A331357 -id:A331357 - cp's OEIS frontend

A331353 Number of achiral colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

1, 28, 387, 2784, 13125, 46836, 137543, 349952, 797769, 1667500, 3248971, 5973408, 10459917, 17571204, 28479375, 44742656, 68393873, 102041532, 148984339, 213340000, 300189141, 415735188, 567481047, 764423424

Offset: 1

Robert A. Russell, Jan 14 2020

A 4-dimensional simplex has 5 vertices and 10 edges. Its Schläfli symbol is {3,3,3}. An achiral coloring is identical to its reflection,
There are 60 elements in the automorphism group of the 4-dimensional simplex that are not in its rotation group.  Each is an odd permutation of the vertices and can be associated with a partition of 5 based on the conjugacy group of the permutation. The first formula is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Partition  Count  Odd Cycle Indices
  41         30     x_2^1x_4^2
  32         20     x_1^1x_3^1x_6^1
  2111       10     x_1^4x_2^3

Colin Barker, Table of n, a(n) for n = 1..1000
G. Royle, Partitions and Permutations
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A331350 (oriented), A063843 (unoriented), A331352 (chiral).
Other polychora: A331361 (8-cell), A331357 (16-cell), A338955 (24-cell), A338967 (120-cell, 600-cell).
Row 4 of A327086 (simplex edges and ridges) and A337886 (simplex faces and peaks).

Mathematica
```
Table[(5 n^3 + n^7)/6, {n, 1, 25}]
```

Vec(x*(1 + 20*x + 191*x^2 + 416*x^3 + 191*x^4 + 20*x^5 + x^6) / (1 - x)^8 + O(x^25)) \\ Colin Barker, Jan 15 2020

a(n) = (5*n^3 + n^7) / 6.
a(n) = C(n,1) + 26*C(n,2) + 306*C(n,3) + 1400*C(n,4) + 2800*C(n,5) + 2520*C(n,6) + 840*C(n,7), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = 2*A063843(n) - A331350(n) = A331350(n) - 2*A331352(n) = A063843(n) - A331352(n).
From Colin Barker, Jan 15 2020: (Start)
G.f.: x*(1 + 20*x + 191*x^2 + 416*x^3 + 191*x^4 + 20*x^5 + x^6) / (1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>8.
(End)

A331354 Number of oriented colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

Original entry on oeis.org

1, 90054, 1471640157, 1466049174160, 310441584462375, 24679078461920106, 997818989210621704, 24595659246351652992, 415450226822646218895, 5208333343963621522750, 51300691059764724112161, 414046079318115654521904

Offset: 1

Robert A. Russell, Jan 14 2020

A regular 4-dimensional orthoplex (also hyperoctahedron or cross polytope) has 8 vertices and 24 edges. Its Schläfli symbol is {3,3,4}. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two. Also the number of oriented colorings of the square faces of a tesseract {4,3,3} with n available colors.
There are 192 elements in the rotation group of the 4-dimensional orthoplex. Each is associated with a partition of 4 based on the conjugacy group of the permutation of the axes. The first formula is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Partition  Count  Even Cycle Indices
  4          6      8x_8^3
  31         8      4x_3^8 + 4x_6^4
  22         3      4x_1^4x_2^10 + 4x_4^6
  211        6      4x_1^2x_2^11 + 2x_1^4x_4^5 + 2x_2^2x_4^5
  1111       1      6x_1^4x_2^10 + x_1^24 + x_2^12

G. Royle, Partitions and Permutations
Index entries for linear recurrences with constant coefficients, signature (25, -300, 2300, -12650, 53130, -177100, 480700, -1081575, 2042975, -3268760, 4457400, -5200300, 5200300, -4457400, 3268760, -2042975, 1081575, -480700, 177100, -53130, 12650, -2300, 300, -25, 1).

Cf. A331355 (unoriented), A331356 (chiral), A331357 (achiral).
Other polychora: A331350 (5-cell), A331358 (8-cell), A338952 (24-cell), A338964 (120-cell, 600-cell).
Row 4 of A337411 (orthoplex edges, orthotope ridges) and A337887 (orthotope faces, orthoplex peaks).

