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A308706
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Number of chiral pairs of set partitions of a primitive cycle of n elements having exactly two different elements.
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2
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0, 0, 0, 0, 0, 0, 0, 1, 2, 7, 12, 31, 58, 126, 233, 484, 904, 1800, 3395, 6643, 12612, 24457, 46655, 90157, 172750, 333498, 641214, 1238664, 2388618, 4620006, 8931536, 17302033, 33521792, 65042495, 126257160, 245361171, 477087772, 928510506, 1808145395, 3523813566
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OFFSET
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0,9
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LINKS
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FORMULA
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a(n) = ((1/(2n)) * Sum_{odd d|n} mu(d)*2^(n/d) - Sum_{d|n} mu(n/d)*2^floor(d/2)) / 2, where mu is the Möbius function at A008683.
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EXAMPLE
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For a(7)=1, the chiral pair is 0001011-0001101. For a(8)=2, the chiral pairs are 00001011-00001101 and 00010011-00011001.
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MATHEMATICA
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Join[{0}, Table[(DivisorSum[NestWhile[#/2 &, n, EvenQ], MoebiusMu[#] 2^(n/#) &]/(2 n) - DivisorSum[n, MoebiusMu[n/#] 2^Floor[#/2] &])/2, {n, 1, 40}]]
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PROG
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(PARI) a(n) = if (n, (sumdiv(n, d, if (d%2, moebius(d)*2^(n/d)))/(2*n) - sumdiv(n, d, moebius(n/d)*2^(d\2)))/2, 0); \\ Michel Marcus, Jun 27 2019; corrected Jun 12 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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