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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).

%I A304975 #28 Sep 08 2022 08:46:21
%S A304975 0,0,0,0,0,1,3,12,34,95,261,630,1700,3801,10143,21672,57414,119155,
%T A304975 314121,639210,1679320,3370301,8832483,17549532,45907994,90541815,
%U A304975 236526381,463889790,1210585740,2364180001,6164760423,11999840592,31271161774,60714998075,158145313041,306438236370,797884712960
%N A304975 Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 5 colors (sets).
%C A304975 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 A304975 G. C. Greubel, <a href="/A304975/b304975.txt">Table of n, a(n) for n = 0..1000</a>
%H A304975 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,11,-11,-38,38,40,-40).
%F A304975 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).
%F A304975 G.f.: x^5 *(1 + x)*(1 - 3*x^2)*(1 + 2*x - 2*x^2) / Product_{k=1..5} (1 - k*x^2).
%F A304975 a(n) = A304972(n,5).
%F A304975 a(2m-1) = A140735(m,5).
%F A304975 a(2m) = A293181(m,5).
%e A304975 For a(6) = 3, the color patterns for both rows and loops are ABCCDE, ABCDBE, and ABCDEA.
%t A304975 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}]
%t A304975 Join[{0}, LinearRecurrence[{1, 11, -11, -38, 38, 40, -40}, {0, 0, 0, 0, 1, 3, 12}, 40]] (* _Robert A. Russell_, Oct 14 2018 *)
%t A304975 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 *)
%o A304975 (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
%o A304975 (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
%Y A304975 Fifth column of A304972.
%Y A304975 Fifth column of A140735 for odd n.
%Y A304975 Fifth column of A293181 for even n.
%Y A304975 Coefficients that determine the first formula and generating function are row 5 of A305008.
%K A304975 nonn,easy
%O A304975 0,7
%A A304975 _Robert A. Russell_, May 22 2018