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 A304976 Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 6 colors (sets). 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Two color patterns are equivalent if we permute the colors. Achiral color patterns must be equivalent if we reverse the order of the pattern. LINKS 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). FORMULA 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). EXAMPLE For a(7) = 3, the color patterns for both rows and loops are ABCDCEF, ABCDEBF, and ABCDEFA. MATHEMATICA 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 *) PROG (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 CROSSREFS 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 KEYWORD nonn,easy AUTHOR Robert A. Russell, May 22 2018 STATUS approved

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Last modified March 25 19:21 EDT 2023. Contains 361528 sequences. (Running on oeis4.)