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

1,6

Links

Table of n, a(n) for n=1..25.

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)).

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}]

Prog

(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

Crossrefs

Sixth column of A305541.

Sequence in context: A223213 A271651 A223348 * A056352 A056346 A283722

Adjacent sequences: A305542 A305543 A305544 * A305546 A305547 A305548

Keyword

nonn,easy

Author

Robert A. Russell, Jun 04 2018

Status

approved