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A208183
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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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12
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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
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OFFSET
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0,12
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LINKS
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Alois P. Heinz, Antidiagonals n = 0..35, flattened
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FORMULA
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A(n,k) = Sum_{d|n} phi(n/d)*(k*d)!/(d!^k*k*n) if n,k>0; A(n,k) = 1 else.
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EXAMPLE
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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, ...
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MAPLE
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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);
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CROSSREFS
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Rows 0-8 give: A000012, A104150, A137729, A208184, A208185, A208186, A208187, A208188, A208189.
Columns 0+1, 2-8 give: A000012, A003239, A118644, A207816, A208190, A208191, A208192, A208193.
Cf. A000010, A000142.
Sequence in context: A180264 A225200 A128706 * A214810 A090737 A204016
Adjacent sequences: A208180 A208181 A208182 * A208184 A208185 A208186
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz, Feb 24 2012
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STATUS
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approved
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