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A304976
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Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 6 colors (sets).
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7
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0, 0, 0, 0, 0, 0, 1, 3, 18, 46, 195, 461, 1696, 3836, 13097, 28819, 94094, 203322, 644911, 1376217, 4279692, 9051592, 27755013, 58319855, 176992090, 370087718, 1114496747, 2321721493, 6950406008, 14437363668, 43021681249, 89162536011, 264732674406, 547676535634
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OFFSET
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0,8
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COMMENTS
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Two color patterns are equivalent if we permute the colors. Achiral color patterns must be equivalent if we reverse the order of the pattern.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,16,-16,-91,91,216,-216,-180, 180).
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FORMULA
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a(n) = [n==0 mod 2] * (S2(n/2+3, 6) - 3*S2(n/2+2, 6) - 8*S2(n/2+1, 6) + 16*S2(n/2, 6)) + [n==1 mod 2] * (3*S2((n+5)/2, 6) - 17*S2((n+3)/2, 6) + 20*S2((n+1)/2, 6 )) where S2(n,k) is the Stirling subset number A008277(n,k).
G.f.: x^6 *(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3) / Product_{k=1..6} (1 - k*x^2).
a(n) = A304972(n,6).
a(2m-1) = A140735(m,6).
a(2m) = A293181(m,6).
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EXAMPLE
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For a(7) = 3, the color patterns for both rows and loops are ABCDCEF, ABCDEBF, and ABCDEFA.
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MATHEMATICA
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Table[If[EvenQ[n], StirlingS2[n/2 + 3, 6] - 3 StirlingS2[n/2 + 2, 6] - 8 StirlingS2[n/2 + 1, 6] + 16 StirlingS2[n/2, 6], 3 StirlingS2[(n + 5)/2, 6] - 17 StirlingS2[(n + 3)/2, 6] + 20 StirlingS2[(n + 1)/2, 6]], {n, 0, 40}]
Join[{0}, LinearRecurrence[{1, 16, -16, -91, 91, 216, -216, -180, 180}, {0, 0, 0, 0, 0, 1, 3, 18, 46}, 40]] (* Robert A. Russell, Oct 14 2018 *)
CoefficientList[Series[x^6 *(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3) / Product[1 - k*x^2, {k, 1, 6}], {x, 0, 50}], x] (* Stefano Spezia, Oct 20 2018 *)
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PROG
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(PARI) m=40; v=concat([0, 0, 0, 0, 0, 1, 3, 18, 46], vector(m-9)); for(n=10, m, v[n] = v[n-1] +16*v[n-2] -16*v[n-3] -91*v[n-4] +91*v[n-5] +216*v[n-6] -216*v[n-7] -180*v[n-8] +180*v[n-9]); concat([0], v) \\ G. C. Greubel, Oct 16 2018
(Magma) I:=[0, 0, 0, 0, 0, 1, 3, 18, 46]; [0] cat [n le 9 select I[n] else Self(n-1) +16*Self(n-2) -16*Self(n-3) -91*Self(n-4) +91*Self(n-5) +216*Self(n-6) -216*Self(n-7) -180*Self(n-8) +180*Self(n-9): n in [1..40]]; // G. C. Greubel, Oct 16 2018
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CROSSREFS
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Sixth column of A304972.
Sixth column of A140735 for odd n.
Sixth column of A293181 for even n.
Coefficients that determine the first formula and generating function are row 6 of A305008.
Sequence in context: A094159 A138976 A275038 * A064043 A267639 A238649
Adjacent sequences: A304973 A304974 A304975 * A304977 A304978 A304979
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KEYWORD
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nonn,easy
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AUTHOR
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Robert A. Russell, May 22 2018
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STATUS
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approved
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