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A305543
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Number of chiral pairs of color loops of length n with exactly 4 different colors.
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3
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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
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OFFSET
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1,4
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LINKS
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FORMULA
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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.
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)).
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EXAMPLE
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For a(4)=3, the chiral pairs of color loops are ABCD-ADCB, ACBD-ADBC, and ABDC-ACDB.
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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