A056293 -id:A056293 - cp's OEIS frontend

A152175 Triangle read by rows: T(n,k) is the number of k-block partitions of an n-set up to rotations.

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 5, 2, 1, 1, 7, 18, 13, 3, 1, 1, 9, 43, 50, 20, 3, 1, 1, 19, 126, 221, 136, 36, 4, 1, 1, 29, 339, 866, 773, 296, 52, 4, 1, 1, 55, 946, 3437, 4281, 2303, 596, 78, 5, 1, 1, 93, 2591, 13250, 22430, 16317, 5817, 1080, 105, 5, 1, 1, 179, 7254, 51075, 115100, 110462, 52376, 13299, 1873, 147, 6, 1

Offset: 1

Vladeta Jovovic, Nov 27 2008

Number of n-bead necklace structures using exactly k different colored beads. Turning over the necklace is not allowed. Permuting the colors does not change the structure. - Andrew Howroyd, Apr 06 2017

			Triangle begins with T(1,1):
  1;
  1,   1;
  1,   1,     1;
  1,   3,     2,      1;
  1,   3,     5,      2,      1;
  1,   7,    18,     13,      3,      1;
  1,   9,    43,     50,     20,      3,      1;
  1,  19,   126,    221,    136,     36,      4,      1;
  1,  29,   339,    866,    773,    296,     52,      4,     1;
  1,  55,   946,   3437,   4281,   2303,    596,     78,     5,    1;
  1,  93,  2591,  13250,  22430,  16317,   5817,   1080,   105   , 5,   1;
  1, 179,  7254,  51075, 115100, 110462,  52376,  13299,  1873,  147,   6, 1;
  1, 315, 20125, 194810, 577577, 717024, 439648, 146124, 27654, 3025, 187, 6, 1;
  ...
For T(4,2)=3, the set partitions are AAAB, AABB, and ABAB.
For T(4,3)=2, the set partitions are 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.
Tilman Piesk, Partition related number triangles

Columns 2-6 are A056295, A056296, A056297, A056298, A056299.
Row sums are A084423.
Partial row sums include A000013, A002076, A056292, A056293, A056294.
Cf. A075195, A087854, A008277 (set partitions), A284949 (up to reflection), A152176 (up to rotation and reflection).
A(1,n,k) in formula is the Stirling subset number A008277.
A(2,n,k) in formula is A293181; A(3,n,k) in formula is A294201.

(* Using recursion formula from Gilbert and Riordan:*)
Adn[d_, n_] := Adn[d, n] = Which[0==n, 1, 1==n, DivisorSum[d, x^# &],
  1==d, Sum[StirlingS2[n, k] x^k, {k, 0, n}],
  True, Expand[Adn[d, 1] Adn[d, n-1] + D[Adn[d, n - 1], x] x]];
Table[CoefficientList[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]/(x n), x],
   {n, 1, 10}] // Flatten (* Robert A. Russell, Feb 23 2018 *)
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[DivisorSum[n,EulerPhi[#]Adnk[#,n/#,k]&]/n,{n,1,12},{k,1,n}] // Flatten (* Robert A. Russell, Oct 16 2018 *)

\\ see A056391 for Polya enumeration functions
T(n,k) = NonequivalentStructsExactly(CyclicPerms(n), k); \\ Andrew Howroyd, Oct 14 2017

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))]))}
{ my(A=R(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Sep 20 2019

T(n,k) = (1/n)*Sum_{d|n} phi(d)*A(d,n/d,k), 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)). - Robert A. Russell, Oct 16 2018

A056355 Number of bracelet structures using a maximum of five different colored beads.

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 36, 89, 322, 1137, 4704, 19839, 88508, 399680, 1839947, 8533488, 39893901, 187393550, 884153396, 4185740195, 19876594537, 94633345608, 451615319433, 2159769331317, 10348546548695, 49672000435724, 238804871206358, 1149792978954373, 5543621482141513

Offset: 0

Marks R. Nester

nonn

Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure.

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 = 0..200

Cf. A056293, A152176.