Table[(48n^3 + 32n^4 + 12n^6 + 12n^7 + 32n^8 + 12n^9 + n^12 + 24n^13 + 18n^14 + n^24)/192, {n, 1, 25}]

a(n) = (48*n^3 + 32*n^4 + 12*n^6 + 12*n^7 + 32*n^8 + 12*n^9 + n^12 + 24*n^13 + 18*n^14 + n^24) / 192.
a(n) = C(n,1) + 90052*C(n,2) + 1471369998*C(n,3) + 1460163153852*C(n,4) + 303126054092610*C(n,5) + 22838390261305920*C(n,6) + 831533453035309605*(n,7) + 17286839341903413240*C(n,8) + 227976665667323280750*C(n,9) + 2046002146009161624900*C(n,10) + 13118524448411114548200*C(n,11) + 62195874413179579657200*C(n,12) + 223421486565003375448800*C(n,13) + 618462331903782130564800*C(n,14) + 1333693289177381452320000*C(n,15) + 2253251792722109699520000*C(n,16) + 2984347082566196867520000*C(n,17) + 3083974243985846090880000*C(n,18) + 2458713052058007064320000*C(n,19) + 1482204734016157831680000*C(n,20) + 653167360418390737920000*C(n,21) + 198468086839148206080000*C(n,22) + 37162274062147153920000*C(n,23) + 3231502092360622080000*C(n,24), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = A331355(n) + A331356(n) = 2*A331355(n) - A331357(n) = 2*A331356(n) +  A331357(n).

A331355 Number of unoriented colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

Original entry on oeis.org

1, 49127, 740360358, 733776248840, 155261523065875, 12340612271439081, 498926608780739307, 12298018390569089088, 207726683413584244680, 2604177120221402303875, 25650403577338260144611, 207023317470352041578712

Offset: 1

Robert A. Russell, Jan 14 2020

A regular 4-dimensional orthoplex (also hyperoctahedron or cross polytope) has 8 vertices and 24 edges. Its Schläfli symbol is {3,3,4}. Two unoriented colorings are the same if congruent; chiral pairs are counted as one. Also the number of unoriented colorings of the square faces of a tesseract {4,3,3} with n available colors.

Index entries for linear recurrences with constant coefficients, signature (25, -300, 2300, -12650, 53130, -177100, 480700, -1081575, 2042975, -3268760, 4457400, -5200300, 5200300, -4457400, 3268760, -2042975, 1081575, -480700, 177100, -53130, 12650, -2300, 300, -25, 1).

Cf. A331354 (oriented), A331356 (chiral), A331357 (achiral).
Other polychora: A063843 (5-cell), A331359 (8-cell), A338953 (24-cell), A338965 (120-cell, 600-cell).
Row 4 of A337412 (orthoplex edges, orthotope ridges) and A337888 (orthotope faces, orthoplex peaks).

Table[(48 n^3 + 64 n^4 + 44 n^6 + 84 n^7 + 56 n^8 + 12 n^9 + 5 n^12 +
    36 n^13 + 18 n^14 + 12 n^15 + 4 n^18 + n^24)/384, {n, 1, 25}]

a(n) = (48*n^3 + 64*n^4 + 44*n^6 + 84*n^7 + 56*n^8 + 12*n^9 + 5*n^12 +
   36*n^13 + 18*n^14 + 12*n^15 + 4*n^18 + n^24) / 384.
a(n) = C(n,1) + 49125*C(n, 2) + 740212980*C(n, 3) + 730815102166*C(n, 4) + 151600044933990*C(n, 5) + 11420034970306170*C(n, 6) + 415777158607920585*C(n, 7) + 8643499341510394200*C(n, 8) + 113988734942055623055*C(n, 9) + 1023002477284840979850*C(n, 10) + 6559265715033958749900*C(n, 11) + 31097943476763200314200*C(n, 12) + 111710751446923209781200*C(n, 13) + 309231173588248964052000*C(n, 14) + 666846649590586048584000*C(n, 15) + 1126625898539640346848000*C(n, 16) + 1492173541849975272288000*C(n, 17) + 1541987122059614438208000*C(n, 18) + 1229356526029003532160000*C(n, 19) + 741102367008078915840000*C(n, 20) + 326583680209195368960000*C(n, 21) + 99234043419574103040000*C(n, 22) + 18581137031073576960000*C(n, 23) + 1615751046180311040000*C(n, 24), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = A331354(n) - A331356(n) = (A331354(n) + A331357(n)) / 2 = A331356(n) + A331357(n).

