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%I #26 Sep 08 2022 08:46:21
%S 0,0,0,0,0,0,1,3,18,46,195,461,1696,3836,13097,28819,94094,203322,
%T 644911,1376217,4279692,9051592,27755013,58319855,176992090,370087718,
%U 1114496747,2321721493,6950406008,14437363668,43021681249,89162536011,264732674406,547676535634
%N Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 6 colors (sets).
%C Two color patterns are equivalent if we permute the colors. Achiral color patterns must be equivalent if we reverse the order of the pattern.
%H G. C. Greubel, <a href="/A304976/b304976.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,16,-16,-91,91,216,-216,-180, 180).
%F 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).
%F 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).
%F a(n) = A304972(n,6).
%F a(2m-1) = A140735(m,6).
%F a(2m) = A293181(m,6).
%e For a(7) = 3, the color patterns for both rows and loops are ABCDCEF, ABCDEBF, and ABCDEFA.
%t 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}]
%t 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 *)
%t 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 *)
%o (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
%o (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
%Y Sixth column of A304972.
%Y Sixth column of A140735 for odd n.
%Y Sixth column of A293181 for even n.
%Y Coefficients that determine the first formula and generating function are row 6 of A305008.
%K nonn,easy
%O 0,8
%A _Robert A. Russell_, May 22 2018
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