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A305544 Number of chiral pairs of color loops of length n with exactly 5 different colors. 2
0, 0, 0, 0, 12, 150, 1200, 7845, 46280, 254676, 1344900, 6892425, 34646220, 171715050, 843004688, 4110478470, 19950471120, 96525524140, 466068873900, 2247609721431, 10832163963860, 52194011649150, 251522234238000, 1212501695554920, 5848043487355752, 28223528190496380, 136307124614215660, 658800774340433025, 3186621527711606940 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Table of n, a(n) for n=1..29.

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=5 different colors used and where S2(n,k) is the Stirling subset number A008277.

a(n) = (A052825(n) - A056491(n)) / 2.

a(n) = A305541(n,5).

G.f.: -30 * x^8 * (1+x)^2 / Product_{j=1..5} (1-j*x^2) - Sum_{d>0} (phi(d)/(2d)) * (log(1-5x^d) - 5*log(1-4x^d) + 10*log(1-3x^3) - 10*log(1-2x^d) + 5*log(1-x^d)).

EXAMPLE

For a(5)=12, the chiral pairs of color loops are ABCDE-AEDCB, ABCED-ADECB, ABDCE-AECDB, ABDEC-ACEDB, ABECD-ADCEB, ABEDC-ACDEB, ACBDE-AEDBC, ACBED-ADEBC, ACDBE-AEBCD, ACEDB-ABDEC, ADBCE-AECBD, ADBEC-ACEBD, and ADCBE-AEBCD.

MATHEMATICA

k=5; 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=5); -(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

Fifth column of A305541.

Sequence in context: A015610 A141326 A154733 * A056351 A056345 A264233

Adjacent sequences:  A305541 A305542 A305543 * A305545 A305546 A305547

KEYWORD

nonn,easy

AUTHOR

Robert A. Russell, Jun 04 2018

STATUS

approved

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Last modified September 24 07:28 EDT 2020. Contains 337317 sequences. (Running on oeis4.)