A331356 Number of chiral pairs of colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

Original entry on oeis.org

0, 40927, 731279799, 732272925320, 155180061396500, 12338466190481025, 498892380429882397, 12297640855782563904, 207723543409061974215, 2604156223742219218875, 25650287482426463967550, 207022761847763612943192

Offset: 1

Robert A. Russell, Jan 14 2020

A regular 4-dimensional orthoplex (also hyperoctahedron or cross polytope) has 8 vertices and 24 edges. Its Schläfli symbol is {3,3,4}. The chiral colorings of its edges come in pairs, each the reflection of the other. Also the number of chiral pairs of colorings of the square faces of a tesseract {4,3,3} with n available colors.

Index entries for linear recurrences with constant coefficients, signature (25, -300, 2300, -12650, 53130, -177100, 480700, -1081575, 2042975, -3268760, 4457400, -5200300, 5200300, -4457400, 3268760, -2042975, 1081575, -480700, 177100, -53130, 12650, -2300, 300, -25, 1).

Cf. A331354 (oriented), A331355 (unoriented), A331357 (achiral).
Other polychora: A331352 (5-cell), A331360 (8-cell), A338954 (24-cell), A338966 (120-cell, 600-cell).
Row 4 of A337413 (orthoplex edges, orthotope ridges) and A337889 (orthotope faces, orthoplex peaks).

Table[(48n^3 - 20n^6 - 60n^7 + 8n^8 + 12n^9 - 3n^12 + 12n^13 + 18n^14 - 12n^15 - 4n^18 + n^24)/384, {n, 1, 25}]

a(n) = (48*n^3 - 20*n^6 - 60*n^7 + 8*n^8 + 12*n^9 - 3*n^12 + 12*n^13 + 18*n^14 - 12*n^15 - 4*n^18 + n^24) / 384.
a(n) = 40927*C(n,2) + 731157018*C(n,3) + 729348051686*C(n,4) + 151526009158620*C(n,5) + 11418355290999750*C(n,6) + 415756294427389020*C(n,7) + 8643340000393019040*C(n,8) + 113987930725267657695*C(n,9) + 1022999668724320645050*C(n,10) + 6559258733377155798300*C(n,11) + 31097930936416379343000*C(n,12) + 111710735118080165667600*C(n,13) + 309231158315533166512800*C(n,14) + 666846639586795403736000*C(n,15) + 1126625894182469352672000*C(n,16) + 1492173540716221595232000*C(n,17) + 1541987121926231652672000*C(n,18) + 1229356526029003532160000*C(n,19) + 741102367008078915840000*C(n,20) + 326583680209195368960000*C(n,21) + 99234043419574103040000*C(n,22) + 18581137031073576960000*C(n,23) + 1615751046180311040000*C(n,24), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = A331354(n) - A331355(n) = (A331354(n) - A331357(n)) / 2 = A331355(n) - A331357(n).

A331361 Number of achiral colorings of the edges of a tesseract with n available colors.

Original entry on oeis.org

1, 93024, 294157089, 91983927296, 7960001890625, 304914963625056, 6652124939544609, 96100248309858304, 1013293206632601441, 8334166666733500000, 56066328722011832961, 319495406392484665344

