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%I #14 Jun 08 2018 11:16:57
%S 0,0,0,0,12,150,1200,7845,46280,254676,1344900,6892425,34646220,
%T 171715050,843004688,4110478470,19950471120,96525524140,466068873900,
%U 2247609721431,10832163963860,52194011649150,251522234238000,1212501695554920,5848043487355752,28223528190496380,136307124614215660,658800774340433025,3186621527711606940
%N Number of chiral pairs of color loops of length n with exactly 5 different colors.
%F 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=5 different colors used and where S2(n,k) is the Stirling subset number A008277.
%F a(n) = (A052825(n) - A056491(n)) / 2.
%F a(n) = A305541(n,5).
%F G.f.: -30 * x^8 * (1+x)^2 / Product_{j=1..5} (1-j*x^2) - Sum_{d>0} (phi(d)/(2d)) * (log(1-5x^d) - 5*log(1-4x^d) + 10*log(1-3x^3) - 10*log(1-2x^d) + 5*log(1-x^d)).
%e For a(5)=12, the chiral pairs of color loops are ABCDE-AEDCB, ABCED-ADECB, ABDCE-AECDB, ABDEC-ACEDB, ABECD-ADCEB, ABEDC-ACDEB, ACBDE-AEDBC, ACBED-ADEBC, ACDBE-AEBCD, ACEDB-ABDEC, ADBCE-AECBD, ADBEC-ACEBD, and ADCBE-AEBCD.
%t k=5; 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 (PARI) a(n) = my(k=5); -(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 Fifth column of A305541.
%K nonn,easy
%O 1,5
%A _Robert A. Russell_, Jun 04 2018
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