A305543 - OEIS

%I #14 Jun 08 2018 11:15:58

%S 0,0,0,3,24,124,588,2487,10240,40488,158220,609078,2333520,8895204,

%T 33864364,128793627,490027200,1865625340,7110959340,27138210888,

%U 103717720000,396965694444,1521562700988,5840509760582,22450188684288,86412088367640,333035003543900,1285108410802038,4964755661788560,19201631174055992

%N Number of chiral pairs of color loops of length n with exactly 4 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=4 different colors used and where S2(n,k) is the Stirling subset number A008277.

%F a(n) = (A052824(n) - A056490(n)) / 2.

%F a(n) = A305541(n,4).

%F G.f.: -6 * x^6 * (1+x)^2 / Product_{j=1..4} (1-j*x^2) - Sum_{d>0} (phi(d)/(2d)) * (log(1-4x^d) - 4*log(1-3x^3) + 6*log(1-2x^d) - 4*log(1-x^d)).

%e For a(4)=3, the chiral pairs of color loops are ABCD-ADCB, ACBD-ADBC, and ABDC-ACDB.

%t k=4; 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=4); -(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 Fourth column of A305541.

%K nonn,easy

%O 1,4

%A _Robert A. Russell_, Jun 04 2018