A081721 -id:A081721 - cp's OEIS frontend

A275554 Number of classes of endofunctions of [n] under vertical translation mod n, rotation and complement to n+1.

1, 1, 2, 3, 14, 65, 680, 8407, 131416, 2391515, 50006040, 1178973851, 30958827996, 896080197025, 28346960490560, 973097534189967, 36028797169965112, 1431211525754907905, 60719765554419645244, 2740193428892401092979, 131072000000281600209176

Offset: 0

Olivier Gérard, Aug 02 2016

nonn

Because of the interaction between the two symmetries indexed by n, classes can be of size from n up to 2*n^2.
.
n  possible class sizes
-------------------------------
1  1
2  2
3  3, 6,                   18
4  4, 8,      16,          32
5  5, 10,                  50
6  6, 12, 18, 24,     36,  72
7  7, 14,                  98
.
but classes of size 2*n^2 account for the bulk of a(n).
n  number of classes
-----------------------------------
1  1
2  2
3  1, 1,                   1
4  2, 3,       4,          5
5  1, 2,                   62
6  2, 4,   2,  2,     48,  622
7  1, 3,                   8403

Andrew Howroyd, Table of n, a(n) for n = 0..100

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. A081721 Classes under rotation and reversal
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. 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

\\ see A056391 for Polya enumeration functions
a(n) = NonequivalentSorts(CyclicPerms(n), DihedralPerms(n)); \\ Andrew Howroyd, Sep 30 2017

Terms a(8) and beyond from Andrew Howroyd, Sep 30 2017

A275555 Number of classes of endofunctions of [n] under vertical translation mod n, rotation and reversal.

Original entry on oeis.org

1, 1, 2, 4, 16, 77, 730, 8578, 132422, 2394795, 50031012, 1179054376, 30959574248, 896082610429, 28346986843640, 973097619619654, 36028798243701780, 1431211529242786625, 60719765604009463866, 2740193429053744941868, 131072000002841600036024

Offset: 0

Olivier Gérard, Aug 05 2016

nonn

Because of the interaction between the two symmetries indexed by n, classes can be of size from n up to 2*n^2.
n  possible class sizes
-----------------------------------
1
2
3, 6,   9
4, 8,      16,               32
5, 10,         25,           50
6, 12, 18, 24,     36,       72
7, 14,                 49,   98
but classes of size 2*n^2 account for the bulk of a(n).
n  number of classes
-----------------------------------
1
2
1, 1,   2
2, 3,       8,               3
1, 2,          24,           50
2, 4,  10,  2,    136,       576
1, 3,                 342,   8232

Andrew Howroyd, Table of n, a(n) for n = 0..100

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. A081721 Classes under rotation and reversal
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. A275556 Classes under translation, rotation, complement and reversal
Cf. A275557 Classes under rotation and complement
Cf. A275558 Classes under rotation, complement and reversal

\\ see A056391 for Polya enumeration functions
a(n) = NonequivalentSorts(DihedralPerms(n), CyclicPerms(n)); \\ Andrew Howroyd, Sep 30 2017

Terms a(8) and beyond from Andrew Howroyd, Sep 30 2017

A275556 Number of classes of endofunctions of [n] under vertical translation mod n, rotation, complement to n+1 and reversal.

Original entry on oeis.org

1, 1, 2, 3, 13, 45, 412, 4375, 66988, 1199038, 25033020, 589567451, 15480284910, 448042511917, 14173510363424, 486548852524671, 18014399792942108, 715605766365332673, 30359882832309625502, 1370096714607544395379, 65536000002956800104588

Offset: 0

Olivier Gérard, Aug 05 2016

nonn

Because of the interaction between the two symmetries indexed by n and the two involutions, classes can be of size from n up to 4*n^2.
.
n  possible class sizes
------------------------------------
1  1
2  2
3  3, 6,                   18
4  4, 8,      16,          32,   64
5  5, 10,                  50,  100
6  6, 12, 18, 24,     36,  72,  144
7  7, 14,                  98,  196
.
but classes of size 4*n^2 account for the bulk of a(n).
n  number of classes
------------------------------------
1  1
2  2
3  1, 1,                    1
4  2, 3,       4,           3,    1
5  1, 2,                   22,   20
6  2, 4,   2,  2,     28, 116,  258
7  1, 3,                  339, 4032

Andrew Howroyd, Table of n, a(n) for n = 0..100

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. A081721 Classes under rotation and reversal
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. A275557 Classes under rotation and complement
Cf. A275558 Classes under rotation, complement and reversal

\\ see A056391 for Polya enumeration functions
a(n) = NonequivalentSorts(DihedralPerms(n), DihedralPerms(n)); \\ Andrew Howroyd, Sep 30 2017

Terms a(8) and beyond from Andrew Howroyd, Sep 30 2017

A275557 Number of classes of endofunctions of [n] under rotation and complement to n+1.