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.

a(n) = Sum_{k=1..5} A152176(n, k) for n > 0. - Andrew Howroyd, Oct 25 2019

a(0)=1 prepended and terms a(25) and beyond from Andrew Howroyd, Oct 25 2019

A056298 Number of n-bead necklace structures using exactly five different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 20, 136, 773, 4281, 22430, 115100, 577577, 2863227, 14051164, 68515514, 332514803, 1608800691, 7767857090, 37460388596, 180536313547, 869901397479, 4192038616700, 20208367895980

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.

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]

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column 5 of A152175.

Cf. A056285, A056292, A056293.

From Robert A. Russell, May 29 2018: (Start)
Adn[d_, n_] := Adn[d, n] = If[1==n, DivisorSum[d, x^# &],
  Expand[Adn[d, 1] Adn[d, n-1] + D[Adn[d, n-1], x] x]];
Table[Coefficient[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]/n , x, 5],
  {n, 1, 40}] (* after Gilbert and Riordan *)
Table[(1/n) DivisorSum[n, EulerPhi[#] Which[Divisible[#, 60], StirlingS2[n/#+4, 5] - 6 StirlingS2[n/#+3, 5] + 11 StirlingS2[n/#+2, 5] - 6 StirlingS2[n/#+1, 5], Divisible[#, 30], StirlingS2[n/#+4, 5] - 8 StirlingS2[n/#+3, 5] + 26 StirlingS2[n/#+2, 5] - 43 StirlingS2[n/#+1, 5] + 30 StirlingS2[n/#, 5], Divisible[#, 20], StirlingS2[n/#+4, 5] - 8 StirlingS2[n/#+3, 5] + 23 StirlingS2[n/#+2, 5] - 24 StirlingS2[n/#+1, 5], Divisible[#, 15], StirlingS2[n/#+4, 5] - 10 StirlingS2[n/#+3, 5] + 38 StirlingS2[n/#+2, 5] - 65 StirlingS2[n/#+1, 5] + 45 StirlingS2[n/#, 5], Divisible[#, 12], 4 StirlingS2[n/#+3, 5] - 24 StirlingS2[n/#+2, 5] + 44 StirlingS2[n/#+1, 5] - 24 StirlingS2[n/#, 5], Divisible[#, 10], StirlingS2[n/#+4, 5] - 10 StirlingS2[n/#+3, 5] + 38 StirlingS2[n/#+2, 5] - 61 StirlingS2[n/#+1, 5] + 30 StirlingS2[n/#, 5], Divisible[#, 6], 2 StirlingS2[n/#+3, 5] - 9 StirlingS2[n/#+2, 5] + 7 StirlingS2[n/#+1, 5] + 6 StirlingS2[n/#, 5], Divisible[#, 5], StirlingS2[n/#+4, 5] - 10 StirlingS2[n/#+3, 5] + 35 StirlingS2[n/#+2, 5] - 50 StirlingS2[n/#+1, 5] + 25 StirlingS2[n/#, 5], Divisible[#, 4], 2 StirlingS2[n/#+3, 5] - 12 StirlingS2[n/#+2, 5] + 26 StirlingS2[n/#+1, 5] - 24 StirlingS2[n/#, 5], Divisible[#, 3], 3 StirlingS2[n/#+2, 5] - 15 StirlingS2[n/#+1, 5] + 21 StirlingS2[n/#, 5], Divisible[#, 2], 3 StirlingS2[n/#+2, 5] - 11 StirlingS2[n/#+1, 5] + 6 StirlingS2[n/#, 5], True, StirlingS2[n/#, 5]] &], {n, 1, 40}]
mx = 40; Drop[CoefficientList[Series[-Sum[(EulerPhi[d] / d) Which[
  Divisible[d, 60], Log[1 - 5x^d] - Log[1 - 4x^d], Divisible[d, 30],
  (3 Log[1 - 5x^d] - 3 Log[1 - 4x^d] + Log[1 - x^d]) / 4, Divisible[d, 20],
  (2 Log[1 - 5x^d] - 2 Log[1 - 4x^d] + Log[1 - 2x^d] - Log[1 - x^d]) / 3,
  Divisible[d, 15], (3 Log[1 - 5x^d] - 3 Log[1 - 4x^d] + 2 Log[1 - 3x^d] -
  2 Log[1 - 2x^d] + 3 Log[1 - x^d]) / 8, Divisible[d, 12],
  (4 Log[1 - 5x^d] - 5 Log[1 - 4x^d]) / 5, Divisible[d, 10],
  (5 Log[1 - 5x^d] - 5 Log[1 - 4x^d] + 4 Log[1 - 2x^d] - Log[1 - x^d]) / 12,
  Divisible[d, 6], (11 Log[1 - 5x^d] - 15 Log[1 - 4x^d] + 5 Log[1 - x^d]) /
  20, Divisible[d, 5], (5 Log[1 - 5x^d] - Log[1 - 4x^d] + 2 Log[1 - 3x^d] -
  2 Log[1 - 2x^d] + Log[1 - x^d]) / 24, Divisible[d, 4], (7 Log[1 - 5x^d] -
  10 Log[1 - 4x^d] + 5 Log[1 - 2x^d] - 5 Log[1 - x^d]) / 15,
  Divisible[d, 3], (7 Log[1 - 5x^d] - 15 Log[1 - 4x^d] + 10 Log[1 - 3x^d] -
  10 Log[1 - 2x^d] + 15 Log[1 - x^d]) / 40, Divisible[d, 2],
  (13 Log[1 - 5x^d] - 25 Log[1 - 4x^d] + 20 Log[1 - 2x^d] -
  5 Log[1 - x^d]) / 60, True, (Log[1 - 5x^d] - 5 Log[1 - 4x^d] +
  10 Log[1 - 3x^d] - 10 Log[1 - 2x^d] + 5 Log[1 - x^d]) / 120], {d, 1, mx}], {x, 0, mx}], x], 1]
(End)

