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

Author

Robert A. Russell, Jun 04 2018

Keywords

• nonn
• easy

Examples

			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.
		

Crossrefs

Fifth column of A305541.

Programs

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}]

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

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)).