A304975 -id:A304975 - cp's OEIS frontend

A304972 Triangle read by rows of achiral color patterns (set partitions) for a row or loop of length n. T(n,k) is the number using exactly k colors (sets).

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 5, 2, 1, 1, 7, 10, 9, 3, 1, 1, 7, 19, 16, 12, 3, 1, 1, 15, 38, 53, 34, 18, 4, 1, 1, 15, 65, 90, 95, 46, 22, 4, 1, 1, 31, 130, 265, 261, 195, 80, 30, 5, 1, 1, 31, 211, 440, 630, 461, 295, 100, 35, 5, 1, 1, 63, 422, 1221, 1700, 1696, 1016, 515, 155, 45, 6, 1, 1, 63, 665, 2002

Offset: 1

Robert A. Russell, May 22 2018

Two color patterns are equivalent if we permute the colors. Achiral color patterns must be equivalent if we reverse the order of the pattern.

			Triangle begins:
1;
1,   1;
1,   1,    1;
1,   3,    2,    1;
1,   3,    5,    2,     1;
1,   7,   10,    9,     3,     1;
1,   7,   19,   16,    12,     3,     1;
1,  15,   38,   53,    34,    18,     4,    1;
1,  15,   65,   90,    95,    46,    22,    4,    1;
1,  31,  130,  265,   261,   195,    80,   30,    5,    1;
1,  31,  211,  440,   630,   461,   295,  100,   35,    5,   1;
1,  63,  422, 1221,  1700,  1696,  1016,  515,  155,   45,   6,  1
1,  63,  665, 2002,  3801,  3836,  3156, 1556,  710,  185,  51,  6, 1;
1, 127, 1330, 5369, 10143, 13097, 10508, 6832, 2926, 1120, 266, 63, 7, 1;
For T(4,2)=3, the row patterns are AABB, ABAB, and ABBA.  The loop patterns are AAAB, AABB, and ABAB.
For T(5,3)=5, the color patterns for both rows and loops are AABCC, ABACA, ABBBC, ABCAB, and ABCBA.

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Ira Gessel, What is the number of achiral color patterns for a row of n colors containing k different colors?, mathoverflow, Jan 30 2018.
Juan B. Gil and Luiz E. Lopez, Enumeration of symmetric arc diagrams, arXiv:2203.10589 [math.CO], 2022.

Columns 1-6 are A057427, A052551(n-2), A304973, A304974, A304975, A304976.
A305008 has coefficients that determine the function and generating function for each column.
Row sums are A080107.

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[Ach[n, k], {n, 1, 15}, {k, 1, n}] // Flatten
Ach[n_, k_] := Ach[n, k] = Which[0==k, Boole[0==n], 1==k, Boole[n>0],
  OddQ[n], Sum[Binomial[(n-1)/2, i] Ach[n-1-2i, k-1], {i, 0, (n-1)/2}],
  True, Sum[Binomial[n/2-1, i] (Ach[n-2-2i, k-1]
  + 2^i Ach[n-2-2i, k-2]), {i, 0, n/2-1}]]
Table[Ach[n, k], {n, 1, 15}, {k, 1, n}] // Flatten

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}
{ my(A=Ach(10)); for(n=1, #A, print(A[n,1..n])) } \\ Andrew Howroyd, Sep 18 2019

T(n,k) = [n>1] * (k*T(n-2,k) + T(n-2,k-1) + T(n-2,k-2)) + [n<2 & n==k & n>=0].
T(2m-1,k) = A140735(m,k).
T(2m,k) = A293181(m,k).
T(n,k) = [k==0 & n==0] + [k==1 & n>0]
  + [k>1 & n==1 mod 2] * Sum_{i=0..(n-1)/2} (C((n-1)/2, i) * T(n-1-2i, k-1))
  + [k>1 & n==0 mod 2] * Sum_{i=0..(n-2)/2} (C((n-2)/2, i) * (T(n-2-2i, k-1)
  + 2^i * T(n-2-2i, k-2))) where C(n,k) is a binomial coefficient.

