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
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
a(n) ~ (n-1)! / (4 * log(2)^(n+1)). - Vaclav Kotesovec, Jul 21 2019
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
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