Original entry on oeis.org

1, 1, 2, 6, 38, 315, 3932, 58828, 1049108, 21523445, 500010024, 12968712306, 371504436220, 11649042561247, 396857394156656, 14596463012746392, 576460752571867208, 24330595937833434249, 1092955779880370116836, 52063675148955620766430, 2621440000000512000336088

Offset: 0

Olivier Gérard, Aug 05 2016

nonn

Classes can be of size 1,2,4, n and 2n.
n   1  2  4      n     2n
--------------------------
1   1
2   0  2
3   1  1                4
4   0  4  4      2     28
5   1  2  0      0    312
6   0  6  6     70   3850
7   1  3  0      0  58824
For n odd, the constant function (n+1)/2 is the only stable by rotation and complement. So #c1=1.
For n even, there is no stable function, so #c1=0, but constant functions are grouped two by two making n/2 classes of size 2. Functions alternating a value and its complement are also grouped two by two, making another n/2 classes. This gives #c2=n.

Andrew Howroyd, Table of n, a(n) for n = 0..100

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. A081721 Classes under rotation and reversal
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. A275558 Classes under rotation, complement and reversal

\\ see A056391 for Polya enumeration functions
a(n) = NonequivalentSorts(CyclicPerms(n), ReversiblePerms(n)); \\ Andrew Howroyd, Sep 30 2017

Terms a(8) and beyond from Andrew Howroyd, Sep 30 2017

A213941 Partition array a(n,k) with the total number of bracelets (D_n symmetry) with n beads, each available in n colors, with color signature given by the k-th partition of n in Abramowitz-Stegun(A-St) order.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 12, 12, 24, 3, 5, 20, 40, 60, 120, 120, 12, 6, 30, 90, 45, 180, 720, 220, 600, 1440, 900, 60, 7, 42, 126, 168, 315, 1890, 1050, 1890, 2100, 12600, 6720, 6300, 18900, 7560, 360, 8, 56, 224, 280, 224, 672, 4032, 6384, 5544, 6384, 5880, 45360

Offset: 1

Wolfdieter Lang, Jul 20 2012

This array is obtained by multiplying the entry of the array A213939(n,k) (number of bracelets (dihedral D_n symmetry) with n beads, each available in n colors, with color representative given by the n-multiset representative obtained from the k-th partition of n in A-St order after 'exponentiation') with the entry of the array A035206(n,k) (number of members in the equivalence class represented by the color multiset considered for A213939(n,k)): a(n,k)=A213939(n,k)*A035206(n,k), k=1..p(n)=A000041(n), n>=1. The row sums then give the total number of bracelets with n beads from n colors, given by A081721(n).
See A212359 for references, the 'exponentiation', and a link. For multiset signatures and representative multisets defining color multinomials see also a link in A213938.
The corresponding triangle with the summed row entries related to partitions of n with fixed number of parts is A214306.

			n\k 1   2    3    4    5     6     7     8     9    10   11
1   1
2   2   1
3   3   6    1
4   4  12   12   24    3
5   5  20   40   60  120   120    12
6   6  30   90   45  180   720   220   600  1440   900   60
...
Row m=7 is: 7 42 126 168 315 1890 1050 1890 2100 12600 6720 6300 18900 7560 360.
For the rows n=1 to n=15 see the link.
a(3,1) = 3 because the 3 bracelets with 3 beads coming in 3 colors have the color multinomials (here monomials) c[1]^3=c[1]*c[1]*c[1], c[2]^3 and c[3]^3. The partition of 3 is [3], the color representative is c[1]^3, and the equivalence class with color signature from the partition [3] has the three given members. There is no difference between necklace and bracelet numbers in this case.
a(3,2) = 6 from the color signature 2,1 with the representative multinomial c[1]^2 c[2] with coefficient A213939(3,2) = 1, the only 3-bracelet cyclic(112) (taking j for the color c[j]), and A035206(3,2) = 6 members of the whole color equivalence class: cyclic(112), cyclic(113), cyclic(221), cyclic(223), cyclic(331) and cyclic(332). There is no difference between necklaces and bracelets numbers in this case.
a(3,3) = 1, color signature 1^3 = 1,1,1 with representative multinomial c[1]*c[2]*c[3] with coefficient A213939(3,3)=1 from the bracelet cyclic(1,2,3). The necklace (1,3,2) becomes equivalent to this one under D_3 operation. There are no other members in this class (A035206(3,3)=1).
The sum of row No. 3 is 10 = A081721(3). The bracelets are 111, 222, 333, 112, 113, 221, 223, 331, 332 and 123, all taken cyclically.

