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A320934
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Number of chiral pairs of color patterns (set partitions) for a row of length n using 4 or fewer colors (subsets).
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3
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0, 0, 1, 4, 20, 80, 336, 1344, 5440, 21760, 87296, 349184, 1397760, 5591040, 22368256, 89473024, 357908480, 1431633920, 5726601216, 22906404864, 91625881600, 366503526400, 1466015154176, 5864060616704, 23456246661120, 93824986644480, 375299963355136, 1501199853420544, 6004799480791040
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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 A124303 and A305750, 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=4 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)).
a(2*m) = (16^m - 4*4^m)/48.
a(2*m-1) = (16^m - 4*4^m)/192.
a(n) = (4^n - 4^floor(n/2+1))/48.
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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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Table[(4^n - 4^Floor[n/2+1])/48, {n, 40}] (* or *)
LinearRecurrence[{4, 4, -16}, {0, 0, 1}, 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=4; Table[Sum[StirlingS2[n, j]-Ach[n, j], {j, k}]/2, {n, 40}]
CoefficientList[Series[x^2/((-1 + 4 x) (-1 + 4 x^2)), {x, 0, 50}], x] (* Stefano Spezia, Oct 29 2018 *)
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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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