Offset: 1

Robert A. Russell, Jan 14 2020

A tesseract is a regular 4-dimensional orthotope or hypercube with 16 vertices and 32 edges. Its Schläfli symbol is {4,3,3}. An achiral coloring is identical to its reflection. Also the number of achiral colorings of the triangular faces of a regular 4-dimensional orthoplex {3,3,4} with n available colors.
There are 192 elements in the automorphism group of the tesseract that are not in its rotation group. Each is associated with a partition of 4 based on the conjugacy group of the permutation of the axes. The first formula is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Partition  Count  Odd Cycle Indices
  4          6      8x_4^8
  31         8      4x_1^2x_3^2x_6^4 + 4x_2^1x_6^5
  22         3      8x_4^8
  211        6      2x_1^8x_2^12 + 2x_2^16 + 4x_4^8
  1111       1      4x_1^8x_2^12 + 4x_2^16

G. Royle, Partitions and Permutations
Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Cf. A331358 (oriented), A331359 (unoriented), A331360 (chiral).

Cf. A331353 (simplex), A331357 (orthoplex), A338955 (24-cell), A338967 (120-cell, 600-cell).

Table[(2n^6 + 8n^8 + n^16 + n^20)/12, {n, 1, 25}]

a(n) = (2*n^6 + 8*n^8 + n^16 + n^20) / 12.
a(n) = C(n,1) + 93022*C(n,2) + 293878020*C(n,3) + 90807857080*C(n,4) + 7503022894800*C(n,5) + 258528829444320*C(n,6) + 4681671089961600*C(n,7) + 50981530073846400*C(n,8) + 363246007692204000*C(n,9) + 1789536284820648000*C(n,10) + 6323058513173001600*C(n,11) + 16406578807069651200*C(n,12) + 31689737477798400000*C(n,13) + 45786987328642560000*C(n,14) + 49291621471572480000*C(n,15) + 38970361271761920000*C(n,16) + 21972146261345280000*C(n,17) + 8363100653107200000*C(n,18) + 1926047423139840000*C(n,19) + 202741834014720000*C(n,20), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = 2*A331359(n) - A331358(n) = A331358(n) - 2*A331360(n) = A331359(n) - A331360(n).

A337414 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the edges of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 18, 70, 1, 5, 40, 1407, 8200, 1, 6, 75, 12480, 9080559, 12804908, 1, 7, 126, 69050, 1503323520, 4906480368591, 304899216832, 1, 8, 196, 281946, 81461669375, 48226825456539776, 187380251418565888983, 103685962258536432, 1

Offset: 1

Robert A. Russell, Aug 26 2020

An achiral arrangement is identical to its reflection. For n=1, the figure is a line segment with one edge. For n=2, the figure is a square with 4 edges. For n=3, the figure is an octahedron with 12 edges. The number of edges is 2n*(n-1) for n>1.

Also the number of achiral colorings of the regular (n-2)-dimensional orthotopes (hypercubes) in a regular n-dimensional orthotope.

			Table begins with T(1,1):
1  2    3     4     5      6      7       8       9       10 ...
1  6   18    40    75    126    196     288     405      550 ...
1 70 1407 12480 69050 281946 931490 2632512 6598935 15041950 ...
For T(2,2)=6, the arrangements are AAAA, AAAB, AABB, ABAB, ABBB, and BBBB.

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. A337411 (oriented), A337412 (unoriented), A337413 (chiral).
Rows 1-4 are A000027, A002411, A331351, A331357.
Cf. A327086 (simplex edges), A337410 (orthotope edges), A325007 (orthoplex vertices).

m=1; (* dimension of color element, here an edge *)
Fi1[p1_] := Module[{g, h}, Coefficient[Product[g = GCD[k1, p1]; h = GCD[2 k1, p1]; (1 + 2 x^(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, m+1]];
FiSum[] := (Do[Fi2[k2] = Fi1[k2], {k2, Divisors[per]}];DivisorSum[per, DivisorSum[d1 = #, MoebiusMu[d1/#] Fi2[#] &]/# &]);
CCPol[r_List] := (r1 = r; r2 = cs - r1; If[EvenQ[Sum[If[EvenQ[j3], r1[[j3]], r2[[j3]]], {j3,n}]],0,(per = LCM @@ Table[If[cs[[j2]] == r1[[j2]], If[0 == cs[[j2]],1,j2], 2j2], {j2,n}]; 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[m]=b;
row[n_Integer] := row[n] = Factor[(Total[(PartPol[#] pc[#])&/@ IntegerPartitions[n]])/(n! 2^(n-1))]
array[n_, k_] := row[n] /. b -> k
Table[array[n,d+m-n], {d,8}, {n,m,d+m-1}] // Flatten

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).

