A320751 -id:A320751 - cp's OEIS frontend

A320750 Array read by antidiagonals: T(n,k) is the number of color patterns (set partitions) in an unoriented row of length n using k or fewer colors (subsets).

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 6, 1, 1, 2, 4, 10, 10, 1, 1, 2, 4, 11, 25, 20, 1, 1, 2, 4, 11, 31, 70, 36, 1, 1, 2, 4, 11, 32, 107, 196, 72, 1, 1, 2, 4, 11, 32, 116, 379, 574, 136, 1, 1, 2, 4, 11, 32, 117, 455, 1451, 1681, 272, 1

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.
In an unoriented row, chiral pairs are counted as one.
T(n,k) = Pi_k(P_n) which is the number of non-equivalent partitions of the path 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. - Mohammad Hadi Shekarriz, Aug 21 2019
From Allan Bickle, Apr 05 2022: (Start)
The columns count unlabeled k-paths with n+k+2 vertices. (A k-path with order n at least k+2 is a k-tree with exactly two k-leaves (vertices of degree k). It can be constructed from a clique with k+1 vertices by iteratively adding a new degree k vertex adjacent to an existing clique containing an existing k-leaf.)
Recurrences for the columns appear in the papers by Bickle, Eckhoff, and Markenzon et al. (End)

			Array begins with T(1,1):
  1   1     1     1      1      1      1      1      1      1      1 ...
  1   2     2     2      2      2      2      2      2      2      2 ...
  1   3     4     4      4      4      4      4      4      4      4 ...
  1   6    10    11     11     11     11     11     11     11     11 ...
  1  10    25    31     32     32     32     32     32     32     32 ...
  1  20    70   107    116    117    117    117    117    117    117 ...
  1  36   196   379    455    467    468    468    468    468    468 ...
  1  72   574  1451   1993   2135   2151   2152   2152   2152   2152 ...
  1 136  1681  5611   9134  10480  10722  10742  10743  10743  10743 ...
  1 272  5002 22187  43580  55091  58071  58461  58486  58487  58487 ...
  1 528 14884 87979 211659 301633 333774 339764 340359 340389 340390 ...
For T(4,3)=10, the patterns are AAAA, AABB, ABAB, ABBA, ABBC, ABCA, AAAB, AABA, AABC, ABAC, the last four being chiral with partners ABBB, ABAA, ABCC, and ABCB.

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.]

B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.
Allan Bickle, How to Count k-Paths, J. Integer Sequences, 25 (2022) Article 22.5.6.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
J. Eckhoff, Extremal interval graphs, J. Graph Theory 17 1 (1993), 117-127.
L. Markenzon, O. Vernet, and P. R. da Costa Pereira, A clique-difference encoding scheme for labelled k-path graphs, Discrete Appl. Math. 156 (2008), 3216-3222.

Columns 1-7 are A000012, A005418, A001998(n-1), A056323, A056324, A056325, A345207.
As k increases, columns converge to A103293(n+1).
Cf. transpose of A278984 (oriented), A320751 (chiral), A305749 (achiral).
Partial column sums of A284949.

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[StirlingS2[n,j] + Ach[n,j], {j,k-n+1}]/2, {k,15}, {n,k}] // Flatten

T(n,k) = Sum_{j=1..k} (S2(n,j) + Ach(n,j))/2, where S2 is the Stirling subset number A008277 and 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)).
T(n,k) = (A278984(k,n) + A305749(n,k)) / 2 = A278984(k,n) - A320751(n,k) = A320751(n,k) + A305749(n,k).
T(n,k) = Sum_{j=1..k} A284949(n,j).

A320955 Square array read by ascending antidiagonals: A(n, k) (n >= 0, k >= 0) = Sum_{j=0..n-1} (!j/j!)*((n - j)^k/(n - j)!) if k > 0 and 1 if k = 0. Here !n denotes the subfactorial of n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 4, 1, 0, 1, 1, 2, 5, 8, 1, 0, 1, 1, 2, 5, 14, 16, 1, 0, 1, 1, 2, 5, 15, 41, 32, 1, 0, 1, 1, 2, 5, 15, 51, 122, 64, 1, 0, 1, 1, 2, 5, 15, 52, 187, 365, 128, 1, 0, 1, 1, 2, 5, 15, 52, 202, 715, 1094, 256, 1, 0

