%I #14 Jun 08 2018 11:12:57

%S 0,0,1,3,12,35,111,318,934,2634,7503,21071,59472,167229,472133,

%T 1333263,3777600,10721837,30516447,87035631,248820816,712751271,

%U 2045784183,5882388956,16942974060,48876617790,141204945463,408495109005,1183247473872,3431451145390,9962348798055,28953196894668

%N Number of chiral pairs of color loops of length n with exactly 3 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=3 different colors used and where S2(n,k) is the Stirling subset number A008277.

%F a(n) = (A052823(n) - A056489(n)) / 2.

%F a(n) = A305541(n,3).

%F G.f.: -(3/2) * x^4 * (1+x)^2 / Product_{j=1..3} (1-j*x^2) - Sum_{d>0} (phi(d)/(2d)) * (log(1-3x^d) - 3*log(1-2x^d) + 3*log(1-x^d)).

%e For a(4)=3, the chiral pairs of color loops are AABC-AACB, ABBC-ACBB, and ABCC-ACCB.

%t k=3; 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=3); -(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 Third column of A305541.

%K nonn,easy

%O 1,4

%A _Robert A. Russell_, Jun 04 2018