A305544 Number of chiral pairs of color loops of length n with exactly 5 different colors.

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

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