Offset: 0

Peter Luschny, Nov 05 2018

Arndt and Sloane (see the link and A278984) identify the sequence to give "the number of words of length n over an alphabet of size b that are in standard order" and provide the formula Sum_{j = 1..b} Stirling_2(n, j) assuming b >= 1 and j >= 1. Compared to the array as defined here this misses the first row and the first column of our array.
The method used here is the special case of a general method described in A320956 applied to the function exp. For applications to other functions see the cross references.
A(k,n) is the number of color patterns (set partitions) for an oriented row of length n using up to k colors (subsets). Two color patterns are equivalent if the colors are permuted. For A(3,4) = 14, the six achiral patterns are AAAA, AABB, ABAB, ABBA, ABBC, and ABCA; the eight chiral patterns are the four chiral pairs AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB. - Robert A. Russell, Nov 10 2018

			Array starts:
n\k   0  1  2  3   4   5    6    7     8      9  ...
----------------------------------------------------
[0]   1, 0, 0, 0,  0,  0,   0,   0,    0,     0, ...  A000007
[1]   1, 1, 1, 1,  1,  1,   1,   1,    1,     1, ...  A000012
[2]   1, 1, 2, 4,  8, 16,  32,  64,  128,   256, ...  A011782
[3]   1, 1, 2, 5, 14, 41, 122, 365, 1094,  3281, ...  A124302
[4]   1, 1, 2, 5, 15, 51, 187, 715, 2795, 11051, ...  A124303
[5]   1, 1, 2, 5, 15, 52, 202, 855, 3845, 18002, ...  A056272
[6]   1, 1, 2, 5, 15, 52, 203, 876, 4111, 20648, ...  A056273, ?A284727
[7]   1, 1, 2, 5, 15, 52, 203, 877, 4139, 21110, ...
[8]   1, 1, 2, 5, 15, 52, 203, 877, 4140, 21146, ...
[9]   1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...
----------------------------------------------------
Seen as a triangle given by the descending antidiagonals:
[0]             1
[1]            0, 1
[2]          0, 1, 1
[3]        0, 1, 1, 1
[4]       0, 1, 2, 1, 1
[5]     0, 1, 4, 2, 1, 1
[6]    0, 1, 8, 5, 2, 1, 1
[7]  0, 1, 16, 14, 5, 2, 1, 1

Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"

Cf. A008290, A000110, A137650, A211561, A000240, A000387, A000449, A103816/A053556.
Cf. A124302, A124303, A056272, A056273, A284727, A278984.
Antidiagonal sums (and row sums of the triangle): A320964.
Cf. this sequence (exp), A320962 (log(x+1)), A320956 (sec+tan), A320958 (arcsin), A320959 (arctanh).
Cf. A320750 (unoriented), A320751 (chiral), A305749 (achiral).

A := (n, k) -> if k = 0 then 1 else add(A008290(n, n-j)*(n-j)^k, j=0..n-1)/n! fi:
seq(lprint(seq(A(n, k), k=0..9)), n=0..9); # Prints the array row-wise.
seq(seq(A(n-k, k), k=0..n), n=0..11); # Gives the array as listed.

T[n_, 0] := 1; T[n_, k_] := Sum[(Subfactorial[j]/Factorial[j])((n - j)^k/(n - j)!), {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten
Table[Sum[StirlingS2[k, j], {j, 0, n-k}], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert A. Russell, Nov 10 2018 *)

A(n, k) = (1/n!)*Sum_{j=0..n-1} A008290(n, n-j)*(n-j)^k if k > 0.
If one drops the special case A(n, 0) = 1 from the definition then column 0 becomes Sum_{k=0..n} (-1)^k/k! = A103816(n)/A053556(n).
Row n is given for k >= 1 by a_n(k), where
a_0(k) = 0^k/0!.
a_1(k) = 1^k/1!.
a_2(k) = (2^k)/2!.
a_3(k) = (3^k + 3)/3!.
a_4(k) = (6*2^k + 4^k + 8)/4!.
a_5(k) = (20*2^k + 10*3^k + 5^k + 45)/5!.
a_6(k) = (135*2^k + 40*3^k + 15*4^k + 6^k + 264)/6!.
a_7(k) = (924*2^k + 315*3^k + 70*4^k + 21*5^k + 7^k + 1855)/7!.
a_8(k) = (7420*2^k + 2464*3^k + 630*4^k + 112*5^k + 28*6^k + 8^k + 14832)/8!.
Note that the coefficients of the generating functions a_n are the recontres numbers A000240, A000387, A000449, ...
Rewriting the formulas with exponential generating functions for the rows we have egf(n) = Sum_{k=0..n} !k*binomial(n,k)*exp(x*(n-k)) and A(n, k) = (k!/n!)*[x^k] egf(n). In this formulation no special rule for the case k = 0 is needed.
The rows converge to the Bell numbers. Convergence here means that for every fixed k the terms in column k differ from A000110(k) only for finitely many indices.
A(n, n) are the Bell numbers A000110(n) for n >= 0.
Let S(n, k) = Bell(n+k+1) - A(n, k+n+1) for n >= 0 and k >= 0, then the square array S(n, k) read by descending antidiagonals equals provable the triangle A137650 and equals empirical the transpose of the array A211561.

