A093566 -id:A093566 - cp's OEIS frontend

A047780 Number of inequivalent ways to color faces of a cube using at most n colors.

0, 1, 10, 57, 240, 800, 2226, 5390, 11712, 23355, 43450, 76351, 127920, 205842, 319970, 482700, 709376, 1018725, 1433322, 1980085, 2690800, 3602676, 4758930, 6209402, 8011200, 10229375, 12937626, 16219035, 20166832, 24885190, 30490050

Offset: 0

Jud McCranie

Here inequivalent means under the action of the rotation group of the cube, of order 24, which in its action on the faces has cycle index (x1^6 + 3*x1^2*x2^2 + 6*x1^2*x4 + 6*x2^3 + 8*x3^2)/24.
a(n) is also the number of inequivalent colorings of the vertices of a regular octahedron using at most n colors. - José H. Nieto S., Jan 19 2012
From Robert A. Russell, Oct 08 2020: (Start)
Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols for the regular octahedron and cube are {3,4} and {4,3} respectively. They are mutually dual.
There are 24 elements in the rotation group of the regular octahedron/cube. They divide into five conjugacy classes. The first formula is obtained by averaging the cube face (octahedron vertex) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Conjugacy Class    Count    Even Cycle Indices
  Identity              1     x_1^6
  Vertex rotation       8     x_3^2
  Edge rotation         6     x_2^3
  Small face rotation   6     x_1^2x_4^1
  Large face rotation   3     x_1^2x_2^2 (End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 254 (corrected).
N. G. De Bruijn, Polya's theory of counting, in E. F. Beckenbach, ed., Applied Combinatorial Mathematics, Wiley, 1964, pp. 144-184 (see p. 147).
M. Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246 (the formula given is incorrect but was corrected in the second printing).
J.-P. Delahaye, 'Le miraculeux "lemme de Burnside"','Le coloriage du cube' p. 147 in 'Pour la Science' (French edition of 'Scientific American') No.350 December 2006 Paris.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Polyhedron Coloring
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1)

Cf. A198833 (unoriented), A093566(n+1) (chiral), A337898 (achiral).
Other elements: A060530 (edges), A000543 (cube vertices, octahedron faces).
Cf. A006008 (tetrahedron), A000545 (dodecahedron faces, icosahedron vertices), A054472 (icosahedron faces, dodecahedron vertices).
Row 3 of A325004 (orthoplex vertices, orthotope facets) and A337887 (orthotope faces, orthoplex peaks).

[(n^6 + 3*n^4 + 12*n^3 + 8*n^2)/24: n in [1..30]]; // Vincenzo Librandi, Apr 27 2012

CoefficientList[Series[x*(1+3*x+8*x^2+16*x^3+2*x^4)/(1-x)^7,{x,0,33}],x] (* Vincenzo Librandi, Apr 27 2012 *)

a(n) = (n^6 + 3*n^4 + 12*n^3 + 8*n^2)/24 = n+8*C(n, 2)+30*C(n, 3)+68*C(n, 4)+75*C(n, 5)+30*C(n, 6). Each term of the RHS indicates the number of ways to use n colors to color the cube faces (octahedron vertices) with exactly 1, 2, 3, 4, 5, or 6 colors.
G.f.: x*(1+3*x+8*x^2+16*x^3+2*x^4)/(1-x)^7. - Colin Barker, Jan 29 2012
a(n) = A198833(n) + A093566(n+1) = 2*A198833(n) - A337898(n) = 2*A093566(n+1) + A337898(n). - Robert A. Russell, Oct 08 2020

Corrected version of A006550 and A006529.

Entry revised by N. J. A. Sloane, Jan 03 2005

A325006 Array read by descending antidiagonals: A(n,k) is the number of chiral pairs of colorings of the facets of a regular n-dimensional orthotope using up to k colors.