a(n) = A056293(n) - A056292(n).
From Robert A. Russell, May 29 2018: (Start)
a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 60] * (S2(n/d+4,5) -
  6*S2(n/d+3,5) + 11*S2(n/d+2,5) - 6*S2(n/d+1,5)) + [d==30 mod 60] *
  (S2(n/d+4,5) - 8*S2(n/d+3,5) + 26*S2(n/d+2,5) - 43*S2(n/d+1,5) +
  30*S2(n/d,5)) + [d==20 mod 60 | d==40 mod 60] * (S2(n/d+4,5) -
  8*S2(n/d+3,5) + 23*S2(n/d+2,5) - 24*S2(n/d+1,5)) + [d==15 mod 60 |
  d==45 mod 60] * (S2(n/d+4,5) - 10*S2(n/d+3,5) + 38*S2(n/d+2,5) -
  65*S2(n/d+1,5) + 45*S2(n/d,5)) + [d mod 60 in {12,24,36,48}] *
  (4*S2(n/d+3,5) - 24*S2(n/d+2,5) + 44*S2(n/d+1,5) - 24*S2(n/d,5)) +
  [d=10 mod 60 | d==50 mod 60] * (S2(n/d+4,5) - 10*S2(n/d+3,5) +
  38*S2(n/d+2,5) - 61*S2(n/d+1,5) + 30*S2(n/d,5)) + [d mod 60 in
  {6,18,42,54}] * (2*S2(n/d+3,5) - 9*S2(n/d+2,5) + 7*S2(n/d+1,5) +
  6*S2(n/d,5)) + [d mod 60 in {5,25,35,55}] * (S2(n/d+4,5) -
  10*S2(n/d+3,5) + 35*S2(n/d+2,5) - 50*S2(n/d+1,5) + 25*S2(n/d,5)) +
  [d mod 60 in {4,8,16,28,32,44,52,56}] * (2*S2(n/d+3,5) - 12*S2(n/d+2,5) +
  26*S2(n/d+1,5) - 24*S2(n/d,5)) + [d mod 60 in {3,9,21,27,33,39,51,57}] *
  (3*S2(n/d+2,5) - 15*S2(n/d+1,5) + 21*S2(n/d,5)) + [d mod 60 in
  {2,14,22,26,34,38,46,58}] * (3*S2(n/d+2,5) - 11*S2(n/d+1,5) +
  6*S2(n/d,5)) + [d mod 60 in {1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,
  59}] * S2(n/d,5)), where S2(n,k) is the Stirling subset number, A008277.
G.f.: -Sum_{d>0} (phi(d) / d) * ([d==0 mod 60] * (log(1-4x^d) -
  log(1-3x^d)) + [d==30 mod 60] * (3*log[1-5x^d) - 3*log(1-4x^d) +
  log(1-x^d)) / 4 + [d==20 mod 60 | d==40 mod 60] * (2*log(1-5x^d) -
  2*log(1-4x^d) + log(1-2x^d) - log(1-x^d)) / 3 +
  [d==15 mod 60 | d==45 mod 60] * (3*log(1-5x^d) - 3*log(1-4x^d) +
  2*log(1-3x^d) - 2*log(1-2x^d) + 3*log(1-x^d)) / 8 + [d mod 60 in
  {12,24,36,48}] * (4*log(1-5x^d) - 5*log(1-4x^d)) / 5 + [d=10 mod 60 |
  d==50 mod 60] * (5*log(1-5x^d) - 5*log(1-4x^d) + 4*log(1-2x^d) -
  log(1-x^d)) / 12 + [d mod 60 in {6,18,42,54}] * (11*log(1-5x^d) -
  15*log(1-4x^d) + 5*log(1-x^d)) / 20 + [d mod 60 in {5,25,35,55}] *
  (5*log(1-5x^d) - log(1-4x^d) + 2*log(1-3x^d) - 2*log(1-2x^d) +
  log(1-x^d)) / 24 + [d mod 60 in {4,8,16,28,32,44,52,56}] *
  (7*log(1-5x^d) - 10*log(1-4x^d) + 5*log(1-2x^d) - 5*log(1-x^d)) /
  15 + [d mod 60 in {3,9,21,27,33,39,51,57}] * (7*log(1-5x^d) -
  15*log(1-4x^d) + 10*log(1-3x^d) - 10*log(1-2x^d) + 15*log(1-x^d)) /
  40 + [d mod 60 in {2,14,22,26,34,38,46,58}] * (13*log(1-5x^d) -
  25*log(1-4x^d) + 20*log(1-2x^d) - 5*log(1-x^d)) / 60 + [d mod 60 in
  {1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59}] * (log(1-5x^d) -
  5*log(1-4x^d) + 10*log(1-3x^d) - 10*log(1-2x^d) + 5*log(1-x^d)) / 120).
(End)