Wolfdieter Lang, Rows n=1 to n=15.

Cf. A213939, A035206, A081721, A214306.

a(n,k) = A213939(n,k)*A035206(n,k), k=1, 2, ..., p(n) = A000041(n), n >= 1.

A051137 Table T(n,k) read by antidiagonals: number of necklaces allowing turnovers (bracelets) with n beads of k colors.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 6, 10, 10, 5, 1, 1, 8, 21, 20, 15, 6, 1, 1, 13, 39, 55, 35, 21, 7, 1, 1, 18, 92, 136, 120, 56, 28, 8, 1, 1, 30, 198, 430, 377, 231, 84, 36, 9, 1, 1, 46, 498, 1300, 1505, 888, 406, 120, 45, 10, 1

Offset: 0

Alford Arnold

Unlike A075195 and A284855, antidiagonals go from bottom-left to top-right.

			Table begins with T[0,1]:
1  1    1     1      1       1        1        1         1         1
1  2    3     4      5       6        7        8         9        10
1  3    6    10     15      21       28       36        45        55
1  4   10    20     35      56       84      120       165       220
1  6   21    55    120     231      406      666      1035      1540
1  8   39   136    377     888     1855     3536      6273     10504
1 13   92   430   1505    4291    10528    23052     46185     86185
1 18  198  1300   5895   20646    60028   151848    344925    719290
1 30  498  4435  25395  107331   365260  1058058   2707245   6278140
1 46 1219 15084 110085  563786  2250311  7472984  21552969  55605670
1 78 3210 53764 493131 3037314 14158228 53762472 174489813 500280022

C. G. Bower, Transforms (2)

Columns 2-6 are A000029, A027671, A032275, A032276, and A056341.
Rows 2-7 are A000217, A000292, A002817, A060446, A027670, and A060532.
Cf. A000031.
Cf. A081720, A081721.
T(n,k) = (A075195(n,k) + A284855(n,k)) / 2.

b[n_, k_] := DivisorSum[n, EulerPhi[#]*k^(n/#) &] / n;
c[n_, k_] := If[EvenQ[n], (k^(n/2) + k^(n/2+1))/2, k^((n+1)/2)];
T[0, ] = 1; T[n, k_] := (b[n, k] + c[n, k])/2;
Table[T[n, k-n], {k, 1, 11}, {n, k-1, 0, -1}] // Flatten
(* Robert A. Russell, Sep 21 2018 after Jean-François Alcover *)

T(n, k) = (k^floor((n+1)/2) + k^ceiling((n+1)/2)) / 4 + (1/(2*n)) * Sum_{d divides n} phi(d) * k^(n/d). - Robert A. Russell, Sep 21 2018
G.f. for column k: (kx/4)*(kx+x+2)/(1-kx^2) - Sum_{d>0} phi(d)*log(1-kx^d)/2d. - Robert A. Russell, Sep 28 2018
T(n, k) = (k^floor((n+1)/2) + k^ceiling((n+1)/2))/4 + (1/(2*n))*Sum_{i=1..n} k^gcd(n,i). (See A075195 formulas.) - Richard L. Ollerton, May 04 2021

A214306 Triangle with entry a(n,m) giving the total number of bracelets of n beads (D_n symmetry) with n colors available for each bead, but only m distinct colors present, with m from {1, 2, ..., n} and n >= 1.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 24, 24, 3, 5, 60, 180, 120, 12, 6, 165, 1120, 2040, 900, 60, 7, 336, 5145, 21420, 25200, 7560, 360, 8, 784, 23016, 183330, 442680, 335160, 70560, 2520, 9, 1584, 91056, 1320480, 5846400, 8890560, 4656960, 725760, 20160, 10