A107767 a(n) = (1 + 3^n - 23^(n/2))/4 if n is even, (1 + 3^n - 43^((n-1)/2))/4 if n odd.

Original entry on oeis.org

0, 1, 4, 16, 52, 169, 520, 1600, 4840, 14641, 44044, 132496, 397852, 1194649, 3585040, 10758400, 32278480, 96845281, 290545684, 871666576, 2615029252, 7845176329, 23535617560, 70607118400, 211821620920, 635465659921

Offset: 1

Emeric Deutsch, Jun 12 2005

a(n-1) is the number of chiral pairs of color patterns (set partitions) for a row of length n using up to 3 colors (subsets). For n=4, a(n-1)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB. - Robert A. Russell, Oct 28 2018

Balaban, A. T., Brunvoll, J., Cyvin, B. N., & Cyvin, S. J. (1988). Enumeration of branched catacondensed benzenoid hydrocarbons and their numbers of Kekulé structures. Tetrahedron, 44(1), 221-228. See Eq. 5.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 60).
Index entries for linear recurrences with constant coefficients, signature (4,0,-12,9).

Cf. A167993 (first differences).
Column 3 of A320751, offset by 1.
Cf. A124302 (oriented), A001998 (unoriented), A182522 (achiral), varying offsets.

a:=[];; for n in [1..30] do if n mod 2 <> 0 then Add(a,(1+3^n-4*3^((n-1)/2))/4); else Add(a,(1+3^n-2*3^(n/2))/4); fi; od; a; # Muniru A Asiru, Oct 30 2018

I:=[0, 1, 4, 16]; [n le 4 select I[n] else 4*Self(n-1)-12*Self(n-3)+9*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 26 2012

a:=proc(n) if n mod 2 = 0 then (1+3^n-2*3^(n/2))/4 else (1+3^n-4*3^((n-1)/2))/4 fi end: seq(a(n),n=1..32);

CoefficientList[Series[-x/((x-1)*(3*x-1)*(3*x^2-1)),{x,0,40}],x] (* or *) LinearRecurrence[{4,0,-12,9},{0,1,4,16},50] (* Vincenzo Librandi, Jun 26 2012 *)
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[StirlingS2[n,j]-Ach[n,j],{j,k}]/2,{n,2,40}] (* Robert A. Russell, Oct 28 2018 *)
CoefficientList[Series[(1/12 E^(-Sqrt[3] x) (-3 + 2 Sqrt[3] - (3 + 2 Sqrt[3]) E^(2 Sqrt[3] x) + 3 E^((3 + Sqrt[3]) x) + 3 E^(x + Sqrt[3] x)))/x, {x, 0, 20}], x]*Table[(k+1)!, {k, 0, 20}] (* Stefano Spezia, Oct 29 2018 *)

x='x+O('x^50); concat(0, Vec(x^2/((1-x)*(3*x-1)*(3*x^2-1)))) \\ Altug Alkan, Sep 23 2018

G.f.: -x^2 / ( (x-1)*(3*x-1)*(3*x^2-1) ). - R. J. Mathar, Dec 16 2010
a(n) = 4*a(n-1) - 12*a(n-3) + 9*a(n-4). - Vincenzo Librandi, Jun 26 2012
From Robert A. Russell, Oct 28 2018: (Start)
a(n-1) = Sum_{j=0..k} (S2(n,j) - Ach(n,j)) / 2, where k=3 is the maximum number of colors, S2 is the Stirling subset number A008277, and 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)).
a(n-1) = (A124302(n) - A182522(n))/2.
a(n-1) = A124302(n) - A001998(n-1).
a(n-1) = A001998(n-1) - A182522(n).
a(n-1) = A122746(n-2) + A320526(n). (End)
E.g.f.: (1/12)*exp(-sqrt(3)*x)*(-3 + 2*sqrt(3) - (3 + 2*sqrt(3))*exp(2*sqrt(3)*x) + 3*exp((3 + sqrt(3))*x) + 3*exp(x + sqrt(3)*x)). - Stefano Spezia, Oct 29 2018
From Bruno Berselli, Oct 31 2018: (Start)
a(n) = (1 + 3^n - 3^((n-1)/2)*(4 + (-2 + sqrt(3))*(1 + (-1)^n)))/4. Therefore:
a(2*k)   = (3^k - 1)^2/4;
a(2*k+1) = (3^k - 1)*(3^(k+1) - 1)/4. (End)