A305751 Number of achiral color patterns (set partitions) in a row or cycle of length n with 5 or fewer colors (subsets).

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 30, 55, 141, 266, 688, 1313, 3407, 6532, 16970, 32595, 84721, 162846, 423348, 813973, 2116227, 4069352, 10580110, 20345735, 52898501, 101726626, 264488408, 508629033, 1322433847, 2543136972, 6612152850, 12715668475

Offset: 0

Robert A. Russell, Jun 09 2018

An equivalent color pattern is obtained when we permute the colors. Thus all permutations of ABCDE are equivalent, as are AABCDE and BBCDEA. A color pattern is achiral if it is equivalent to its reversal. Rotations of the colors of a cycle are equivalent, so for cycles AABBCDE = BBCDEAA = CDEAABB.

			For a(5) = 12, the achiral patterns for both rows and cycles are AAAAA, AABAA, ABABA, ABBBA, AABCC, ABACA, ABBBC, ABCAB, ABCBA, ABCBD, ABCDA, and ABCDE.

Index entries for linear recurrences with constant coefficients, signature (1,7,-7,-10,10).

Fifth column of A305749.
Cf. A056272 (oriented), A056324 (unoriented), A320935 (chiral), for rows.
Cf. A056293 (oriented), A056355 (unoriented), A320745 (chiral), for cycles.

seq(coeff(series((1-2*x)*(1+2*x-2*x^2-3*x^3+x^4)/((1-x)*(1-2*x^2)*(1-5*x^2)),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Oct 30 2018

Table[If[EvenQ[n], StirlingS2[(n+10)/2, 5] - 13 StirlingS2[(n+8)/2, 5] + 62 StirlingS2[(n+6)/2, 5] - 130 StirlingS2[(n+4)/2, 5] + 110 StirlingS2[(n+2)/2, 5] - 24 StirlingS2[n/2, 5], StirlingS2[(n+9)/2, 5] - 12 StirlingS2[(n+7)/2, 5] + 52 StirlingS2[(n+5)/2, 5] - 95 StirlingS2[(n+3)/2, 5] + 60 StirlingS2[(n+1)/2, 5]], {n, 0, 40}]
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[Ach[n, j], {j, 0, k}], {n, 0, 40}]
CoefficientList[Series[(1-2x)(1+2x-2x^2-3x^3+x^4) / ((1- x)(1-2x^2)(1-5x^2)), {x,0,40}], x]
Join[{1},LinearRecurrence[{1,7,-7,-10,10},{1,2,3,7,12},40]]
Join[{1}, Table[If[EvenQ[n], (15 + 20 2^(n/2) + 13 5^(n/2)) / 60, (3 + 2 2^((n+1)/2) + 5^((n+1)/2)) / 12], {n,40}]]

a(n) = Sum_{j=0..5} Ach(n,j), where Ach(n,k) = [n>1] * (k*T(n-2,k) + T(n-2,k-1) + T(n-2,k-2)) + [0<=n<2 & n==k].
G.f.: (1 - 2x)*(1+2x-2x^2-3x^3+x^4) / ((1-x)*(1-2x^2)*(1-5x^2)).
a(2m) = S2(m+5,5) - 13*S2(m+4,5) + 62*S2(m+3,5) - 130*S2(m+2,5) + 110*S2(m+1,5) - 24*S2(m,5);
a(2m-1) = S2(m+4,5) - 12*S2(m+3,5) + 52*S2(m+2,5) - 95*S2(m+1,5) + 60*S2(m,5), where S2(n,k) is the Stirling subset number A008277.
For n>0, a(2m) = (15 + 20*2^m + 13*5^m) / 60.
a(2m-1) = (3 + 2*2^m + 5^m) / 12.
a(n) = 2*A056324(n) - A056272(n) = A056272(n) - 2*A320935(n) = A056324(n) - A320935(n).
a(n) = 2*A056355(n) - A056293(n) = A056293(n) - 2*A320745(n) = A056355(n) - A320745(n).
a(n) = A057427(n) + A052551(n-2) + A304973(n) + A304974(n) + A304975(n).
a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - 10*a(n-4) + 10*a(n-5). - Muniru A Asiru, Oct 30 2018

A056299 Number of n-bead necklace structures using exactly six different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 36, 296, 2303, 16317, 110462, 717024, 4532105, 28046285, 170938814, 1029749994, 6149327905, 36477979041, 215304158916, 1265984738264, 7422971231829, 43433472086235, 253759842223290

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.

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]

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column 6 of A152175.

Cf. A056286, A056293, A056294.

