A320742 -id:A320742 - cp's OEIS frontend

A059053 Number of chiral pairs of necklaces with n beads and two colors (color complements being equivalent); i.e., turning the necklace over neither leaves it unchanged nor simply swaps the colors.

0, 0, 0, 0, 0, 0, 0, 1, 2, 7, 12, 31, 58, 126, 234, 484, 906, 1800, 3402, 6643, 12624, 24458, 46686, 90157, 172810, 333498, 641340, 1238671, 2388852, 4620006, 8932032, 17302033, 33522698, 65042526, 126258960, 245361172, 477091232

Offset: 0

Henry Bottomley, Dec 21 2000

nonn

Number of chiral pairs of set partitions of a cycle of n elements using exactly two different elements. - Robert A. Russell, Oct 02 2018

			For a(7) = 1, the chiral pair is AAABABB-AAABBAB.
For a(8) = 2, the chiral pairs are AAAABABB-AAAABBAB and AAABAABB-AAABBAAB.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Illustration of initial terms
Index entries for sequences related to necklaces

Column 2 of A320647 and A320742.

Cf. A056295 (oriented), A056357 (unoriented), A052551(n-2) (achiral).

Prepend[Table[DivisorSum[n, EulerPhi[#] StirlingS2[n/# + If[Divisible[#,2],1,0], 2] &] / (2n) - StirlingS2[1+Floor[n/2],2] / 2, {n, 1, 40}],0] (* Robert A. Russell, Oct 02 2018 *)

a(n) = {if(n<1, 0, (sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (2*n) - 2^(n\2))/2)}; \\ Andrew Howroyd, Nov 03 2019

a(n) = A000013(n) - A000011(n) = A000011(n) - A016116(n) = (A000013(n) - A016116(n))/2.
From Robert A. Russell, Oct 02 2018: (Start)
a(n) = (A056295(n)-A052551(n-2)) / 2 = A056295(n) - A056357(n) = A056357(n) - A052551(n-2).
a(n) = -S2(1+floor(n/2),2) + (1/2n) * Sum_{d|n} phi(d) * S2(n/d+[2|d],2), where S2 is a Stirling subset number A008277.
G.f.: -x(1+2x)/(2-4x^2) - Sum_{d>0} phi(d) * log(1-2x^d) / (2d*(2-[2|d])).
(End)

Name clarified by Robert A. Russell, Oct 02 2018

A320748 Array read by antidiagonals: T(n,k) is the number of color patterns (set partitions) in an unoriented cycle of length n using k or fewer colors (subsets).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 1, 1, 2, 3, 6, 4, 1, 1, 2, 3, 7, 9, 8, 1, 1, 2, 3, 7, 11, 22, 9, 1, 1, 2, 3, 7, 12, 33, 40, 18, 1, 1, 2, 3, 7, 12, 36, 73, 100, 23, 1, 1, 2, 3, 7, 12, 37, 89, 237, 225, 44, 1, 1, 2, 3, 7, 12, 37, 92, 322, 703, 582, 63, 1, 1, 2, 3, 7, 12, 37, 93, 349, 1137, 2433, 1464, 122, 1, 1, 2, 3, 7, 12, 37, 93, 353, 1308, 4704, 8309, 3960, 190, 1, 1, 2, 3, 7, 12, 37, 93, 354, 1345, 5953, 19839, 30108, 10585, 362, 1

Offset: 1

Robert A. Russell, Oct 21 2018

Two color patterns are equivalent if the colors are permuted. An unoriented cycle counts each chiral pair as one, i.e., they are equivalent.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
T(n,k)=Pi_k(C_n) which is the number of non-equivalent partitions of the cycle on n vertices, with at most k parts. Two partitions P1 and P2 of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. - Bahman Ahmadi, Aug 21 2019
In other words, the number of n-bead bracelet structures using a maximum of k different colored beads. - Andrew Howroyd, Oct 30 2019

