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A305543 Number of chiral pairs of color loops of length n with exactly 4 different colors. 3
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=4 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}]

PROG

(PARI) a(n) = my(k=4); -(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

Fourth column of A305541.

Sequence in context: A009137 A319097 A326789 * A183900 A001089 A069515

Adjacent sequences:  A305540 A305541 A305542 * A305544 A305545 A305546

KEYWORD

nonn,easy

AUTHOR

Robert A. Russell, Jun 04 2018

STATUS

approved

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Last modified September 30 05:03 EDT 2020. Contains 337435 sequences. (Running on oeis4.)