From Robert A. Russell, May 29 2018: (Start)
Adn[d_, n_] := Adn[d, n] = If[1==n, DivisorSum[d, x^# &],
  Expand[Adn[d, 1] Adn[d, n-1] + D[Adn[d, n-1], x] x]];
Table[Coefficient[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]/n , x, 6],
  {n, 1, 40}] (* after Gilbert and Riordan *)
Table[(1/n) DivisorSum[n, EulerPhi[#] Which[Divisible[#, 60], StirlingS2[n/#+5, 6] - 10 StirlingS2[n/#+4, 6] + 35 StirlingS2[n/#+3, 6] - 50 StirlingS2[n/#+2, 6] + 24 StirlingS2[n/#+1, 6], Divisible[#, 30], StirlingS2[n/#+5, 6] - 12 StirlingS2[n/#+4, 6] + 56 StirlingS2[n/#+3, 6] - 123 StirlingS2[n/#+2, 6] + 108 StirlingS2[n/#+1, 6], Divisible[#, 20], 4 StirlingS2[n/#+4, 6] - 44 StirlingS2[n/#+3, 6] + 176 StirlingS2[n/#+2, 6] - 296 StirlingS2[n/#+1, 6] + 160 StirlingS2[n/#, 6], Divisible[#, 15], 3 StirlingS2[n/#+4, 6] - 36 StirlingS2[n/#+3, 6] + 159 StirlingS2[n/#+2, 6] - 306 StirlingS2[n/#+1, 6] + 225 StirlingS2[n/#, 6], Divisible[#, 12], StirlingS2[n/#+5, 6] - 12 StirlingS2[n/#+4, 6] + 59 StirlingS2[n/#+3, 6] - 156 StirlingS2[n/#+2, 6] + 228 StirlingS2[n/#+1, 6] - 144 StirlingS2[n/#, 6], Divisible[#, 10], 2 StirlingS2[n/#+4, 6] - 23 StirlingS2[n/#+3, 6] + 103 StirlingS2[n/#+2, 6] - 212 StirlingS2[n/#+1, 6] + 160 StirlingS2[n/#, 6], Divisible[#, 6], StirlingS2[n/#+5, 6] - 14 StirlingS2[n/#+4, 6] + 80 StirlingS2[n/#+3, 6] - 229 StirlingS2[n/#+2, 6] + 312 StirlingS2[n/#+1, 6] - 144 StirlingS2[n/#, 6], Divisible[#, 5], 2 StirlingS2[n/#+4, 6] - 24 StirlingS2[n/#+3, 6] + 106 StirlingS2[n/#+2, 6] - 204 StirlingS2[n/#+1, 6] + 145 StirlingS2[n/#, 6], Divisible[#, 4], 2 StirlingS2[n/#+4, 6] - 20 StirlingS2[n/#+3, 6] + 70 StirlingS2[n/#+2, 6] - 92 StirlingS2[n/#+1, 6] + 16 StirlingS2[n/#, 6], Divisible[#, 