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A081721
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Number of bracelets of n beads in up to n colors.
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19
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1, 3, 10, 55, 377, 4291, 60028, 1058058, 21552969, 500280022, 12969598086, 371514016094, 11649073935505, 396857785692525, 14596464294191704, 576460770691256356, 24330595997127372497, 1092955780817066765469, 52063675152021153895330, 2621440000054016000176044
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OFFSET
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1,2
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COMMENTS
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Number of classes of functions of [n] to [n] under rotation and reversal.
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Classes can be of size between 1 and 2n
depending on divisibility properties of n.
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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)
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LINKS
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FORMULA
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A000169 Classes under translation mod n
Cf. A168658 Classes under complement to n+1
Cf. A130293 Classes under translation and rotation
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.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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