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 A304975 Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 5 colors (sets). 8
 0, 0, 0, 0, 0, 1, 3, 12, 34, 95, 261, 630, 1700, 3801, 10143, 21672, 57414, 119155, 314121, 639210, 1679320, 3370301, 8832483, 17549532, 45907994, 90541815, 236526381, 463889790, 1210585740, 2364180001, 6164760423, 11999840592, 31271161774, 60714998075, 158145313041, 306438236370, 797884712960 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 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 (1,11,-11,-38,38,40,-40). FORMULA a(n) = [n==0 mod 2] * (3*S2(n/2+2, 5) - 11*S2(n/2+1, 5) + 6*S2(n/2, 5)) + [n==1 mod 2] * (S2((n+5)/2, 5) - 3*S2((n+3)/2, 5)) where S2(n,k) is the Stirling subset number A008277(n,k). G.f.: x^5 *(1 + x)*(1 - 3*x^2)*(1 + 2*x - 2*x^2) / Product_{k=1..5} (1 - k*x^2). a(n) = A304972(n,5). a(2m-1) = A140735(m,5). a(2m) = A293181(m,5). EXAMPLE For a(6) = 3, the color patterns for both rows and loops are ABCCDE, ABCDBE, and ABCDEA. MATHEMATICA Table[If[EvenQ[n], 3 StirlingS2[n/2+2, 5] - 11 StirlingS2[n/2+1, 5] + 6 StirlingS2[n/2, 5], StirlingS2[(n+5)/2, 5] - 3 StirlingS2[(n+3)/2, 5]], {n, 0, 40}] Join[{0}, LinearRecurrence[{1, 11, -11, -38, 38, 40, -40}, {0, 0, 0, 0, 1, 3, 12}, 40]] (* Robert A. Russell, Oct 14 2018 *) CoefficientList[Series[x^5 *(1 + x)*(1 - 3*x^2)*(1 + 2*x - 2*x^2) / Product[1 - k*x^2, {k, 1, 5}], {x, 0, 50}], x] (* Stefano Spezia, Oct 16 2018 *) PROG (PARI) m=40; v=concat([0, 0, 0, 0, 1, 3, 12], vector(m-7)); for(n=8, m, v[n] = v[n-1] +11*v[n-2] -11*v[n-3] -38*v[n-4] +38*v[n-5] +40*v[n-6] -40*v[n-7] ); concat([0], v) \\ G. C. Greubel, Oct 16 2018 (MAGMA) I:=[0, 0, 0, 0, 1, 3, 12]; [0] cat [n le 7 select I[n] else Self(n-1) +11*Self(n-2) -11*Self(n-3) -38*Self(n-4) +38*Self(n-5) +40*Self(n-6) -40*Self(n-7): n in [1..40]]; // G. C. Greubel, Oct 16 2018 CROSSREFS Fifth column of A304972. Fifth column of A140735 for odd n. Fifth column of A293181 for even n. Coefficients that determine the first formula and generating function are row 5 of A305008. Sequence in context: A184705 A257890 A060298 * A226546 A073372 A305023 Adjacent sequences:  A304972 A304973 A304974 * A304976 A304977 A304978 KEYWORD nonn,easy AUTHOR Robert A. Russell, May 22 2018 STATUS approved

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Last modified September 19 08:05 EDT 2021. Contains 347556 sequences. (Running on oeis4.)