Original entry on oeis.org

0, 1, 0, 3, 0, 0, 6, 3, 0, 0, 10, 15, 1, 0, 0, 15, 45, 20, 0, 0, 0, 21, 105, 120, 15, 0, 0, 0, 28, 210, 455, 210, 6, 0, 0, 0, 36, 378, 1330, 1365, 252, 1, 0, 0, 0, 45, 630, 3276, 5985, 3003, 210, 0, 0, 0, 0, 55, 990, 7140, 20475, 20349, 5005, 120, 0, 0, 0, 0, 66, 1485, 14190, 58905, 98280, 54264, 6435, 45, 0, 0, 0, 0

Offset: 1

Robert A. Russell, May 27 2019

Also called hypercube, n-dimensional cube, and measure polytope. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is a cube with six square faces. For n=4, the figure is a tesseract with eight cubic facets. The Schläfli symbol, {4,3,...,3}, of the regular n-dimensional orthotope (n>1) consists of a four followed by n-2 threes. Each of its 2n facets is an (n-1)-dimensional orthotope. The chiral colorings of its facets come in pairs, each the reflection of the other.

Also the number of chiral pairs of colorings of the vertices of a regular n-dimensional orthoplex using up to k colors.

			Array begins with A(1,1):
0 1 3  6  10   15     21       28        36         45          55 ...
0 0 3 15  45  105    210      378       630        990        1485 ...
0 0 1 20 120  455   1330     3276      7140      14190       26235 ...
0 0 0 15 210 1365   5985    20475     58905     148995      341055 ...
0 0 0  6 252 3003  20349    98280    376992    1221759     3478761 ...
0 0 0  1 210 5005  54264   376740   1947792    8145060    28989675 ...
0 0 0  0 120 6435 116280  1184040   8347680   45379620   202927725 ...
0 0 0  0  45 6435 203490  3108105  30260340  215553195  1217566350 ...
0 0 0  0  10 5005 293930  6906900  94143280  886163135  6358402050 ...
0 0 0  0   1 3003 352716 13123110 254186856 3190187286 29248649430 ...
For a(2,3)=3, each chiral pair consists of two adjacent edges of the square with one of the three colors.

Robert A. Russell, Table of n, a(n) for n = 1..325
Robin Chapman, answer to Coloring the faces of a hypercube, Math StackExchange, September 30, 2010.

Cf. A325004 (oriented), A325005 (unoriented), A325007 (achiral), A325010 (exactly k colors)
Other n-dimensional polytopes: A007318(k,n+1) (simplex), A325014 (orthoplex)
Rows 1-3 are A161680, A050534, A093566(n+1), A234249(n-1)

Table[Binomial[Binomial[d-n+1,2],n],{d,1,12},{n,1,d}] // Flatten

a(n, k) = binomial(binomial(k, 2), n)
array(rows, cols) = for(x=1, rows, for(y=1, cols, print1(a(x, y), ", ")); print(""))
/* Print initial 10 rows and 11 columns of array as follows: */
array(10, 11) \\ Felix Fröhlich, May 30 2019

A(n,k) = binomial(binomial(k,2),n).
A(n,k) = Sum_{j=1..2*n} A325010(n,j) * binomial(k,j).
A(n,k) = A325004(n,k) - A325005(n,k) = (A325004(n,k) - A325007(n,k)) / 2 = A325005(n,k) - A325007(n,k).
G.f. for row n: Sum{j=1..2*n} A325010(n,j) * x^j / (1-x)^(j+1).
Linear recurrence for row n: T(n,k) = Sum_{j=0..2*n} binomial(-2-j,2*n-j) * T(n,k-1-j).
G.f. for column k: (1+x)^binomial(k,2) - 1.

A234249 Number of ways to choose 4 points in an n X n X n triangular grid.

Original entry on oeis.org

15, 210, 1365, 5985, 20475, 58905, 148995, 341055, 720720, 1426425, 2672670, 4780230, 8214570, 13633830, 21947850, 34389810, 52602165, 78738660, 115584315, 166695375, 236561325, 330791175, 456326325, 621682425, 837222750, 1115465715, 1471429260, 1923014940

Offset: 3

Heinrich Ludwig, Feb 02 2014

Sequence is column #5 of A084546: a(n) = A084546(n+1, 4).
All elements of the sequence are multiples of 15.
a(n-1) is the number of chiral pairs of colorings of the 8 cubic facets of a tesseract (hypercube) with Schläfli symbol {4,3,3} or of the 8 vertices of a hyperoctahedron with Schläfli symbol {3,3,4}. Both figures are regular 4-D polyhedra and they are mutually dual. Each member of a chiral pair is a reflection, but not a rotation, of the other. - Robert A. Russell, Oct 20 2020