3], StirlingS2[n/#+4, 6] - 12 StirlingS2[n/#+3, 6] + 53 StirlingS2[n/#+2, 6] - 102 StirlingS2[n/#+1, 6] + 81 StirlingS2[n/#, 6], Divisible[#, 2], StirlingS2[n/#+3, 6] - 3 StirlingS2[n/#+2, 6] - 8 StirlingS2[n/#+1, 6] + 16 StirlingS2[n/#, 6], True, StirlingS2[n/#, 6]] &], {n, 1, 40}]
mx = 40; Drop[CoefficientList[Series[-Sum[(EulerPhi[d] / d) Which[
  Divisible[d, 60], Log[1 - 6x^d] - Log[1 - 5x^d], Divisible[d, 30],
  (3 Log[1 - 6x^d] - 3 Log[1 - 5x^d] + Log[1 - 2x^d] - Log[1 - x^d]) / 4,
  Divisible[d, 20], (5 Log[1 - 6x^d] - 6 Log[1 - 5x^d] + 2 Log[1 - 3x^d] -
  3 Log[1 - 2x^d]) / 9, Divisible[d, 15], (5 Log[1 - 6x^d] -
  6 Log[1 - 5x^d] + 3 Log[1 - 4x^d] - 4 Log[1 - 3x^d] + 3 Log[1 - 2x^d] -
  6 Log[1 - x^d]) / 16, Divisible[d, 12], (4 Log[1 - 6x^d] -
  4 Log[1 - 5x^d] + Log[1 - x^d]) / 5, Divisible[d, 10], (11 Log[1 - 6x^d] -
  15 Log[1 - 5x^d] + 8 Log[1 - 3x^d] - 3 Log[1 - 2x^d] - 9 Log[1 - x^d]) /
  36, Divisible[d, 6], (11 Log[1 - 6x^d] - 11 Log[1 - 5x^d] +
  5 Log[1 - 2x^d] - Log[1 - x^d]) / 20, Divisible[d, 5], (29 Log[1 - 6x^d] -
  30 Log[1 - 5x^d] + 3 Log[1 - 4x^d] - 4 Log[1 - 3x^d] + 3 Log[1 - 2x^d] -
  30 Log[1 - x^d]) / 144, Divisible[d, 4], (16 Log[1 - 6x^d] -
  21 Log[1 - 5x^d] + 10 Log[1 - 3x^d] - 15 Log[1 - 2x^d] + 9 Log[1 - x^d]) /
  45, Divisible[d, 3], (9 Log[1 - 6x^d] - 14 Log[1 - 5x^d] +
  15 Log[1 - 4x^d] - 20 Log[1 - 3x^d] + 15 Log[1 - 2x^d] -
  14 Log[1 - x^d]) / 80, Divisible[d, 2], (19 Log[1 - 6x^d] -
  39 Log[1 - 5x^d] + 40 Log[1 - 3x^d] - 15 Log[1 - 2x^d] - 9 Log[1 - x^d]) /
  180, True, (Log[1 - 6x^d] - 6 Log[1 - 5x^d] + 15 Log[1 - 4x^d] -
  20 Log[1 - 3x^d] + 15 Log[1 - 2x^d] - 6 Log[1 - x^d]) / 720],
  {d, 1, mx}], {x, 0, mx}], x], 1]
(End)