T(n,k) = 2*A337412(n,k) - A337411(n,k) = A337411(n,k) - 2*A337413(n,k) = A337412(n,k) - A337413(n,k).

A338955 Number of achiral colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

Original entry on oeis.org

1, 24124751133507584, 883287060208158070437496209, 27692675763559261523047959805034496, 18070082615414169898334284655914306640625, 1018202231744161700740376040914469837333037056

Offset: 1

Robert A. Russell, Nov 17 2020

An achiral coloring is identical to its reflection. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. There are 576 elements in the automorphism group of the 24-cell that are not in its rotation group. They divide into 10 conjugacy classes. The first formula is obtained by averaging the edge (or face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Count    Odd Cycle Indices     Count    Odd Cycle Indices
     12    x_1^24x_2^36             96    x_1^2x_2^2x_3^2x_6^14
     12    x_1^8x_2^44              96    x_3^8x_6^12
  12+12    x_3^48                   96    x_2^3x_6^15
  72+72    x_4^24                   96    x_6^16

Robert A. Russell, Table of n, a(n) for n = 1..30
Index entries for linear recurrences with constant coefficients, order 61.

Cf. A338952 (oriented), A338953 (unoriented), A338954 (chiral), A338959 (exactly n colors), A338951 (vertices, facets), A331353 (5-cell), A331361 (8-cell edges, 16-cell faces), A331357 (16-cell edges, 8-cell faces), A338967 (120-cell, 600-cell).

Table[(8n^16+8n^18+16n^20+12n^24+2n^48+n^52+n^60)/48,{n,15}]

a(n) = (8*n^16 + 8*n^18 + 16*n^20 + 12*n^24 + 2*n^48 + n^52 + n^60) / 48.
a(n) = Sum_{j=1..Min(n,60)} A338959(n) * binomial(n,j).
a(n) = 2*A338953(n) - A338952(n) = A338952(n) - 2*A338954(n) = A338953(n) - A338954(n).

A337955 Number of achiral colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract.

Original entry on oeis.org

1, 308, 34128, 1056576, 15303750, 136236276, 865711763, 4296782848, 17656466751, 62510672500, 196174554026, 557301826368, 1456216515468, 3543525156276, 8109415963125, 17592637669376, 36414622551373

Offset: 1

Robert A. Russell, Oct 03 2020

An achiral coloring is identical to its reflection. The Schläfli symbols for the tesseract and the hyperoctahedron are {4,3,3} and {3,3,4} respectively. Both figures are regular 4-D polyhedra and they are mutually dual. There are 192 elements in the automorphism group of the tesseract that are not in its rotation group. Each involves a permutation of the axes that can be associated with a partition of 4 based on the conjugacy class of the permutation. This table shows the hyperoctahedron facet (tesseract vertex) cycle indices for each member of such a class. The first formula is obtained by averaging these cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Partition  Count  Odd Cycle Indices
        6      8x_1^2x_2^1x_4^3
       8      8x_2^2x_6^2
       3      8x_4^4
      6      2x_1^8x_2^4 + 2x_2^8 + 4x_4^4
     1      8x_2^8

Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A337952 (oriented), A128767 (unoriented), A337954 (chiral).
Other elements: A331361 (tesseract edges, hyperoctahedron faces), A331357 (tesseract faces, hyperoctahedron edges), A337958 (tesseract facets, hyperoctahedron vertices).
Other polychora: A132366(n-1) (4-simplex facets/vertices), A338951 (24-cell), A338967 (120-cell, 600-cell).
Row 4 of A325015 (orthoplex facets, orthotope vertices).