A305008 Triangle read by rows of coefficients for functions and generating functions for the number of achiral color patterns (set partitions) for a row or loop of varying length using exactly n colors (sets).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 2, -1, -2, 1, 2, -1, -4, -2, 1, 3, -3, -11, 0, 6, 1, 3, -3, -17, -8, 20, 16, 1, 4, -6, -32, 1, 64, 20, -20, 1, 4, -6, -44, -19, 140, 136, -120, -132, 1, 5, -10, -70, 5, 301, 152, -396, -280, 28, 1, 5, -10, -90, -35, 541, 608, -1228, -1752, 800, 1216, 1, 6, -15, -130, 15, 966, 643, -2798

Offset: 0

Robert A. Russell, May 23 2018

Triangle begins with T(0,0).
Two color patterns are equivalent if we permute the colors. Achiral color patterns must be equivalent if we reverse the order of the pattern.
The generating function for exactly n colors (column n of A304972) is
  x^n * Sum_{k=0..n} (T(n, k) * x^k) / Product_{k=1..n} (1 - k*x^2).
Both the numerator and denominator of this g.f. have factors of (1+x) and (1-(n-2)*x^2) when n > 2.
Letting S2(m,n) be the Stirling subset number A008277(m,n), the function for exactly n colors for a row or loop of length m, A304972(m,n), n even, is
  [m==0 mod 2] * Sum_{k=0..n/2} T(n, 2k) * S2((m+n)/2-k, n) +
  [m==1 mod 2] * Sum_{k=1..n/2} T(n, 2k-1) * S2((m+n+1)/2-k, n).
When n is odd, the function for A304972(m,n) is
  [m==0 mod 2] * Sum_{k=0..(n-1)/2} T(n, 2k+1) * S2((m+n-1)-k, n) +
  [m==1 mod 2] * Sum_{k=0..(n-1)/2} T(n, 2k) * S2((m+n)/2-k, n).

			Triangle begins:
1;
1, 1;
1, 1,   0;
1, 2,  -1,   -2;
1, 2,  -1,   -4,  -2;
1, 3,  -3,  -11,   0,   6;
1, 3,  -3,  -17,  -8,  20,  16;
1, 4,  -6,  -32,   1,  64,  20,   -20;
1, 4,  -6,  -44, -19, 140, 136,  -120,  -132;
1, 5, -10,  -70,   5, 301, 152,  -396,  -280,   28;
1, 5, -10,  -90, -35, 541, 608, -1228, -1752,  800, 1216;
1, 6, -15, -130,  15, 966, 643, -2798, -3028, 2236, 3600, 936;

Coefficients for functions and generating functions of A304973, A304974, A304975, A304976, which are columns 3-6 of A304972.

Coef[n_, -1] := Coef[n, -1] = 0; Coef[n_, 0] := Coef[n, 0] = Boole[n>=0];
Coef[n_, k_] := Coef[n, k] = If[k > n, 0, Coef[n-1, k-1] + Coef[n-2, k] - (n-1) Coef[n-2, k-2]]
Table[Coef[n, k], {n, 0, 30}, {k, 0, n}] // Flatten

T(n,k) = [1 <= k <= n] * (T(n-1, k-1) + T(n-2, k) - (n-1) * T(n-2, k-2)) + [k==0 & n>=0].

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

A305752 Number of achiral color patterns (set partitions) in a row or cycle of length n with 6 or fewer colors (subsets).

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 31, 58, 159, 312, 883, 1774, 5103, 10368, 30067, 61414, 178815, 366168, 1068259, 2190190, 6395919, 13120944, 38335123, 78665590, 229890591, 471814344, 1378985155, 2830350526, 8272839855, 16980500640, 49633834099, 101878204486

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 AABCDEF and BBCDEFA. A color pattern is achiral if it is equivalent to its reversal. Rotations of the colors of a cycle are equivalent, so for cycles AABCCDEF = BCCDEFAA = CCDEFAAB.

			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 (1,11,-11,-36,36,36,-36).

Sixth column of A305749.
Cf. A056273 (oriented), A056325 (unoriented), A320936 (chiral), for rows.
Cf. A056294 (oriented), A056356 (unoriented), A320746 (chiral), for cycles.

seq(coeff(series((1-10*x^2+x^3+29*x^4-6*x^5-25*x^6+8*x^7)/((1-x)*(1-2*x^2)*(1-3*x^2)*(1-6*x^2)),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 30 2018

Table[If[EvenQ[n], StirlingS2[(n+12)/2, 6] - 19 StirlingS2[(n+10)/2, 6] + 140 StirlingS2[(n+8)/2, 6] - 501 StirlingS2[(n+6)/2, 6] + 887 StirlingS2[(n+4)/2, 6] - 692 StirlingS2[(n+2)/2, 6] + 160 StirlingS2[n/2, 6], StirlingS2[(n+11)/2, 6] - 18 StirlingS2[(n+9)/2, 6] + 124 StirlingS2[(n+7)/2, 6] - 404 StirlingS2[(n+5)/2, 6] + 613 StirlingS2[(n+3)/2, 6] - 340 StirlingS2[(n+1)/2, 6]], {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=6; Table[Sum[Ach[n, j], {j, 0, k}], {n, 0, 40}]
CoefficientList[Series[(1-10x^2+x^3+29x^4-6x^5-25x^6+8x^7) / ((1-x)(1-2x^2)(1-3x^2)(1-6 x^2)), {x, 0, 40}], x]
LinearRecurrence[{1,11,-11,-36,36,36,-36},{1,1,2,3,7,12,31,58},40]
Join[{1}, Table[If[EvenQ[n], (36 + 45 2^(n/2) + 40 3^(n/2) + 19 6^(n/2)) / 180, (72 + 45 2^((n+1)/2) + 40 3^((n+1)/2) + 13 6^((n+1)/2)) / 360], {n,40}]]

