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

0, 0, 0, 0, 0, 60, 1080, 11970, 105840, 821592, 5873760, 39705630, 258121080, 1631169900, 10096542792, 61535329380, 370709045280, 2213740488600, 13132064237040, 77509384111278, 455754440462040, 2672268921657540, 15636049474529880, 91353538645037220, 533180401444362672

Offset: 1

Author

Robert A. Russell, Jun 04 2018

Keywords

• nonn
• easy

Examples

			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.
		

Crossrefs

Sixth column of A305541.

Programs

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}]

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

Formula

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.

a(n) = (A052826(n) - A056492(n)) / 2.

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)).