1,  1,     2,         6,             24, ...

%e   1, 1,  2,    16,       318,          11352, ...

%e   1, 1,  4,   188,     30804,       11211216, ...

%e   1, 1, 10,  2896,   3941598,    15277017432, ...

%e   1, 1, 26, 50452, 586637256, 24934429725024, ...

%p with(numtheory):

%p A:= (n, k)-> `if`(n=0 or k=0, 1,

%p               add(phi(n/d) *(k*d)!/(d!^k *k*n), d=divisors(n))):

%p seq(seq(A(n, d-n), n=0..d), d=0..10);

%t 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 *)

%Y Rows n=0-8 give: A000012, A104150, A137729, A208184, A208185, A208186, A208187, A208188, A208189.

%Y Columns k=0+1, 2-8 give: A000012, A003239, A118644, A207816, A208190, A208191, A208192, A208193.

%Y Main diagonal gives A252765.

%Y Cf. A000010, A000142.

%K nonn,tabl

%O 0,12

%A _Alois P. Heinz_, Feb 24 2012