a(n) = Sum_{j=0..6} 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.: (1-10x^2+x^3+29x^4-6x^5-25x^6+8x^7) / ((1-x)*(1-2x^2)*(1-3x^2)*(1-6x^2)).
a(2m) = S2(m+6,6) - 19*S2(m+5,6) + 140*S2(m+4,6) - 501*S2(m+3,6) + 887*S2(m+2,6) - 692*S2(m+1,6) + 160*S2(m,6);
a(2m-1) = S2(m+5,6) - 18*S2(m+4,6) + 124*S2(m+3,6) - 404*S2(m+2,6) + 613*S2(m+1,6) - 340*S2(m,6), where S2(n,k) is the Stirling subset number A008277.
For n>0, a(2m) = (36 + 45*2^m + 40*3^m + 19*6^m) / 180.
a(2m-1) = (72 + 45*2^m + 40*3^m + 13*6^m) / 360.
a(n) = 2*A056325(n) - A056273(n) = A056273(n) - 2*A320936(n) = A056325(n) - A320936(n).
a(n) = 2*A056356(n) - A056294(n) = A056294(n) - 2*A320746(n) = A056356(n) - A320936(n).
a(n) = A057427(n) + A052551(n-2) + A304973(n) + A304974(n) + A304975(n) + A304976(n).
a(n) = a(n-1) + 11*a(n-2) - 11*a(n-3) - 36*a(n-4) + 36*a(n-5) + 36*a(n-6) - 36*a(n-7). - Muniru A Asiru, Oct 30 2018

A320528 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 5 colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132, 123050, 688850, 3752350, 20032446, 105372624, 548066568, 2826316248, 14478890712, 73794322750, 374602205590, 1895629599050, 9568906539786, 48208435317284, 242500368793628, 1218342441784468, 6115097961883092, 30669103347259650, 153720181809997530, 770100204404335350, 3856500105221902326

Offset: 1

Robert A. Russell, Oct 14 2018

Two color patterns are equivalent if we permute the colors. Chiral color patterns must not be equivalent if we reverse the order of the pattern.

			For a(6)=6, the chiral pairs are AABCDE-ABCDEE, ABACDE-ABCDED, ABCADE-ABCDEC, ABCDAE-ABCDEB, ABBCDE-ABCDDE, and ABCBDE-ABCDCE.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-48,-36,551,-683,-1542,3546,80,-4280,2400).

Col. 5 of A320525.

Cf. A000481 (oriented), A056329 (unoriented), A304975 (achiral).

I:=[0,0,0,0,0,6,64,508,3428,21132]; [n le 10 select I[n] else 13*Self(n-1)-48*Self(n-2)-36*Self(n-3)+551*Self(n-4)-683*Self(n-5) -1542*Self(n-6)+3546*Self(n-7)+80*Self(n-8)-4280*Self(n-9) +2400*Self(n-10): n in [1..30]]; // G. C. Greubel, Oct 20 2018

k=5; Table[(StirlingS2[n,k] - If[EvenQ[n], 3StirlingS2[n/2+2,5] - 11StirlingS2[n/2+1,5] + 6StirlingS2[n/2,5], StirlingS2[(n+5)/2,5] - 3StirlingS2[(n+3)/2,5]])/2, {n,30}]
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[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{13, -48, -36, 551, -683, -1542, 3546, 80, -4280, 2400}, {0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132}, 30]

m=30; v=concat([0,0,0,0,0,6,64,508,3428,21132], vector(m-10)); for(n=11, m, v[n] = 13*v[n-1]-48*v[n-2]-36*v[n-3]+551*v[n-4]-683*v[n-5] -1542*v[n-6] +3546*v[n-7] +80*v[n-8] -4280*v[n-9] +2400*v[n-10]); v \\ G. C. Greubel, Oct 20 2018

a(n) = (S2(n,k) - A(n,k))/2, where k=5 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].
G.f.: (x^5 / Product_{k=1..5} (1 - k*x) - x^5 (1 + x) (1 - 3 x^2) (1 + 2 x - 2 x^2) / Product_{k=1..5} (1 - k*x^2)) / 2.
a(n) = (A000481(n) - A304975(n)) / 2 = A000481(n) - A056329(n) = A056329(n) - A304975(n).
a(n) = 13*a(n-1) - 48*a(n-2) - 36*a(n-3) + 551*a(n-4) - 683*a(n-5) - 1542*a(n-6) + 3546*a(n-7) + 80*a(n-8) - 4280*a(n-9) + 2400*a(n-10) for n>10. - Colin Barker, May 12 2020

A056329 Number of reversible string structures with n beads using exactly five different colors.

