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A320935
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Number of chiral pairs of color patterns (set partitions) for a row of length n using 5 or fewer colors (subsets).
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4
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0, 0, 1, 4, 20, 86, 400, 1852, 8868, 42892, 210346, 1038034, 5150110, 25623486, 127740880, 637539592, 3184224728, 15910524632, 79520923966, 397508610454, 1987255480650, 9935410066186, 49674450471460, 248364429410332, 1241798688445588, 6208922948527572, 31044403310614786
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OFFSET
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1,4
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COMMENTS
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Two color patterns are equivalent if the colors are permuted.
A chiral row is not equivalent to its reverse.
There are nonrecursive formulas, generating functions, and computer programs for A056272 and A305751, which can be used in conjunction with the first formula.
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LINKS
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FORMULA
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a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=5 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
G.f.: x^3*(1 - 7*x + 10*x^2 + 18*x^3 - 49*x^4 + 25*x^5)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 5*x)*(1 - 5*x^2)*(1 - 2*x^2)). - Bruno Berselli, Oct 31 2018
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EXAMPLE
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For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.
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MATHEMATICA
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LinearRecurrence[{11, -34, -16, 247, -317, -200, 610, -300}, {0, 0, 1, 4, 20, 86, 400, 1852}, 40] (* or *)
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]] (* A304972 *)
k=5; Table[Sum[StirlingS2[n, j]-Ach[n, j], {j, k}]/2, {n, 40}]
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CROSSREFS
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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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