cp's OEIS Frontend

A081721 Number of bracelets of n beads in up to n colors.

%I A081721 #49 Nov 26 2020 16:15:53
%S A081721 1,3,10,55,377,4291,60028,1058058,21552969,500280022,12969598086,
%T A081721 371514016094,11649073935505,396857785692525,14596464294191704,
%U A081721 576460770691256356,24330595997127372497,1092955780817066765469,52063675152021153895330,2621440000054016000176044
%N A081721 Number of bracelets of n beads in up to n colors.
%C A081721 T(n,n), T given in A081720.
%C A081721 From _Olivier Gérard_, Aug 01 2016: (Start)
%C A081721 Number of classes of functions of [n] to [n] under rotation and reversal.
%C A081721 .
%C A081721 Classes can be of size between 1 and 2n
%C A081721 depending on divisibility properties of n.
%C A081721 .
%C A081721 n   1   2   3   4    5         n     2n
%C A081721 ----------------------------------------
%C A081721 1   1
%C A081721 2   2   1
%C A081721 3   3   0   6                         1
%C A081721 4   4   6   0                 30     15
%C A081721 5   5   0   0                120    252
%C A081721 6   6  15  30                725   3515
%C A081721 7   7   0   0               2394  57627
%C A081721 .
%C A081721 (End)
%H A081721 Alois P. Heinz, <a href="/A081721/b081721.txt">Table of n, a(n) for n = 1..200</a>
%F A081721 a(n) ~ n^(n-1) / 2. - _Vaclav Kotesovec_, Mar 18 2017
%t A081721 Table[CycleIndex[DihedralGroup[n],s]/.Table[s[i]->n,{i,1,n}],{n,1,20}] (* _Geoffrey Critzer_, Jun 18 2013 *)
%t A081721 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 A081721 Cf. A000312 All endofunctions
%Y A081721 Cf. A000169 Classes under translation mod n
%Y A081721 Cf. A001700 Classes under sort
%Y A081721 Cf. A056665 Classes under rotation
%Y A081721 Cf. A168658 Classes under complement to n+1
%Y A081721 Cf. A130293 Classes under translation and rotation
%Y A081721 Cf. A275549 Classes under reversal
%Y A081721 Cf. A275550 Classes under reversal and complement
%Y A081721 Cf. A275551 Classes under translation and reversal
%Y A081721 Cf. A275552 Classes under translation and complement
%Y A081721 Cf. A275553 Classes under translation, complement and reversal
%Y A081721 Cf. A275554 Classes under translation, rotation and complement
%Y A081721 Cf. A275555 Classes under translation, rotation and reversal
%Y A081721 Cf. A275556 Classes under translation, rotation, complement and reversal
%Y A081721 Cf. A275557 Classes under rotation and complement
%Y A081721 Cf. A275558 Classes under rotation, complement and reversal
%Y A081721 Row sums of partition array A213941.
%Y A081721 Main diagonal of A321791.
%K A081721 nonn
%O A081721 1,2
%A A081721 _N. J. A. Sloane_, based on information supplied by _Gary W. Adamson_, Apr 05 2003
%E A081721 Name changed by _Olivier Gérard_, Aug 05 2016
%E A081721 Name revised by _Álvar Ibeas_, Apr 20 2018