Heinrich Ludwig, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A084546, A050534 (number of ways to choose 2 points), A093566 (3 points), A231653.
Cf. A337956 (oriented), A337956 (unoriented), A337956 (achiral) colorings, A331356 (hyperoctahedron edges, tesseract faces), A331360 (hyperoctahedron faces, tesseract edges), A337954 (hyperoctahedron facets, tesseract vertices).
Other polychora: A000389 (5-cell), A338950 (24-cell), A338966 (120-cell, 600-cell).
Row 4 of  A325006 (orthotope facets, orthoplex vertices).

A234249:=n->n*(n + 1)*(n - 1)*(n + 2)*(n - 2)*(n + 3)*(n^2 + n - 4)/384: seq(A234249(n), n=3..40); # Wesley Ivan Hurt, Jan 10 2017

Table[Binomial[Binomial[n,2],4], {n,4,30}] (* Robert A. Russell, Oct 20 2020 *)

Vec(-15*x^3*(x^2+5*x+1)/(x-1)^9 + O(x^100)) \\ Colin Barker, Feb 02 2014

a(n) = n*(n + 1)*(n - 1)*(n + 2)*(n - 2)*(n + 3)*(n^2 + n - 4)/384.
a(n) = C(C(n + 1, 2), 4).
G.f.: -15*x^3*(x^2+5*x+1) / (x-1)^9. - Colin Barker, Feb 02 2014
From Robert A. Russell, Oct 20 2020: (Start)
a(n-1) = 15*C(n,4) + 135*C(n,5) + 330*C(n,6) + 315*C(n,7) + 105*C(n,8), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
a(n-1) = A337956(n) - A337957(n) = (A337956(n) - A337958(n)) / 2 = A337957(n) - A337958(n).
a(n-1) = A325006(4,n). (End)

A198833 The number of inequivalent ways to color the vertices of a regular octahedron using at most n colors.

Original entry on oeis.org

1, 10, 56, 220, 680, 1771, 4060, 8436, 16215, 29260, 50116, 82160, 129766, 198485, 295240, 428536, 608685, 848046, 1161280, 1565620, 2081156, 2731135, 3542276, 4545100, 5774275, 7268976, 9073260, 11236456, 13813570, 16865705, 20460496, 24672560, 29583961

Offset: 1

Geoffrey Critzer, Oct 30 2011

The cycle index: 1/48 (s_1^6 + 3 s_1^4 s_2 + 9 s_1^2 s_2^2 +7 s_2^3 + 8 s_3^2 + 6 s_1^2 s_4 + 6 s_2 s_4 + 8 s_6) is returned in Mathematica by CycleIndex[ Automorphisms[ OctahedralGraph ], s].
One-sixth the area of the right triangles with sides 2b+2, b^2+2b, and b^2+2b+2 with b = A000217(n), the n-th triangular number. - J. M. Bergot, Aug 02 2013
Also the number of ways to color the faces of a cube with n colors, counting each pair of mirror images as one.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1)

Cf. A047780 (oriented), A093566(n+1) (chiral), A337898 (achiral), A199406 (edges), A128766 (octahedron faces, cube vertices), A000332(n+3) (tetrahedron), A128766 (octahedron faces, cube vertices), A252705 (dodecahedron faces, icosahedron vertices), A252704 (icosahedron faces, dodecahedron vertices), A000217 (triangular numbers).

Row 3 of A325005 (orthotope facets, orthoplex vertices) and A337888 (orthotope faces, orthoplex peaks).