Entry revised by N. J. A. Sloane, Jul 29 2011

A320936 Number of chiral pairs of color patterns (set partitions) for a row of length n using 6 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 1, 4, 20, 86, 409, 1976, 10168, 54208, 299859, 1699012, 9808848, 57335124, 338073107, 2004955824, 11936998016, 71253827696, 426061036747, 2550545918300, 15280090686256, 91588065861292, 549159350303235, 3293482358956552, 19755007003402944

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.
A chiral row is not equivalent to its reverse.
There are nonrecursive formulas, generating functions, and computer programs for A056273 and A305752, which can be used in conjunction with the first formula.

			For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (16,-84,84,685,-2140,180,7200,-8244, -4176,11664,-5184).

Column 6 of A320751.

Cf. A056273 (oriented), A056325 (unoriented), A305752 (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 *)
k=6; Table[Sum[StirlingS2[n,j]-Ach[n,j],{j,k}]/2,{n,40}]
LinearRecurrence[{16, -84, 84, 685, -2140, 180, 7200, -8244, -4176, 11664, -5184}, {0, 0, 1, 4, 20, 86, 409, 1976, 10168, 54208, 299859}, 40]

concat([0,0], Vec(x^3*(1 - 12*x + 40*x^2 + 18*x^3 - 308*x^4 + 376*x^5 + 364*x^6 - 882*x^7 + 378*x^8) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*x)*(1 - 2*x^2)*(1 - 3*x^2)*(1 - 6*x^2)) + O(x^40))) \\ Colin Barker, Nov 22 2018

a(n) = (A056273(n) - A305752(n))/2.
a(n) = A056273(n) - A056325(n).
a(n) = A056325(n) - A305752(n).
a(n) = A122746(n-2) + A320526(n) + A320527(n) + A320528(n) + A320529(n).
a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=6 is the maximum number of colors, S2 is the Stirling subset number A008277, and 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)).
From Colin Barker, Nov 22 2018: (Start)
G.f.: x^3*(1 - 12*x + 40*x^2 + 18*x^3 - 308*x^4 + 376*x^5 + 364*x^6 - 882*x^7 + 378*x^8) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*x)*(1 - 2*x^2)*(1 - 3*x^2)*(1 - 6*x^2)).
a(n) = 16*a(n-1) - 84*a(n-2) + 84*a(n-3) + 685*a(n-4) - 2140*a(n-5) + 180*a(n-6) + 7200*a(n-7) - 8244*a(n-8) - 4176*a(n-9) + 11664*a(n-10) - 5184*a(n-11) for n>11.
(End)

A320935 Number of chiral pairs of color patterns (set partitions) for a row of length n using 5 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 1, 4, 20, 86, 400, 1852, 8868, 42892, 210346, 1038034, 5150110, 25623486, 127740880, 637539592, 3184224728, 15910524632, 79520923966, 397508610454, 1987255480650, 9935410066186, 49674450471460, 248364429410332, 1241798688445588, 6208922948527572, 31044403310614786

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.
A chiral row is not equivalent to its reverse.
There are nonrecursive formulas, generating functions, and computer programs for A056272 and A305751, which can be used in conjunction with the first formula.

			For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

Index entries for linear recurrences with constant coefficients, signature (11,-34,-16,247,-317,-200,610,-300).

Column 5 of A320751.

Cf. A056272 (oriented), A056324 (unoriented), A305751 (achiral).

LinearRecurrence[{11, -34, -16, 247, -317, -200, 610, -300}, {0, 0, 1, 4, 20, 86, 400, 1852}, 40] (* or *)
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[StirlingS2[n,j]-Ach[n,j],{j,k}]/2,{n,40}]

a(n) = (A056272(n) - A305751(n))/2.
a(n) = A056272(n) - A056324(n).
a(n) = A056324(n) - A305751(n).
a(n) = A122746(n-2) + A320526(n) + A320527(n) + A320528(n).
a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=5 is the maximum number of colors, S2 is the Stirling subset number A008277, and 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)).
G.f.: x^3*(1 - 7*x + 10*x^2 + 18*x^3 - 49*x^4 + 25*x^5)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 5*x)*(1 - 5*x^2)*(1 - 2*x^2)). - Bruno Berselli, Oct 31 2018

A320934 Number of chiral pairs of color patterns (set partitions) for a row of length n using 4 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 1, 4, 20, 80, 336, 1344, 5440, 21760, 87296, 349184, 1397760, 5591040, 22368256, 89473024, 357908480, 1431633920, 5726601216, 22906404864, 91625881600, 366503526400, 1466015154176, 5864060616704, 23456246661120, 93824986644480, 375299963355136, 1501199853420544, 6004799480791040

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.
A chiral row is not equivalent to its reverse.
There are nonrecursive formulas, generating functions, and computer programs for A124303 and A305750, which can be used in conjunction with the first formula.

