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A321791
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Table read by descending antidiagonals: T(n,k) is the number of unoriented cycles (bracelets) of length n using up to k available colors.
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5
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1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 1, 0, 1, 5, 10, 10, 6, 1, 0, 1, 6, 15, 20, 21, 8, 1, 0, 1, 7, 21, 35, 55, 39, 13, 1, 0, 1, 8, 28, 56, 120, 136, 92, 18, 1, 0, 1, 9, 36, 84, 231, 377, 430, 198, 30, 1, 0
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OFFSET
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0,8
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LINKS
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FORMULA
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T(n,k) = [n==0] + [n>0] * (k^floor((n+1)/2) + k^ceiling((n+1)/2)) / 4 + (1/(2*n)) * Sum_{d|n} phi(d) * k^(n/d)).
Linear recurrence for row n: T(n,k) = Sum_{j=0..n} -binomial(j-n-1,j+1) * T(n,k-1-j) for k >= n + 1.
O.g.f. for column k >= 0: Sum_{n>=0} T(n,k)*x^n = 3/4 + (1 + k*x)^2/(4*(1 - k*x^2)) - (1/2) * Sum_{d >= 1} (phi(d)/d) * log(1 - k*x^d). - Petros Hadjicostas, Feb 07 2021
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EXAMPLE
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Table begins with T(0,0):
1 1 1 1 1 1 1 1 1 1 1 ...
0 1 2 3 4 5 6 7 8 9 10 ...
0 1 3 6 10 15 21 28 36 45 55 ...
0 1 4 10 20 35 56 84 120 165 220 ...
0 1 6 21 55 120 231 406 666 1035 1540 ...
0 1 8 39 136 377 888 1855 3536 6273 10504 ...
0 1 13 92 430 1505 4291 10528 23052 46185 86185 ...
0 1 18 198 1300 5895 20646 60028 151848 344925 719290 ...
0 1 30 498 4435 25395 107331 365260 1058058 2707245 6278140 ...
0 1 46 1219 15084 110085 563786 2250311 7472984 21552969 55605670 ...
0 1 78 3210 53764 493131 3037314 14158228 53762472 174489813 500280022 ...
For T(3,3)=10, the unoriented cycles are 9 achiral (AAA, AAB, AAC, ABB, ACC, BBB, BBC, BCC, CCC) and one chiral pair (ABC-ACB).
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MATHEMATICA
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Table[If[k>0, DivisorSum[k, EulerPhi[#](n-k)^(k/#)&]/(2k) + ((n-k)^Floor[(k+1)/2]+(n-k)^Ceiling[(k+1)/2])/4, 1], {n, 0, 12}, {k, 0, n}] // Flatten
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CROSSREFS
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Cf. A051137 (ascending antidiagonals).
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KEYWORD
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AUTHOR
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STATUS
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approved
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