[n*(n+1)*(n^2+n+2)*(n^2+n+4)/48: n in [1..35]]; // Vincenzo Librandi, Aug 04 2013

Table[(n^6 + 3 n^5 + 9 n^4 + 13 n^3 + 14 n^2 + 8 n)/48, {n, 25}]
CoefficientList[Series[-(1 + 3 x + 7 x^2 + 3 x^3 + x^4) / (x - 1)^7, {x, 0, 35}], x] (* Vincenzo Librandi, Aug 04 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,10,56,220,680,1771,4060},40] (* Harvey P. Dale, Nov 06 2024 *)

a(n)=n*(n+1)*(n^2+n+2)*(n^2+n+4)/48 \\ Charles R Greathouse IV, Aug 02 2013

a(n) = n*(n+1)*(n^2+n+2)*(n^2+n+4)/48.
G.f.: x*(1+3*x+7*x^2+3*x^3+x^4) / (1-x)^7. - R. J. Mathar, Oct 30 2011
a(n) = Sum_{i=1..A000217(n)} A000217(i). [Bruno Berselli, Sep 06 2013]
a(n) = 1*C(n,1) + 8*C(n,2) + 29*C(n,3) + 52*C(n,4) + 45*C(n,5) + 15*C(n,6), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors.
a(n) = A047780(n) - A093566(n+1) = (A047780(n) + A337898(n)) / 2 = A093566(n+1) + A337898(n). - Robert A. Russell, Oct 19 2020

A337898 Number of achiral colorings of the 6 square faces of a cube or the 6 vertices of a regular octahedron using n or fewer colors.

Original entry on oeis.org

1, 10, 55, 200, 560, 1316, 2730, 5160, 9075, 15070, 23881, 36400, 53690, 77000, 107780, 147696, 198645, 262770, 342475, 440440, 559636, 703340, 875150, 1079000, 1319175, 1600326, 1927485, 2306080, 2741950, 3241360

Offset: 1

Robert A. Russell, Sep 28 2020

An achiral coloring is identical to its reflection. The Schläfli symbols for the cube and regular octahedron are {4,3} and {3,4} respectively. They are mutually dual.
There are 24 elements in the automorphism group of the regular octahedron/cube that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the cube face (octahedron vertex) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Conjugacy Class     Count    Odd Cycle Indices
  Inversion              1     x_2^3
  Vertex rotation*       8     x_6^1           Asterisk indicates that the
  Edge rotation*         6     x_1^2x_2^2      operation is followed by an
  Small face rotation*   6     x_2^1x_4^1      inversion.
  Large face rotation*   3     x_1^4x_2^1

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A047780 (oriented), A198833 (unoriented), A093566(n+1) (chiral).
Other elements: A331351 (edges), A337897 (cube vertices/octahedron faces).
Other polyhedra: A006003 (simplex), A337962 (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices).
Row 3 of A325007 (orthotope facets, orthoplex vertices) and A337890 (orthotope faces, orthoplex peaks).

Table[n(1+n)(2+n)(4-3n+3n^2)/24, {n, 35}]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,10,55,200,560,1316},40] (* Harvey P. Dale, Feb 15 2022 *)

a(n)=n*(n+1)*(n+2)*(3*n^2-3*n+4)/24 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = n * (n+1) * (n+2) * (3*n^2 - 3*n + 4) / 24.
a(n) = 1*C(n,1) + 8*C(n,2) + 28*C(n,3) + 36*C(n,4) + 15*C(n,5), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A198833(n) - A047780(n) = A047780(n) - 2*A093566(n+1) = A198833(n) - A093566(n+1).
G.f.: x * (x + 4*x^2 + 10*x^3) / (1-x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Wesley Ivan Hurt, Sep 30 2020

A337889 Array read by descending antidiagonals: T(n,k) is the number of chiral pairs of colorings of the square faces of a regular n-dimensional orthotope (hypercube) using k or fewer colors.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 40927, 0, 0, 20, 731279799, 314824333015938998688, 0, 0, 120, 732272925320, 38491882659300767730994725249684096, 38343035259947576596859560773963975000551460473665493534170658111488, 0

Offset: 2

Robert A. Russell, Sep 28 2020

Each member of a chiral pair is a reflection, but not a rotation, of the other. Each face is a square bounded by four edges. For n=2, the figure is a square with one face. For n=3, the figure is a cube with 6 faces. For n=4, the figure is a tesseract with 24 faces. The number of faces is 2^(n-2)*C(n,2).
Also the number of chiral pairs of colorings of peaks of an n-dimensional orthoplex. A peak is an (n-3)-dimensional simplex.
The algorithm used in the Mathematica program below assigns each permutation of the axes to a partition of n and then considers separate conjugacy classes for axis reversals. It uses the formulas in Balasubramanian's paper. If the value of m is increased, one can enumerate colorings of higher-dimensional elements beginning with T(m,1).

