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 A305545 Number of chiral pairs of color loops of length n with exactly 6 different colors. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 LINKS 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

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Last modified August 9 22:52 EDT 2020. Contains 336335 sequences. (Running on oeis4.)