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A019537 Number of special orbits for dihedral group of degree n. 2
1, 2, 4, 14, 61, 414, 3416, 34274, 394009, 5113712, 73758368, 1170495180, 20263806277, 380048113202, 7676106638884, 166114210737254, 3834434327929981, 94042629562443206, 2442147034770292496, 66942194906543381336, 1931543452346146410965, 58519191359170883258606 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is the number of ways to color a necklace of n beads using at most n colors. Turning the necklace over does not count as different. - Robert A. Russell, May 31 2018

LINKS

Table of n, a(n) for n=1..22.

M. Goebel, On the number of special permutation-invariant orbits and terms, in Applicable Algebra in Engin., Comm. and Comp. (AAECC 8), Volume 8, Number 6, 1997, pp. 505-509 (Lect. Notes Comp. Sci.)

FORMULA

a(n) = Sum_{k=1..n} ((k!/4)*(S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/(2 n))*Sum_{d|n} phi(d)*S2(n/d,k)), where S2(n,k) is the Stirling subset number A008277. - Robert A. Russell, May 31 2018

EXAMPLE

For a(3) = 4, the necklaces are AAA, AAB, ABB, and ABC. Last one is chiral. For a(4) = 14, the necklacess are AAAA, AAAB, AABB, ABAB, ABBB, ABAC, ABCB, ACBC, AABC, ABBC, ABCC, ABCD, ABDC, and ACBD. Last six are chiral. - Robert A. Russell, May 31 2018

MATHEMATICA

Table[Sum[(k!/(2n)) DivisorSum[n, EulerPhi[#] StirlingS2[n/#, k] &] + (k!/4) (StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k]), {k, 1, n}], {n, 1, 40}] (* Robert A. Russell, May 31 2018 *)

PROG

(PARI) a(n) = sum(k=1, n, (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

Cf. A019536.

Row sums of A273891.

Sequence in context: A047009 A027740 A132880 * A046911 A089127 A132852

Adjacent sequences:  A019534 A019535 A019536 * A019538 A019539 A019540

KEYWORD

nonn

AUTHOR

Manfred Goebel (goebel(AT)informatik.uni-tuebingen.de)

EXTENSIONS

More terms (using A273891) from Alois P. Heinz, Jun 02 2016

STATUS

approved

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Last modified October 16 13:07 EDT 2018. Contains 316263 sequences. (Running on oeis4.)