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A305545 Number of chiral pairs of color loops of length n with exactly 6 different colors.

%I A305545 #14 Jun 08 2018 11:19:15
%S A305545 0,0,0,0,0,60,1080,11970,105840,821592,5873760,39705630,258121080,
%T A305545 1631169900,10096542792,61535329380,370709045280,2213740488600,
%U A305545 13132064237040,77509384111278,455754440462040,2672268921657540,15636049474529880,91353538645037220,533180401444362672
%N A305545 Number of chiral pairs of color loops of length n with exactly 6 different colors.
%F A305545 a(n) = -(k!/4)*(S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/(2n))*Sum_{d|n} phi(d)*S2(n/d,k), with k=6 different colors used and where S2(n,k) is the Stirling subset number A008277.
%F A305545 a(n) = (A052826(n) - A056492(n)) / 2.
%F A305545 a(n) = A305541(n,6).
%F A305545 G.f.: -180 * x^10 * (1+x)^2 / Product_{j=1..6} (1-j*x^2) - Sum_{d>0} (phi(d)/(2d)) * (log(1-6x^d) - 6*log(1-5x^d) + 15*log(1-4x^d) - 20*log(1-3x^3) + 15*log(1-2x^d) - 5*log(1-x^d)).
%e A305545 For a(6) = 60, we pair up the 5! = 120 permutations of BCDEF, each with its reversal.  Then put an A before each to end up with 60 chiral pairs such as ABCDEF-AFEDCB.
%t A305545 k=6; Table[(k!/(2n)) DivisorSum[n, EulerPhi[#] StirlingS2[n/#, k] &] - (k!/4) (StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k]), {n, 1, 40}]
%o A305545 (PARI) a(n) = my(k=6); -(k!/4)*(stirling(floor((n+1)/2),k,2) + stirling(ceil((n+1)/2),k,2)) + (k!/(2*n))*sumdiv(n, d, eulerphi(d)*stirling(n/d,k,2)); \\ _Michel Marcus_, Jun 06 2018
%Y A305545 Sixth column of A305541.
%K A305545 nonn,easy
%O A305545 1,6
%A A305545 _Robert A. Russell_, Jun 04 2018