			Array begins with T(1,1):
1   1    1     1     1      1      1      1      1      1      1      1 ...
1   2    2     2     2      2      2      2      2      2      2      2 ...
1   2    3     3     3      3      3      3      3      3      3      3 ...
1   4    6     7     7      7      7      7      7      7      7      7 ...
1   4    9    11    12     12     12     12     12     12     12     12 ...
1   8   22    33    36     37     37     37     37     37     37     37 ...
1   9   40    73    89     92     93     93     93     93     93     93 ...
1  18  100   237   322    349    353    354    354    354    354    354 ...
1  23  225   703  1137   1308   1345   1349   1350   1350   1350   1350 ...
1  44  582  2433  4704   5953   6291   6345   6350   6351   6351   6351 ...
1  63 1464  8309 19839  28228  31284  31874  31944  31949  31950  31950 ...
1 122 3960 30108 88508 144587 171283 178190 179204 179300 179306 179307 ...
For T(7,2)=9, the patterns are AAAAAAB, AAAAABB, AAAABAB, AAAABBB, AAABAAB, AAABABB, AABAABB, AABABAB, and AAABABB; only the last is chiral, paired with AAABBAB.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1275
B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Partial row sums of A152176.
Columns 1-6 are A057427, A000011, A056353, A056354, A056355, A056356
For increasing k, columns converge to A084708.
Cf. A320747 (oriented), A320742 (chiral), A305749 (achiral).

Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d, Adnk[d,n-1,k-#]&], Boole[n == 0 && k == 0]]
Ach[n_,k_] := Ach[n,k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
Table[Sum[(DivisorSum[n, EulerPhi[#] Adnk[#,n/#,j]&]/n + Ach[n,j])/2, {j,k-n+1}], {k,15}, {n,k}] // Flatten

\\ Ach is A304972 and R is A152175 as square matrices.
Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
R(n)={Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
T(n)={my(M=(R(n) + Ach(n))/2); for(i=2, n, M[,i] += M[,i-1]); M}
{ my(A=T(12)); for(n=1, #A, print(A[n, ])) } \\ Andrew Howroyd, Nov 03 2019

T(n,k) = Sum_{j=1..k} Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k)+Ach(n-2,k-1)+Ach(n-2,k-2)) and A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).

T(n,k) = (A320747(n,k) + A305749(n,k)) / 2 = A320747(n,k) - A320742(n,k) = A320742(n,k) + A305749(n,k).

A320743 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 3 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 13, 46, 144, 420, 1221, 3474, 9856, 27794, 78632, 222156, 629760, 1787440, 5087797, 14509580, 41479867, 118811286, 341009901, 980488510, 2824029648, 8146494860, 23534997912, 68084154502, 197211336576, 571915188840, 1660405181149, 4825559508106, 14038010213051, 40875403561680, 119122661856133, 347441159864556, 1014152747485696

Offset: 1

Robert A. Russell, Oct 21 2018

Two color patterns are equivalent if the colors are permuted.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
There are nonrecursive formulas, generating functions, and computer programs for A002076 and A182522, which can be used in conjunction with the first formula.

			For a(6)=4, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, and AABACC-AABBAC.

Andrew Howroyd, Table of n, a(n) for n = 1..200
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column 3 of A320742.

Cf. A002076 (oriented), A056353 (unoriented), A182522 (achiral).

Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d, Adnk[d,n-1,k-#]&], Boole[n == 0 && k == 0]]
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=3; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]

a(n) = (A002076(n) - A182522(n)) / 2 = A002076(n) - A056353(n) = A056353(n) - A182522(n).
a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=3 is the maximum number of colors, Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)), and A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).
a(n) = A059053(n) + A320643(n).

A320749 Number of chiral pairs of color patterns (set partitions) in a cycle of length n.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 34, 190, 1011, 5352, 29740, 172466, 1055232, 6793791, 46034940, 327303819, 2436650368, 18944771253, 153488081102, 1293086505784, 11306373089104, 102425178180769, 959825673145688, 9290807818971900, 92771800581171418, 954447025978145744, 10105871186441842623, 110009631951698573068, 1229996584263621368224, 14112483571723367245825, 166021918475962174194914, 2001010469483653602192695

Offset: 1

Robert A. Russell, Oct 22 2018

Two color patterns are equivalent if the colors are permuted.

Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.

			For a(6)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD, and AABCBD-AABCDC.

Andrew Howroyd, Table of n, a(n) for n = 1..200
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Row sums of A320647.
Columns of A320742 converge to this as k increases.
Cf. A084423 (oriented), A084708 (unoriented), A080107 (achiral).

Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d,Adnk[d,n-1,k-#]&], Boole[n==0 && k==0]]
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]]
Table[Sum[(DivisorSum[n, EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j,n}], {n,40}]

a(n) = Sum_{j=1..n} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)) and A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).

a(n) = (A084423(n) - A080107(n)) / 2 = A084423(n) - A084708(n) = A084708(n) - A080107(n).

A320744 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 4 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 30, 130, 532, 2006, 7626, 28401, 106260, 396435, 1486147, 5580130, 21032880, 79486763, 301317282, 1145123672, 4362804633, 16658456825, 63738451998, 244332656201, 938244497740, 3608640426930, 13899977105315, 53614228550220, 207061964668740, 800639722002163, 3099251007215286, 12009598156277090, 46582685655751645, 180850428684482360

Offset: 1

Robert A. Russell, Oct 21 2018

Two color patterns are equivalent if the colors are permuted.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
There are nonrecursive formulas, generating functions, and computer programs for A056292 and A305750, which can be used in conjunction with the first formula.

			For a(6)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD, and AABCBD-AABCDC.

Andrew Howroyd, Table of n, a(n) for n = 1..200
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column 4 of A320742.

Cf. A056292 (oriented), A056354 (unoriented), A305750 (achiral).

Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d, Adnk[d,n-1,k-#]&], Boole[n == 0 && k == 0]]
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=4; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]

a(n) = (A056292(n) - A305750(n)) / 2 = A056292(n) - A056354(n) = A056354(n) - A305750(n).
a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=4 is the maximum number of colors, Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)), and A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).
a(n) = A059053(n) + A320643(n) + A320644(n).

A320745 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 5 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 34, 181, 871, 4016, 18526, 85101, 393148, 1822977, 8500893, 39809180, 187230704, 883730048, 4184926222, 19874478310, 94629276256, 451604739323, 2159748985582, 10348493650194, 49671898709098, 238804606717950, 1149792470325340, 5543620159707666, 26762240285558924, 129350640352555296, 625889650880647630, 3031651402693863747, 14698911258326292182, 71332938143655936584, 346474231506471943759

Offset: 1

Robert A. Russell, Oct 21 2018

Two color patterns are equivalent if the colors are permuted.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
There are nonrecursive formulas, generating functions, and computer programs for A056293 and A305751, which can be used in conjunction with the first formula.

			For a(6)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD, and AABCBD-AABCDC.

Andrew Howroyd, Table of n, a(n) for n = 1..200
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column 5 of A320742.

Cf. A056293 (oriented), A056355 (unoriented), A305751 (achiral).

Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d, Adnk[d,n-1,k-#]&], Boole[n == 0 && k == 0]]
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=5; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]

a(n) = (A056293(n) - A305751(n)) / 2 = A056293(n) - A056355(n) = A056355(n) - A305751(n).
a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=5 is the maximum number of colors, Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)), and A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).
a(n) = A059053(n) + A320643(n) + A320644(n) + A320645(n).

A320746 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 6 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 34, 190, 996, 5070, 26454, 139484, 749742, 4082481, 22509626, 125231540, 702004040, 3958071545, 22423227634, 127524417922, 727617119592, 4163076477731, 23876455868772, 137228326265794, 790200053665362, 4557942281943078, 26331297198477874, 152331940294133402, 882422871962784662, 5117852332008063806, 29715786649820358328, 172720006045619486686, 1004904748993330281274, 5852047136464153694312

Offset: 1

Robert A. Russell, Oct 21 2018

Two color patterns are equivalent if the colors are permuted.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
There are nonrecursive formulas, generating functions, and computer programs for A056294 and A305752, which can be used in conjunction with the first formula.

			For a(6)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD, and AABCBD-AABCDC.

Andrew Howroyd, Table of n, a(n) for n = 1..200
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column 6 of A320742.

Cf. A056294 (oriented), A056356 (unoriented), A305752 (achiral).

Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d, Adnk[d,n-1,k-#]&], Boole[n == 0 && k == 0]]
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=6; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]

a(n) = (A056294(n) - A305752(n)) / 2 = A056294(n) - A056356(n) = A056356(n) - A305752(n).
a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=6 is the maximum number of colors, Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)), and A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).
a(n) = A059053(n) + A320643(n) + A320644(n) + A320645(n) + A320646(n).

