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A051137
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Table T(n,k) read by antidiagonals: number of necklaces allowing turnovers (bracelets) with n beads of k colors.
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6
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1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 6, 10, 10, 5, 1, 1, 8, 21, 20, 15, 6, 1, 1, 13, 39, 55, 35, 21, 7, 1, 1, 18, 92, 136, 120, 56, 28, 8, 1, 1, 30, 198, 430, 377, 231, 84, 36, 9, 1, 1, 46, 498, 1300, 1505, 888, 406, 120, 45, 10, 1
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OFFSET
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0,5
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COMMENTS
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Unlike A075195 and A284855, antidiagonals go from bottom-left to top-right.
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LINKS
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FORMULA
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T(n, k) = (k^floor((n+1)/2) + k^ceiling((n+1)/2)) / 4 + (1/(2*n)) * Sum_{d divides n} phi(d) * k^(n/d). - Robert A. Russell, Sep 21 2018
G.f. for column k: (kx/4)*(kx+x+2)/(1-kx^2) - Sum_{d>0} phi(d)*log(1-kx^d)/2d. - Robert A. Russell, Sep 28 2018
T(n, k) = (k^floor((n+1)/2) + k^ceiling((n+1)/2))/4 + (1/(2*n))*Sum_{i=1..n} k^gcd(n,i). (See A075195 formulas.) - Richard L. Ollerton, May 04 2021
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EXAMPLE
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Table begins with T[0,1]:
1 1 1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 10
1 3 6 10 15 21 28 36 45 55
1 4 10 20 35 56 84 120 165 220
1 6 21 55 120 231 406 666 1035 1540
1 8 39 136 377 888 1855 3536 6273 10504
1 13 92 430 1505 4291 10528 23052 46185 86185
1 18 198 1300 5895 20646 60028 151848 344925 719290
1 30 498 4435 25395 107331 365260 1058058 2707245 6278140
1 46 1219 15084 110085 563786 2250311 7472984 21552969 55605670
1 78 3210 53764 493131 3037314 14158228 53762472 174489813 500280022
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MATHEMATICA
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b[n_, k_] := DivisorSum[n, EulerPhi[#]*k^(n/#) &] / n;
c[n_, k_] := If[EvenQ[n], (k^(n/2) + k^(n/2+1))/2, k^((n+1)/2)];
T[0, _] = 1; T[n_, k_] := (b[n, k] + c[n, k])/2;
Table[T[n, k-n], {k, 1, 11}, {n, k-1, 0, -1}] // Flatten
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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