			Array begins with T(2,1):
0     0         0            0               0                 0 ...
0     0         1           20             120               455 ...
0 40927 731279799 732272925320 155180061396500 12338466190481025 ...

K. Balasubramanian, Computational enumeration of colorings of hyperplanes of hypercubes for all irreducible representations and applications, J. Math. Sci. & Mod. 1 (2018), 158-180.

Cf. A337887 (oriented), A337888 (unoriented), A337890 (achiral).
Other elements: A325014 (vertices), A337409 (edges).
Other polytopes: A337885 (simplex), A337893 (orthoplex).
Rows 2-4 are A000004, A093566(n+1), A331356.

m=2; (* dimension of color element, here a square face *)
Fi1[p1_] := Module[{g, h}, Coefficient[Product[g = GCD[k1, p1]; h = GCD[2 k1, p1]; (1+2x^(k1/g))^(r1[[k1]] g) If[Divisible[k1, h], 1, (1+2x^(2 k1/h))^(r2[[k1]] h/2)], {k1, Flatten[Position[cs, n1_ /; n1 > 0]]}], x, n-m]];
FiSum[] := (Do[Fi2[k2] = Fi1[k2], {k2, Divisors[per]}]; DivisorSum[per, DivisorSum[d1 = #, MoebiusMu[d1/#] Fi2[#] &]/# &]);
CCPol[r_List] := (r1 = r; r2 = cs - r1; per = LCM @@ Table[If[cs[[j2]] == r1[[j2]], If[0 == cs[[j2]],1,j2], 2j2], {j2,n}]; If[EvenQ[Sum[If[EvenQ[j3], r1[[j3]], r2[[j3]]], {j3,n}]],1,-1]Times @@ Binomial[cs, r1] 2^(n-Total[cs]) b^FiSum[]);
PartPol[p_List] := (cs = Count[p, #]&/@ Range[n]; Total[CCPol[#]&/@ Tuples[Range[0,cs]]]);
pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #]&/@ mb; n!/(Times@@(ci!) Times@@(mb^ci))] (*partition count*)
row[n_Integer] := row[n] = Factor[(Total[(PartPol[#] pc[#])&/@ IntegerPartitions[n]])/(n! 2^n)]
array[n_, k_] := row[n] /. b -> k
Table[array[n,d+m-n], {d,6}, {n,m,d+m-1}] // Flatten

T(n,k) = A337887(n,k) - A337888(n,k) = (A337887(n,k) - A337890(n,k)) / 2 = A337888(n,k) - A337890(n,k).

A337896 Number of chiral pairs of colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.

Original entry on oeis.org

0, 1, 66, 920, 6350, 29505, 106036, 317856, 832140, 1961025, 4248310, 8590296, 16398746, 29814785, 51983400, 87399040, 142333656, 225359361, 347978730, 525376600, 777308070, 1129138241, 1613050076, 2269437600

Offset: 1

Robert A. Russell, Sep 28 2020

nonn

Each member of a chiral pair is a reflection, but not a rotation, of the other.

			For a(2)=1, centering the octahedron (cube) at the origin and aligning the diagonals (edges) with the axes, color the faces (vertices) in the octants ---, --+, -++, and +++ with one color and the other 4 elements with the other color.

Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A000543 (oriented), A128766(unoriented), A337897 (achiral).
Other elements: A337406 (edges), A093566(n+1) (cube faces, octahedron vertices).
Other polyhedra: A000332 (simplex), A093566(n+1) (cube/octahedron).
Row 3 of A325014 (chiral pairs of colorings of orthoplex facets or orthotope vertices).
Row 3 of A337893 (chiral pairs of colorings of orthoplex faces or orthotope peaks).

Table[(n-1)n^2(n+1)(8-5n^2+n^4)/48, {n,30}]

a(n) = (n-1) * n^2 * (n+1) * (8 - 5*n^2 + n^4) / 48.
a(n) = 1*C(n,2) + 63*C(n,3) + 662*C(n,4) + 2400*C(n,5) + 3900*C(n,6) + 2940*C(n,7) + 840*C(n,8), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
G.f.: x^2 * (1+x) * (1+56*x+306*x^2+56*x^3+x^4) / (1-x)^9.
a(n) = A000543(n) - A128766(n) = (A000543(n) - A337897(n)) / 2 = A128766(n) - A337897(n).