Offset: 1

Wolfdieter Lang, Jul 20 2012

This triangle is obtained from the array A213941 by summing in row n, for n >= 1, all entries related to partitions of n with the same number of parts m.
a(n,m) is the total number of necklaces of n beads (dihedral D_n symmetry) corresponding to all the color multinomials obtained from all p(n,m) = A008284(n,m) partitions of n with m parts, written in nonincreasing form, by 'exponentiation'. Therefore only m from the available n colors are present, and a(n,m) gives the number of bracelets with n beads with only m of the n available colors present, for m from 1,2,...,n, and n >= 1. All of the possible color assignments are counted.
See the comments on A212359 for the Abramowitz-Stegun (A-St) order of partitions, and the 'exponentiation' to obtain multisets, used to encode color multinomials, from partitions. See a link in A213938 for representative multisets for given signature used to define color multinomials.
The row sums of this triangle coincide with the ones of array A213941, and they are given by A081721.

			n\m 1    2     3       4       5       8       7      8     9
1   1
2   2    1
3   3    6     1
4   4   24    24       3
5   5   60   180     120      12
6   6  165  1120    2040     900     60
7   7  336  5145   21420   25200   7560      360
8   8  784 23016  183330  442680  33516    70560   2520
9   9 1584 91056 1320480 5846400 8890560 4656960 725760 20160
...
Row n=10:  10, 3420, 357480, 8691480, 64420272, 172609920, 177811200, 68040000, 8164800, 181440;
Row n=11:  11, 6820, 1327095, 52727400, 622175400, 2714009760, 4837417200, 3592512000, 1047816000, 99792000, 1814400.
a(2,2) = 1 from the color monomial c[1]^1*c[2]^1 = c[1]*c[2] (from the m=2 partition [1,1] of n=2). The bracelet in question is cyclic(12) (we use j for color c[j] in these examples). The same holds for the necklace case.
a(5,3) = 60 + 120 = 180, from A213941(5,4) + A213941(5,5), because k(5,3,1) = A214314(5,3) = 4 and p(5,3)=2.
a(3,1) = 3 from the color monomials c[1]^3, c[2]^3 and c[3]^3. The three bracelets are cyclic(111), cyclic(222) and cyclic(333). The same holds for the necklace case.
In general a(n,1)=n from the partition [n] providing the color signature (exponent), and the n color choices.
a(3,2) = 6 from the color signature c[.]^2 c[.]^1, (from the m=2 partition [2,1] of n=3), and there are 6 choices for the color indices. The 6 bracelets are cyclic(112), cyclic(113), cyclic(221), cyclic(223), cyclic(331) and cyclic(332). The same holds for the necklace case.
a(3,3) = 1. The color multinomial is c[1]*c[2]*c[3] (from the m=3 partition [1,1,1]). All three available colors are used. There is only one bracelet: cyclic(1,2,3). The necklace cyclic(1,3,2) becomes equivalent under D_3 operation.
a(4,2) = 24 from two color signatures c[.]^3 c[.] and c[.]^2 c[.]^2 (from the two m=2 partitions of n=4: [3,1] and [2,2]). The first one produces 4*3=12 bracelets, namely 1112, 1113, 1114, 2221, 2223, 2224, 3331, 3332, 3334, 4441, 4442 and 4443, all taken cyclically. The second color signature leads to another 6*2=12 bracelets: 1122, 1133, 1144, 2233, 2244, 3344, 1212, 1313, 1414, 2323, 2424 and 3434, all taken cyclically. Together they provide the 24 bracelets counted by a(4,2). The same holds for the necklace case.
a(4,3) = 24 from the color signature c[.]^2 c[.]c[.]. There are 4*3 =12 color choices each with two bracelets: 1123, 1213, 1124, 1214, 1134, 1314, 2213, 2123, 2214, 2124, 2234, 2324, 3312, 3132, 3314, 3134, 3324, 3234, 4412, 4142, 4413, 4143, 4423 and 4243, each taken cyclically.

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Cf. A081721, A212360 (necklaces), A213941, A273891.

