A081721 - OEIS

%I #49 Nov 26 2020 16:15:53

%S 1,3,10,55,377,4291,60028,1058058,21552969,500280022,12969598086,

%T 371514016094,11649073935505,396857785692525,14596464294191704,

%U 576460770691256356,24330595997127372497,1092955780817066765469,52063675152021153895330,2621440000054016000176044

%N Number of bracelets of n beads in up to n colors.

%C T(n,n), T given in A081720.

%C From _Olivier Gérard_, Aug 01 2016: (Start)

%C Number of classes of functions of [n] to [n] under rotation and reversal.

%C .

%C Classes can be of size between 1 and 2n

%C depending on divisibility properties of n.

%C .

%C n   1   2   3   4    5         n     2n

%C ----------------------------------------

%C 1   1

%C 2   2   1

%C 3   3   0   6                         1

%C 4   4   6   0                 30     15

%C 5   5   0   0                120    252

%C 6   6  15  30                725   3515

%C 7   7   0   0               2394  57627

%C .

%C (End)

%H Alois P. Heinz, <a href="/A081721/b081721.txt">Table of n, a(n) for n = 1..200</a>

%F a(n) ~ n^(n-1) / 2. - _Vaclav Kotesovec_, Mar 18 2017

%t Table[CycleIndex[DihedralGroup[n],s]/.Table[s[i]->n,{i,1,n}],{n,1,20}] (* _Geoffrey Critzer_, Jun 18 2013 *)

%t t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]); a[n_] := t[n, n]; Array[a, 20] (* _Jean-François Alcover_, Nov 02 2017, after Maple code for A081720 *)

%Y Cf. A000312 All endofunctions

%Y Cf. A000169 Classes under translation mod n

%Y Cf. A001700 Classes under sort

%Y Cf. A056665 Classes under rotation

%Y Cf. A168658 Classes under complement to n+1

%Y Cf. A130293 Classes under translation and rotation

%Y Cf. A275549 Classes under reversal

%Y Cf. A275550 Classes under reversal and complement

%Y Cf. A275551 Classes under translation and reversal

%Y Cf. A275552 Classes under translation and complement

%Y Cf. A275553 Classes under translation, complement and reversal

%Y Cf. A275554 Classes under translation, rotation and complement

%Y Cf. A275555 Classes under translation, rotation and reversal

%Y Cf. A275556 Classes under translation, rotation, complement and reversal

%Y Cf. A275557 Classes under rotation and complement

%Y Cf. A275558 Classes under rotation, complement and reversal

%Y Row sums of partition array A213941.

%Y Main diagonal of A321791.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, based on information supplied by _Gary W. Adamson_, Apr 05 2003

%E Name changed by _Olivier Gérard_, Aug 05 2016

%E Name revised by _Álvar Ibeas_, Apr 20 2018