A144228 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes with k edges where each maximally connected subgraph has at most one cycle.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 3, 1, 1, 6, 15, 20, 15, 1, 10, 45, 120, 210, 222, 1, 15, 105, 455, 1365, 2913, 3670, 1, 21, 210, 1330, 5985, 20139, 49294, 68820, 1, 28, 378, 3276, 20475, 97860, 362670, 976560, 1456875, 1, 36, 630, 7140, 58905, 376236, 1914276, 7663500, 22089870, 34506640

Offset: 0

Alois P. Heinz, Sep 15 2008

			T(4,4) = 15, because there are 15 simple graphs on 4 labeled nodes with 4 edges where each maximally connected subgraph has at most one cycle:
  1-2  1-2  1-2  1-2  1-2  1-2  1 2  1 2  1-2  1 2  1 2  1-2  1-2  1-2  1 2
  |/|  |X   |/   |\|   X|   \|  |/|   X|   /|  |\|  |X   |\   | |   X   |X|
  4 3  4 3  4-3  4 3  4 3  4-3  4-3  4-3  4-3  4-3  4-3  4-3  4-3  4-3  4 3
Triangle begins:
  1;
  1,  0;
  1,  1,  0;
  1,  3,  3,   1;
  1,  6, 15,  20,  15;
  1, 10, 45, 120, 210, 222;
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..140, flattened

Columns k=0-3 give: A000012, A000217, A050534, A093566.
Main diagonal gives A137916.
Row sums give: A133686.
T(2n,n) gives A369828.
Cf. A000272, A057500, A007318, A000142, A129271.

cy:= proc(n) option remember; local t; binomial(n-1, 2) *add((n-3)! /(n-2-t)! *n^(n-2-t), t=1..n-2) end: T:= proc(n,k) option remember; local j; if k=0 then 1 elif k<0 or n

t[, 0] = 1; t[n, k_] /; (k<0 || nJean-François Alcover, Jan 15 2014, after Maple *)

T(n,0) = 1, T(n,k) = 0 if k<0 or nA000272(j+1) T(n-j-1,k-j) + A057500(j+1) T(n-j-1,k-j-1)).

E.g.f.: exp(B(x,y)), where B(x,y) = Sum(Sum(A062734(n,k)*y^k*x^n/n!, k=0..n), n=1..infinity) = -1/2*log(1+LambertW(-x*y))+1/2*LambertW(-x*y) -1/4*LambertW(-x*y)^2-1/y *(LambertW(-x*y)+1/2 *LambertW(-x*y)^2). - Vladeta Jovovic, Sep 16 2008

A194485 T(n,k) = number of ways to arrange k indistinguishable points on an n X n X n triangular grid so that no four points are in the same row or diagonal.

Original entry on oeis.org

1, 0, 3, 0, 3, 6, 0, 1, 15, 10, 0, 0, 20, 45, 15, 0, 0, 15, 120, 105, 21, 0, 0, 6, 207, 455, 210, 28, 0, 0, 1, 234, 1347, 1330, 378, 36, 0, 0, 0, 165, 2817, 5922, 3276, 630, 45, 0, 0, 0, 63, 4135, 19362, 20307, 7140, 990, 55, 0, 0, 0, 9, 4080, 47010, 94584, 58527, 14190, 1485

Offset: 1

R. H. Hardin, Aug 26 2011

Table starts
...1....0......0.......0.........0..........0...........0............0
...3....3......1.......0.........0..........0...........0............0
...6...15.....20......15.........6..........1...........0............0
..10...45....120.....207.......234........165..........63............9
..15..105....455....1347......2817.......4135........4080.........2463
..21..210...1330....5922.....19362......47010.......83745.......105663
..28..378...3276...20307.....94584.....337860......927471......1931571
..36..630...7140...58527....365904....1790472.....6924357.....21123489
..45..990..14190..148239...1193283....7622340....39196161....162957252
..55.1485..26235..339669...3413619...27489825...180512640....974497260
..66.2145..45760..718344...8800704...87018360...708150465...4794685500
..78.3003..76076.1422564..20845968..247874770..2442836682..20207649891
..91.4095.121485.2666664..46017972..647091588..7582054194..75074999142
.105.5460.187460.4771221..95710797.1569661600.21540941994.251128663929
.120.7140.280840.8201466.189154056.3576049620.56763356130.768641935191