Original entry on oeis.org

0, 0, 0, 0, 1, 9, 76, 542, 3523, 21393, 123680, 690550, 3756151, 20042589, 105394296, 548123982, 2826435403, 14479204833, 73794961960, 374603884910, 1895632969351, 9568915372269, 48208452866816

Offset: 1

Marks R. Nester

nonn

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

Number of set partitions for an unoriented row of n elements using exactly five different elements. An unoriented row is equivalent to its reverse. - Robert A. Russell, Oct 14 2018

			For a(6)=9, the color patterns are ABCDEA, ABCDBA, ABCCDE, AABCDE, ABACDE, ABCADE, ABCDAE, ABBCDE, and ABCBDE. The first three are achiral. - _Robert A. Russell_, Oct 14 2018

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]

Index entries for linear recurrences with constant coefficients, signature (13, -48, -36, 551, -683, -1542, 3546, 80, -4280, 2400).

Column 5 of A284949.
Cf. A056312.
Cf. A000481 (oriented), A320528 (chiral), A304975 (achiral).

k=5; Table[(StirlingS2[n,k] + If[EvenQ[n], 3StirlingS2[n/2+2,5] - 11StirlingS2[n/2+1,5] + 6StirlingS2[n/2,5], StirlingS2[(n+5)/2,5] - 3StirlingS2[(n+3)/2,5]])/2, {n,30}] (* Robert A. Russell, Oct 14 2018 *)
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]]
k=5; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 1, 30}] (* Robert A. Russell, Oct 14 2018 *)
LinearRecurrence[{13, -48, -36, 551, -683, -1542, 3546, 80, -4280, 2400}, {0, 0, 0, 0, 1, 9, 76, 542, 3523, 21393}, 30] (* Robert A. Russell, Oct 14 2018 *)

a(n) = A056324(n) - A056323(n).
Empirical g.f.: -x^5*(70*x^5 - 102*x^4 + 22*x^3 + 7*x^2 - 4*x + 1) / ((x-1)*(2*x-1)*(2*x+1)*(3*x-1)*(4*x-1)*(5*x-1)*(2*x^2-1)*(5*x^2-1)). - Colin Barker, Nov 25 2012
From Robert A. Russell, Oct 14 2018: (Start)
a(n) = (S2(n,k) + A(n,k))/2, where k=5 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].
a(n) = (A000481(n) + A304975(n)) / 2 = A000481(n) - A320528(n) = A320528(n) + A304975(n). (End)

A320645 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 5 colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 51, 339, 2010, 10900, 56700, 286888, 1426542, 7014746, 34229050, 166197824, 804243285, 3883608940, 18729354638, 90266471623, 434946282498, 2096010533584, 10104160993993, 48733654211358, 235195966291020, 1135892493220025, 5490005931157446, 26555178320890184, 128550000630553133, 622790399873432344, 3019641804537586657

Offset: 1

Robert A. Russell, Oct 18 2018

Two color patterns are the same if the colors are permuted. A chiral cycle is different from its reverse.
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 A056298 and A304975, which can be used in conjunction with the first formula.

			For a(7)=4, the chiral pairs are AABACDE-AABCDAE, AABCBDE-AABCDED, AABCDBE-AABCDEC, and ABACBDE-ABACDBE.

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 A320647.

Cf. A056298 (oriented), A056360 (unoriented), A304975 (achiral).

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 *)
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]]
k=5; Table[DivisorSum[n,EulerPhi[#]Adnk[#,n/#,k]&]/(2n) - Ach[n,k]/2,{n,40}]

a(n) = (A056298(n) - A304975(n)) / 2 = A056298(n) - A056360(n) = A056360(n) - A304975(n).

a(n) = -Ach(n,k)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,k), where k=5 is number of colors or sets, 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)).

cp's OEIS Frontend

A304972 Triangle read by rows of achiral color patterns (set partitions) for a row or loop of length n. T(n,k) is the number using exactly k colors (sets).

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A305008 Triangle read by rows of coefficients for functions and generating functions for the number of achiral color patterns (set partitions) for a row or loop of varying length using exactly n colors (sets).

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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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A305752 Number of achiral color patterns (set partitions) in a row or cycle of length n with 6 or fewer colors (subsets).

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A320528 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 5 colors (subsets).

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A056329 Number of reversible string structures with n beads using exactly five different colors.

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A320645 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 5 colors (subsets).

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