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A305625
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Number of chiral pairs of rows of n colors with exactly 5 different colors.
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2
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0, 0, 0, 0, 60, 900, 8400, 63000, 417000, 2551440, 14802900, 82763100, 450501660, 2404493700, 12645952200, 65771370000, 339164682000, 1737485315640, 8855354531100, 44952362878500, 227475739300260, 1148269299919500, 5785013208282000, 29100046926951000, 146201097996135000, 733811769167043840, 3680292427100043300, 18446421887430345900, 92412024657725026860, 462780012983867889300, 2316780309783100387800
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OFFSET
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1,5
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COMMENTS
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If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.
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LINKS
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FORMULA
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a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=5 colors used and where S2(n,k) is the Stirling subset number A008277.
G.f.: -(k!/2) * (x^(2k-1) + x^(2k)) / Product_{j=1..k} (1 - j*x^2) + (k!/2) * x^k / Product_{j=1..k} (1 - j*x) with k=5 colors used.
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EXAMPLE
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For a(5) = 60, the chiral pairs are the 5! = 120 permutations of ABCDE, each paired with its reverse.
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MATHEMATICA
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k=5; Table[(k!/2) (StirlingS2[n, k] - StirlingS2[Ceiling[n/2], k]), {n, 1, 40}]
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PROG
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(PARI) a(n) = 60*(stirling(n, 5, 2) - stirling(ceil(n/2), 5, 2)); \\ Altug Alkan, Sep 26 2018
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CROSSREFS
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A056456(n) is number of achiral rows of n colors with exactly 5 different colors.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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