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A308706 Number of chiral pairs of set partitions of a primitive cycle of n elements having exactly two different elements. 2
0, 0, 0, 0, 0, 0, 0, 1, 2, 7, 12, 31, 58, 126, 233, 484, 904, 1800, 3395, 6643, 12612, 24457, 46655, 90157, 172750, 333498, 641214, 1238664, 2388618, 4620006, 8931536, 17302033, 33521792, 65042495, 126257160, 245361171, 477087772, 928510506, 1808145395, 3523813566 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

Table of n, a(n) for n=0..39.

FORMULA

a(n) = ((1/(2n)) * Sum_{odd d|n} mu(d)*2^(n/d) - Sum_{d|n} mu(n/d)*2^floor(d/2)) / 2, where mu is the Möbius function at A008683.

a(n) = A000048(n) - A000046(n) = (A000048(n) - A179781(n))/2 = A000046(n) - A179781(n).

A059053(n) = Sum_{d|n} a(d).

EXAMPLE

For a(7)=1, the chiral pair is 0001011-0001101.  For a(8)=2, the chiral pairs are 00001011-00001101 and 00010011-00011001.

MATHEMATICA

Join[{0}, Table[(DivisorSum[NestWhile[#/2 &, n, EvenQ], MoebiusMu[#] 2^(n/#) &]/(2 n) - DivisorSum[n, MoebiusMu[n/#] 2^Floor[#/2] &])/2, {n, 1, 40}]]

PROG

(PARI) a(n) = (sumdiv(n, d, if (d%2, moebius(d)*2^(n/d)))/(2*n) - sumdiv(n, d, moebius(n/d)*2^(d\2)))/2; \\ Michel Marcus, Jun 27 2019

CROSSREFS

Cf. A000048 (oriented), A000046 (unoriented), A179781 (achiral), A059053 (not primitive).

Sequence in context: A102371 A007230 A290234 * A059053 A032025 A088662

Adjacent sequences:  A308703 A308704 A308705 * A308707 A308709 A308710

KEYWORD

nonn

AUTHOR

Robert A. Russell, Jun 18 2019

STATUS

approved

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Last modified August 21 06:54 EDT 2019. Contains 326162 sequences. (Running on oeis4.)