A320747 Array read by antidiagonals: T(n,k) is the number of color patterns (set partitions) in an oriented cycle of length n using k or fewer colors (subsets).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 1, 1, 2, 3, 6, 4, 1, 1, 2, 3, 7, 9, 8, 1, 1, 2, 3, 7, 11, 26, 10, 1, 1, 2, 3, 7, 12, 39, 53, 20, 1, 1, 2, 3, 7, 12, 42, 103, 146, 30, 1, 1, 2, 3, 7, 12, 43, 123, 367, 369, 56, 1, 1, 2, 3, 7, 12, 43, 126, 503, 1235, 1002, 94, 1, 1, 2, 3, 7, 12, 43, 127, 539, 2008, 4439, 2685, 180, 1, 1, 2, 3, 7, 12, 43, 127, 543, 2304, 8720, 15935, 7434, 316, 1, 1, 2, 3, 7, 12, 43, 127, 544, 2356, 11023, 38365, 58509, 20441, 596, 1

Offset: 1

Robert A. Russell, Oct 21 2018

Two color patterns are equivalent if the colors are permuted. An oriented cycle counts each chiral pair as two.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
In other words, the number of n-bead necklace structures using a maximum of k different colored beads. - Andrew Howroyd, Oct 30 2019

			Array begins with T(1,1):
1   1    1     1      1      1      1      1      1      1      1      1 ...
1   2    2     2      2      2      2      2      2      2      2      2 ...
1   2    3     3      3      3      3      3      3      3      3      3 ...
1   4    6     7      7      7      7      7      7      7      7      7 ...
1   4    9    11     12     12     12     12     12     12     12     12 ...
1   8   26    39     42     43     43     43     43     43     43     43 ...
1  10   53   103    123    126    127    127    127    127    127    127 ...
1  20  146   367    503    539    543    544    544    544    544    544 ...
1  30  369  1235   2008   2304   2356   2360   2361   2361   2361   2361 ...
1  56 1002  4439   8720  11023  11619  11697  11702  11703  11703  11703 ...
1  94 2685 15935  38365  54682  60499  61579  61684  61689  61690  61690 ...
1 180 7434 58509 173609 284071 336447 349746 351619 351766 351772 351773 ...
For T(4,2)=4, the patterns are AAAA, AAAB, AABB, and ABAB.
For T(4,3)=6, the patterns are the above four, AABC and ABAC.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1275
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Partial row sums of A152175.
Columns 1-6 are A057427, A000013, A002076, A056292, A056293, A056294.
For increasing k, columns converge to A084423.
Cf. A320748 (unoriented), A320742 (chiral), A305749 (achiral).

Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d, Adnk[d,n-1,k-#] &], Boole[n==0 && k==0]]
Table[Sum[DivisorSum[n, EulerPhi[#] Adnk[#,n/#,j] &], {j,k-n+1}]/n, {k,15}, {n,k}] // Flatten

\\ R is A152175 as square matrix
R(n) = {Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
T(n)={my(M=R(n)); for(i=2, n, M[,i] += M[,i-1]); M}
{ my(A=T(12)); for(n=1, #A, print(A[n, ])) } \\ Andrew Howroyd, Nov 03 2019

T(n,k) = (1/n)*Sum_{j=1..k} Sum_{d|n} phi(d)*A(d,n/d,j), where A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).

T(n,k) = A320748(n,k) + A320742(n,k) = 2*A320748(n,k) - A305749(n,k) = A305749(n,k) + 2*A320742(n,k).

cp's OEIS Frontend

A059053 Number of chiral pairs of necklaces with n beads and two colors (color complements being equivalent); i.e., turning the necklace over neither leaves it unchanged nor simply swaps the colors.

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A320748 Array read by antidiagonals: T(n,k) is the number of color patterns (set partitions) in an unoriented cycle of length n using k or fewer colors (subsets).

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A320743 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 3 or fewer colors (subsets).

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A320749 Number of chiral pairs of color patterns (set partitions) in a cycle of length n.

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A320744 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 4 or fewer colors (subsets).

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A320745 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 5 or fewer colors (subsets).

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A320746 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 6 or fewer colors (subsets).

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