			Some solutions for n=5, k=4:
......0..........0..........0..........1..........0..........0..........0
.....0.0........0.1........0.1........1.0........0.0........0.1........0.0
....0.0.0......0.0.1......1.0.0......0.0.0......0.0.0......0.0.0......1.1.0
...0.0.1.1....0.0.0.0....0.0.1.0....0.0.1.0....0.0.1.1....0.0.0.0....0.0.0.0
..0.1.0.0.1..0.1.1.0.0..0.1.0.0.0..0.1.0.0.0..0.0.1.1.0..0.1.1.0.1..1.0.1.0.0

R. H. Hardin, Table of n, a(n) for n = 1..264

Column 1 is A000217.
Column 2 is A050534(n+1).
Column 3 is A093566(n+2).

Empirical:
T(n,1) = (1/2)*n^2 + (1/2)*n
T(n,2) = (1/8)*n^4 + (1/4)*n^3 - (1/8)*n^2 - (1/4)*n
T(n,3) = (1/48)*n^6 + (1/16)*n^5 - (1/16)*n^4 - (11/48)*n^3 + (1/24)*n^2 + (1/6)*n
T(n,4) = (1/384)*n^8 + (1/96)*n^7 - (1/64)*n^6 - (13/120)*n^5 + (19/128)*n^4 + (7/96)*n^3 - (13/96)*n^2 + (1/40)*n
T(n,5) = (1/3840)*n^10 + (1/768)*n^9 - (1/384)*n^8 - (59/1920)*n^7 + (281/3840)*n^6 + (149/3840)*n^5 - (5/24)*n^4 + (29/320)*n^3 + (11/80)*n^2 - (1/10)*n
T(n,6) = (1/46080)*n^12 + (1/7680)*n^11 - (1/3072)*n^10 - (137/23040)*n^9 + (871/46080)*n^8 + (3107/161280)*n^7 - (5573/46080)*n^6 + (1157/23040)*n^5 + (2627/11520)*n^4 - (1121/5760)*n^3 - (181/1440)*n^2 + (11/84)*n
T(n,7) = (1/645120)*n^14 + (1/92160)*n^13 - (1/30720)*n^12 - (79/92160)*n^11 + (101/30720)*n^10 + (757/129024)*n^9 - (3049/92160)*n^8 - (34099/645120)*n^7 + (6613/15360)*n^6 - (16859/23040)*n^5 + (1043/3840)*n^4 + (2759/5040)*n^3 - (753/1120)*n^2 + (13/56)*n
Empirical: general T(n,k,z) for fewer than z points in any row or diagonal is polynomial in n of degree 2k with lead coefficient 1/(2^k*k!) for small k.
T(n,1,2) = (1/2)*n^2 + (1/2)*n
T(n,1,3) = (1/2)*n^2 + (1/2)*n
T(n,1,4) = (1/2)*n^2 + (1/2)*n
T(n,2,2) = (1/8)*n^4 - (1/4)*n^3 - (1/8)*n^2 + (1/4)*n
T(n,2,3) = (1/8)*n^4 + (1/4)*n^3 - (1/8)*n^2 - (1/4)*n
T(n,2,4) = (1/8)*n^4 + (1/4)*n^3 - (1/8)*n^2 - (1/4)*n
T(n,3,3) = (1/48)*n^6 + (1/16)*n^5 - (3/16)*n^4 + (1/48)*n^3 + (1/6)*n^2 - (1/12)*n
T(n,3,4) = (1/48)*n^6 + (1/16)*n^5 - (1/16)*n^4 - (11/48)*n^3 + (1/24)*n^2 + (1/6)*n
T(n,4,3) = (1/384)*n^8 + (1/96)*n^7 - (5/64)*n^6 + (13/240)*n^5 + (27/128)*n^4 - (23/96)*n^3 - (13/96)*n^2 + (7/40)*n
T(n,4,4) = (1/384)*n^8 + (1/96)*n^7 - (1/64)*n^6 - (13/120)*n^5 + (19/128)*n^4 + (7/96)*n^3 - (13/96)*n^2 + (1/40)*n
T(n,5,3) = (1/3840)*n^10 + (1/768)*n^9 - (7/384)*n^8 + (37/1920)*n^7 + (737/3840)*n^6 - (2347/3840)*n^5 + (101/192)*n^4 + (93/320)*n^3 - (7/10)*n^2 + (3/10)*n
T(n,5,4) = (1/3840)*n^10 + (1/768)*n^9 - (1/384)*n^8 - (59/1920)*n^7 + (281/3840)*n^6 + (149/3840)*n^5 - (5/24)*n^4 + (29/320)*n^3 + (11/80)*n^2 - (1/10)*n
T(n,6,4) = (1/46080)*n^12 + (1/7680)*n^11 - (1/3072)*n^10 - (137/23040)*n^9 + (871/46080)*n^8 + (3107/161280)*n^7 - (5573/46080)*n^6 + (1157/23040)*n^5 + (2627/11520)*n^4 - (1121/5760)*n^3 - (181/1440)*n^2 + (11/84)*n
T(n,7,4) = (1/645120)*n^14 + (1/92160)*n^13 - (1/30720)*n^12 - (79/92160)*n^11 + (101/30720)*n^10 + (757/129024)*n^9 - (3049/92160)*n^8 - (34099/645120)*n^7 + (6613/15360)*n^6 - (16859/23040)*n^5 + (1043/3840)*n^4 + (2759/5040)*n^3 - (753/1120)*n^2 + (13/56)*n