(* t = A081720 *) 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)]);
T[n_, k_] := Binomial[n, k]*Sum[(-1)^i * Binomial[k, i]*t[n, k - i], {i, 0, k - 1}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)

a(n,m) = Sum_{j=1..p(n,m)} A213941(n, k(n,m,1)+j-1), with k(n,m,1) = A214314(n,m) the position where in the list of partitions of n in A-St order the first with m parts appears, and p(n,m) is the number of partitions of n with m parts shown in the array A008284. E.g., n=5, m=3: k(5,3,1) = A214314(5,3) = 4, p(5,3) = 2.

a(n,m) = binomial(n,m) * A273891(n,m). - Andrew Howroyd, Mar 25 2017

A321791 Table read by descending antidiagonals: T(n,k) is the number of unoriented cycles (bracelets) of length n using up to k available colors.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 1, 0, 1, 5, 10, 10, 6, 1, 0, 1, 6, 15, 20, 21, 8, 1, 0, 1, 7, 21, 35, 55, 39, 13, 1, 0, 1, 8, 28, 56, 120, 136, 92, 18, 1, 0, 1, 9, 36, 84, 231, 377, 430, 198, 30, 1, 0

Offset: 0

Robert A. Russell, Dec 18 2018

			Table begins with T(0,0):
  1 1  1    1     1      1       1        1        1         1         1 ...
  0 1  2    3     4      5       6        7        8         9        10 ...
  0 1  3    6    10     15      21       28       36        45        55 ...
  0 1  4   10    20     35      56       84      120       165       220 ...
  0 1  6   21    55    120     231      406      666      1035      1540 ...
  0 1  8   39   136    377     888     1855     3536      6273     10504 ...
  0 1 13   92   430   1505    4291    10528    23052     46185     86185 ...
  0 1 18  198  1300   5895   20646    60028   151848    344925    719290 ...
  0 1 30  498  4435  25395  107331   365260  1058058   2707245   6278140 ...
  0 1 46 1219 15084 110085  563786  2250311  7472984  21552969  55605670 ...
  0 1 78 3210 53764 493131 3037314 14158228 53762472 174489813 500280022 ...
For T(3,3)=10, the unoriented cycles are 9 achiral (AAA, AAB, AAC, ABB, ACC, BBB, BBC, BCC, CCC) and 1 chiral pair (ABC-ACB).

Index entries for sequences related to bracelets

Cf. A075195 (oriented), A293496(chiral), A284855 (achiral).
Cf. A051137 (ascending antidiagonals).
Columns 0-6 are A000007, A000012, A000029, A027671, A032275, A032276, and A056341.
Rows 0-7 are A000012, A001477, A000217, A000292, A002817, A060446, A027670, and A060532.
Main diagonal gives A081721.

Table[If[k>0, DivisorSum[k, EulerPhi[#](n-k)^(k/#)&]/(2k) + ((n-k)^Floor[(k+1)/2]+(n-k)^Ceiling[(k+1)/2])/4, 1], {n, 0, 12}, {k, 0, n}] // Flatten

T(n,k) = [n==0] + [n>0] * (k^floor((n+1)/2) + k^ceiling((n+1)/2)) / 4 + (1/(2*n)) * Sum_{d|n} phi(d) * k^(n/d).
T(n,k) = (A075195(n,k) + A284855(n,k)) / 2.
T(n,k) = A075195(n,k) - A293496(n,k) = A293496(n,k) + A284855(n,k).
Linear recurrence for row n: T(n,k) = Sum_{j=0..n} -binomial(j-n-1,j+1) * T(n,k-1-j) for k >= n + 1.
O.g.f. for column k >= 0: Sum_{n>=0} T(n,k)*x^n = 3/4 + (1 + k*x)^2/(4*(1 - k*x^2)) - (1/2) * Sum_{d >= 1} (phi(d)/d) * log(1 - k*x^d). - Petros Hadjicostas, Feb 07 2021

cp's OEIS Frontend

A275554 Number of classes of endofunctions of [n] under vertical translation mod n, rotation and complement to n+1.

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A275555 Number of classes of endofunctions of [n] under vertical translation mod n, rotation and reversal.

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A275556 Number of classes of endofunctions of [n] under vertical translation mod n, rotation, complement to n+1 and reversal.

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A275557 Number of classes of endofunctions of [n] under rotation and complement to n+1.

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A213941 Partition array a(n,k) with the total number of bracelets (D_n symmetry) with n beads, each available in n colors, with color signature given by the k-th partition of n in Abramowitz-Stegun(A-St) order.

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A051137 Table T(n,k) read by antidiagonals: number of necklaces allowing turnovers (bracelets) with n beads of k colors.

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A214306 Triangle with entry a(n,m) giving the total number of bracelets of n beads (D_n symmetry) with n colors available for each bead, but only m distinct colors present, with m from {1, 2, ..., n} and n >= 1.

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A321791 Table read by descending antidiagonals: T(n,k) is the number of unoriented cycles (bracelets) of length n using up to k available colors.

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