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A305543 revision #10

A305543

Number of chiral pairs of color loops of length n with exactly 4 different colors.

0, 0, 0, 3, 24, 124, 588, 2487, 10240, 40488, 158220, 609078, 2333520, 8895204, 33864364, 128793627, 490027200, 1865625340, 7110959340, 27138210888, 103717720000, 396965694444, 1521562700988, 5840509760582, 22450188684288, 86412088367640, 333035003543900, 1285108410802038, 4964755661788560, 19201631174055992

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Table of n, a(n) for n=1..30.

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) = (A052824(n) - A056490(n)) / 2.

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

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)).

EXAMPLE

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

MATHEMATICA

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}]

CROSSREFS

Fourth column of A305541.

Sequence in context: A009137 A319097 A326789 * A356363 A183900 A001089

Adjacent sequences: A305540 A305541 A305542 * A305544 A305545 A305546

KEYWORD

nonn,easy

AUTHOR

Robert A. Russell, Jun 04 2018

STATUS

editing