a(n) = A056294(n) - A056293(n).
From Robert A. Russell, May 29 2018: (Start)
a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 60] * (S2(n/d+5,6) -
  10*S2(n/d+4,6) + 35*S2(n/d+3,6) - 50*S2(n/d+2,6) + 24*S2(n/d+1,6)) +
  [d==30 mod 60] * (S2(n/d+5,6) - 12*S2(n/d+4,6) + 56*S2(n/d+3,6) -
  123*S2(n/d+2,6) + 108*S2(n/d+1,6)) + [d==20 mod 60 | d==40 mod 60] *
  (4*S2(n/d+4,6) - 44*S2(n/d+3,6) + 176*S2(n/d+2,6) - 296*S2(n/d+1,6) +
  160*S2(n/d,6)) + [d==15 mod 60 | d==45 mod 60] * (3*S2(n/d+4,6) -
  36*S2(n/d+3,6) + 159*S2(n/d+2,6) - 306*S2(n/d+1,6) + 225*S2(n/d,6)) +
  [d mod 60 in {12,24,36,48}] * (S2(n/d+5,6) - 12*S2(n/d+4,6) +
  59*S2(n/d+3,6) - 156*S2(n/d+2,6) + 228*S2(n/d+1,6) - 144*S2(n/d,6)) +
  [d=10 mod 60 | d==50 mod 60] * (2*S2(n/d+4,6) - 23*S2(n/d+3,6) +
  103*S2(n/d+2,6) - 212*S2(n/d+1,6) + 160*S2(n/d,6)) + [d mod 60 in
  {6,18,42,54}] * (S2(n/d+5,6) - 14*S2(n/d+4,6) + 80*S2(n/d+3,6) -
  229*S2(n/d+2,6) + 312*S2(n/d+1,6) - 144*S2(n/d,6)) + [d mod 60 in
  {5,25,35,55}] * (2*S2(n/d+4,6) - 24*S2(n/d+3,6) + 106*S2(n/d+2,6) -
  204*S2(n/d+1,6) + 145*S2(n/d,6)) + [d mod 60 in {4,8,16,28,32,44,52,56}] *
  (2*S2(n/d+4,6) - 20*S2(n/d+3,6) + 70*S2(n/d+2,6) - 92*S2(n/d+1,6) +
  16*S2(n/d,6)) + [d mod 60 in {3,9,21,27,33,39,51,57}] * (S2(n/d+4,6) -
  12*S2(n/d+3,6) + 53*S2(n/d+2,6) - 102*S2(n/d+1,6) + 81*S2(n/d,6)) +
  [d mod 60 in {2,14,22,26,34,38,46,58}] * (S2(n/d+3,6) - 3*S2(n/d+2,6) -
  8*S2(n/d+1,6) + 16*S2(n/d,6)) + [d mod 60 in {1,7,11,13,17,19,23,29,31,37,
  41,43,47,49,53,59}] * S2(n/d,6)), where S2(n,k) is the Stirling subset
  number, A008277.
G.f.: -Sum_{d>0} (phi(d) / d) * ([d==0 mod 60] * (log(1-6x^d) -
  log(1-5x^d)) + [d==30 mod 60] * (3*log(1-6x^d) - 3*log(1-5x^d) +
  log(1-2x^d) - log(1-x^d)) / 4 + [d==20 mod 60 | d==40 mod 60] *
  (5*log(1-6x^d) - 6*log(1-5x^d) + 2*log(1-3x^d) - 3*log(1-2x^d)) / 9 +
  [d==15 mod 60 | d==45 mod 60] * (5*log(1-6x^d) - 6*log(1-5x^d) +
  3*log(1-4x^d) - 4*log(1-3x^d) + 3*log(1-2x^d) - 6*log(1-x^d)) / 16 +
  [d mod 60 in {12,24,36,48}] * (4*log(1-6x^d) - 4*log(1-5x^d) +
  log(1-x^d)) / 5 + [d=10 mod 60 | d==50 mod 60] * (11*log(1-6x^d) -
  15*log(1-5x^d) + 8*log(1-3x^d) - 3*log(1-2x^d) - 9*log(1-x^d)) / 36 +
  [d mod 60 in {6,18,42,54}] * (11*log(1-6x^d) - 11*log(1-5x^d) +
  5*log(1-2x^d) - log(1-x^d)) / 20 + [d mod 60 in {5,25,35,55}] *
  (29*log(1-6x^d) - 30*log(1-5x^d) + 3*log(1-4x^d) - 4*log(1-3x^d) +
