OFFSET
1,6
FORMULA
k=6 different colors used; a(n) = -(k!/2)*(S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/(2n))*Sum_{d|n} phi(d)*S2(n/d,k), where S2(n,k) is the Stirling subset number A008277.
a(n) = A305541(n,6).
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)).
EXAMPLE
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.
MATHEMATICA
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}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Jun 04 2018
STATUS
proposed