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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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,12 COMMENTS 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) LINKS Alois P. Heinz, Antidiagonals n = 0..35, flattened 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. 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,               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); 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 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. Sequence in context: A128706 A253586 A318191 * A214810 A257248 A090737 Adjacent sequences:  A208180 A208181 A208182 * A208184 A208185 A208186 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Feb 24 2012 STATUS approved

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Last modified March 21 22:19 EDT 2019. Contains 321382 sequences. (Running on oeis4.)