  3*log(1-2x^d) - 30*log(1-x^d)) / 144 + [d mod 60 in {4,8,16,28,32,44,52,
  56}] * (16*log(1-6x^d) - 21*log(1-5x^d) + 10*log(1-3x^d) -
  15*log(1-2x^d) + 9*log(1-x^d)) / 45 + [d mod 60 in {3,9,21,27,33,39,51, 57}] * (9*log(1-6x^d) - 14*log(1-5x^d) + 15*log(1-4x^d) - 20*log(1-3x^d) +
  15*log(1-2x^d) - 14*log(1-x^d)) / 80 + [d mod 60 in {2,14,22,26,34,38,46,
  58}] * (19*log(1-6x^d) - 39*log(1-5x^d) + 40*log(1-3x^d) -
  15*log(1-2x^d) - 9*log(1-x^d)) / 180 + [d mod 60 in {1,7,11,13,17,19,23, 29,31,37,41,43,47,49,53,59}] * (log(1-6x^d) - 6 log(1-5x^d) +
  15 log(1-4x^d) - 20 log(1-3x^d) + 15 log(1-2x^d) - 6 log(1-x^d)) / 720).
(End)

A056301 Number of primitive (period n) n-bead necklace structures using a maximum of five different colored beads.

Original entry on oeis.org

1, 1, 2, 5, 11, 38, 122, 496, 2005, 8707, 38364, 173562, 792827, 3662800, 17034367, 79702578, 374624253, 1767881397, 8370666416, 39751064122, 189262621739, 903220020390, 4319518316898, 20697040024784

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.

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]

Cf. A001692, A008683.

a(n) = Sum_{d|n} mu(d) * A056293(n/d); mu = A008683.

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).

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

A152175 Triangle read by rows: T(n,k) is the number of k-block partitions of an n-set up to rotations.

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A056355 Number of bracelet structures using a maximum of five different colored beads.

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A056298 Number of n-bead necklace structures using exactly five different colored beads.

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A305751 Number of achiral color patterns (set partitions) in a row or cycle of length n with 5 or fewer colors (subsets).

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A056299 Number of n-bead necklace structures using exactly six different colored beads.

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A056301 Number of primitive (period n) n-bead necklace structures using a maximum of five different colored beads.

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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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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).

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