			For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

Index entries for linear recurrences with constant coefficients, signature (4,4,-16).

Column 4 of A320751.

Cf. A124303 (oriented), A056323 (unoriented), A305750 (achiral).

Table[(4^n - 4^Floor[n/2+1])/48, {n, 40}] (* or *)
LinearRecurrence[{4, 4, -16}, {0, 0, 1}, 40] (* or *)
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[StirlingS2[n,j]-Ach[n,j],{j,k}]/2,{n,40}]
CoefficientList[Series[x^2/((-1 + 4 x) (-1 + 4 x^2)), {x, 0, 50}], x] (* Stefano Spezia, Oct 29 2018 *)

a(n) = (A124303(n) - A305750(n))/2.
a(n) = A124303(n) - A056323(n).
a(n) = A056323(n) - A305750(n).
a(n) = A122746(n-2) + A320526(n) + A320527(n).
a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=4 is the maximum number of colors, S2 is the Stirling subset number A008277, and 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)).
a(2*m) = (16^m - 4*4^m)/48.
a(2*m-1) = (16^m - 4*4^m)/192.
a(n) = (4^n - 4^floor(n/2+1))/48.
G.f.: x^2/((-1 + 4*x)*(-1 + 4*x^2)). - Stefano Spezia, Oct 29 2018
a(n) = 2^n*(2^n - (-1)^n - 3)/48. - Bruno Berselli, Oct 31 2018

A320937 Number of chiral pairs of color patterns (set partitions) for a row of length n.

Original entry on oeis.org

0, 0, 1, 4, 20, 86, 409, 1988, 10404, 57488, 338180, 2103378, 13814202, 95423766, 691415451, 5239857008, 41431883216, 341036489096, 2916365967707, 25862060748614, 237434856965694, 2253357681164288, 22076002386446896, 222979432604192844, 2319295160051570620

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.

A chiral row is not equivalent to its reverse.

			For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row sums of triangle A320525.
Limit as k increases of column k of array A320751.
Cf. A000110 (oriented), A103293 (unoriented), A080107 (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 *)
Table[Sum[StirlingS2[n,j]-Ach[n,j],{j,n}]/2,{n,40}]

\\ Ach is A304972 as square matrix.
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}
seq(n)={my(A=Ach(n)); vector(n, n, sum(k=1, n, stirling(n,k,2) - A[n,k])/2)} \\ Andrew Howroyd, Sep 18 2019

a(n) = (A000110(n) + A080107(n)) / 2 = A000110(n) - A103293(n+1) = A103293(n+1) - A080107(n).

a(n) = Sum_{j=1..n} (S2(n,j) - Ach(n,j)) / 2, where S2 is the Stirling subset number A008277, and 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)).

cp's OEIS Frontend

A320750 Array read by antidiagonals: T(n,k) is the number of color patterns (set partitions) in an unoriented row of length n using k or fewer colors (subsets).

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A320955 Square array read by ascending antidiagonals: A(n, k) (n >= 0, k >= 0) = Sum_{j=0..n-1} (!j/j!)*((n - j)^k/(n - j)!) if k > 0 and 1 if k = 0. Here !n denotes the subfactorial of n.

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A107767 a(n) = (1 + 3^n - 2*3^(n/2))/4 if n is even, (1 + 3^n - 4*3^((n-1)/2))/4 if n odd.

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A320936 Number of chiral pairs of color patterns (set partitions) for a row of length n using 6 or fewer colors (subsets).

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A320935 Number of chiral pairs of color patterns (set partitions) for a row of length n using 5 or fewer colors (subsets).

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A320934 Number of chiral pairs of color patterns (set partitions) for a row of length n using 4 or fewer colors (subsets).

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A320937 Number of chiral pairs of color patterns (set partitions) for a row of length n.

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A107767 a(n) = (1 + 3^n - 23^(n/2))/4 if n is even, (1 + 3^n - 43^((n-1)/2))/4 if n odd.