OFFSET
1,2
COMMENTS
T(n,n), T given in A081720.
From Olivier Gérard, Aug 01 2016: (Start)
Number of classes of functions of [n] to [n] under rotation and reversal.
.
Classes can be of size between 1 and 2n
depending on divisibility properties of n.
.
n 1 2 3 4 5 n 2n
----------------------------------------
1 1
2 2 1
3 3 0 6 1
4 4 6 0 30 15
5 5 0 0 120 252
6 6 15 30 725 3515
7 7 0 0 2394 57627
.
(End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..200
FORMULA
a(n) ~ n^(n-1) / 2. - Vaclav Kotesovec, Mar 18 2017
MATHEMATICA
Table[CycleIndex[DihedralGroup[n], s]/.Table[s[i]->n, {i, 1, n}], {n, 1, 20}] (* Geoffrey Critzer, Jun 18 2013 *)
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 *)
CROSSREFS
Cf. A000312 All endofunctions
Cf. A000169 Classes under translation mod n
Cf. A001700 Classes under sort
Cf. A056665 Classes under rotation
Cf. A168658 Classes under complement to n+1
Cf. A130293 Classes under translation and rotation
Cf. A275549 Classes under reversal
Cf. A275550 Classes under reversal and complement
Cf. A275551 Classes under translation and reversal
Cf. A275552 Classes under translation and complement
Cf. A275553 Classes under translation, complement and reversal
Cf. A275554 Classes under translation, rotation and complement
Cf. A275555 Classes under translation, rotation and reversal
Cf. A275556 Classes under translation, rotation, complement and reversal
Cf. A275557 Classes under rotation and complement
Cf. A275558 Classes under rotation, complement and reversal
Row sums of partition array A213941.
Main diagonal of A321791.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, based on information supplied by Gary W. Adamson, Apr 05 2003
EXTENSIONS
Name changed by Olivier Gérard, Aug 05 2016
Name revised by Álvar Ibeas, Apr 20 2018
STATUS
approved