A337961 Number of chiral pairs of colorings of the 12 pentagonal faces of a regular dodecahedron or the 12 vertices of a regular icosahedron using n or fewer colors.

Original entry on oeis.org

0, 14, 3720, 132184, 1987720, 17935806, 114638048, 570597216, 2348263008, 8320953630, 26126986952, 74247445272, 194049316552, 472265688622, 1080900468480, 2345089916288, 4854316187136, 9638888023278, 18442173583176

Offset: 1

Robert A. Russell, Oct 03 2020

Each member of a chiral pair is a reflection, but not a rotation, of the other. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A000545 (oriented), A252705 (unoriented), A337962 (achiral).
Other elements: A337959 (dodecahedron vertices, icosahedron faces), A337964 (edges).
Other polyhedra: A000332 (tetrahedron), A093566(n+1) (cube faces, octahedron vertices), A337896 (octahedron faces, cube vertices).

Table[(n^12-15n^8+14n^6+44n^4-44n^2)/120,{n,30}]
LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{0,14,3720,132184,1987720,17935806,114638048,570597216,2348263008,8320953630,26126986952,74247445272,194049316552},20] (* Harvey P. Dale, Nov 17 2024 *)

a(n) = (n-1) * n^2 * (n+1) * (n^8 + n^6 - 14*n^4 + 44) / 120.
a(n) = 14*C(n,2) + 3678*C(n,3) + 117388*C(n,4) + 1363860*C(n,5) + 7918056*C(n,6) + 26332992*C(n,7) + 53428032*C(n,8) + 67359600*C(n,9) + 51559200*C(n,10) + 21954240*C(n,11) + 3991680*C(n,12), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
a(n) = A000545(n) - A252705(n) = (A000545(n) - A337962(n)) / 2 = A252705(n) - A337962(n).

cp's OEIS Frontend

A047780 Number of inequivalent ways to color faces of a cube using at most n colors.

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A325006 Array read by descending antidiagonals: A(n,k) is the number of chiral pairs of colorings of the facets of a regular n-dimensional orthotope using up to k colors.

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A234249 Number of ways to choose 4 points in an n X n X n triangular grid.

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A198833 The number of inequivalent ways to color the vertices of a regular octahedron using at most n colors.

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A337898 Number of achiral colorings of the 6 square faces of a cube or the 6 vertices of a regular octahedron using n or fewer colors.

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A337889 Array read by descending antidiagonals: T(n,k) is the number of chiral pairs of colorings of the square faces of a regular n-dimensional orthotope (hypercube) using k or fewer colors.

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A337896 Number of chiral pairs of colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.

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