Table[(3n^12+5n^8+12n^6+28n^4)/48,{n,30}]

a(n) = n^4 * (3*n^8 + 5*n^4 + 12*n^2 + 28) / 48.
a(n) = 1*C(n,1) + 306*C(n,2) + 33207*C(n,3) + 921908*C(n,4) + 10359075*C(n,5) + 59584470*C(n,6) + 197644440*C(n,7) + 400752240*C(n,8) + 505197000*C(n,9) + 386694000*C(n,10) + 164656800*C(n,11) + 29937600*C(n,12), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A128767(n) - A337952(n) = A337952(n) - 2*A337954(n) = A128767(n) - A337954(n).

A337958 Number of achiral colorings of the 8 cubic facets of a tesseract or of the 8 vertices of a hyperoctahedron.

Original entry on oeis.org

1, 15, 126, 700, 2850, 9261, 25480, 61776, 135675, 275275, 523446, 943020, 1623076, 2686425, 4298400, 6677056, 10104885, 14942151, 21641950, 30767100, 43008966, 59208325, 80378376, 107730000, 142699375, 186978051, 242545590

Offset: 1

Robert A. Russell, Oct 03 2020

An achiral coloring is identical to its reflection. The Schläfli symbols for the tesseract and the hyperoctahedron are {4,3,3} and {3,3,4} respectively. Both figures are regular 4-D polyhedra and they are mutually dual.

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A337956 (oriented), A337956 (unoriented), A234249(n+1) (chiral).
Other elements: A331357 (hyperoctahedron edges, tesseract faces), A331361 (hyperoctahedron faces, tesseract edges), A337955 (hyperoctahedron facets, tesseract vertices).
Other polychora: A132366(n-1) (5-cell), A338951 (24-cell), A338967 (120-cell, 600-cell).
Row 4 of A325007 (orthotope facets, orthoplex vertices).

Table[Binomial[Binomial[n+1,2]+3,4] - Binomial[Binomial[n,2],4],{n,30}]

a(n) = binomial(binomial(n+1,2)+3,4) - binomial(binomial(n,2),4).
a(n) = n^2 * (n+1)^2 * (n+3) * (n^2 -2n +4) / 48.
a(n) = 1*C(n,1) + 13*C(n,2) + 84*C(n,3) + 282*C(n,4) + 465*C(n,5) + 360*C(n,6) + 105*C(n,7), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A337957(n) - A337956(n) = A337956(n) - 2 * A234249(n+1) = A337957(n) - A234249(n+1).
From Stefano Spezia, Oct 04 2020: (Start)
G.f.: x*(1 + 7*x + 34*x^2 + 56*x^3 + 8*x^4 - x^5)/(1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n > 8.
(End)

A337890 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the square faces of a regular n-dimensional orthotope (hypercube) using k or fewer colors.

Original entry on oeis.org

1, 2, 1, 3, 10, 1, 4, 55, 8200, 1, 5, 200, 9080559, 199556208371776, 1, 6, 560, 1503323520, 1370366433970979158839987, 388032967149969852957120195660938882809069568, 1

Offset: 2

Robert A. Russell, Sep 28 2020

An achiral arrangement is identical to its reflection. 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):
1    2       3          4           5             6              7 ...
1   10      55        200         560          1316           2730 ...
1 8200 9080559 1503323520 81461669375 2146080958056 34228350856910 ...

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), A337889 (chiral).
Other elements: A325015 (vertices), A337410 (edges).
Other polytopes: A337886 (simplex), A337894 (orthoplex).
Rows 2-4 are A000027, A337897, A331357.

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 + 2 x^(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; If[EvenQ[Sum[If[EvenQ[j3], r1[[j3]], r2[[j3]]], {j3,n}]],0,(per = LCM @@ Table[If[cs[[j2]] == r1[[j2]], If[0 == cs[[j2]],1,j2], 2j2], {j2,n}]; 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-1))]
array[n_, k_] := row[n] /. b -> k
Table[array[n,d+m-n], {d,6}, {n,m,d+m-1}] // Flatten

T(n,k) = 2*A337888(n,k) - A337887(n,k) = A337887(n,k) - 2*A337889(n,k) = A337888(n,k) - A337889(n,k).

cp's OEIS Frontend

A331353 Number of achiral colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

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A331354 Number of oriented colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

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A331355 Number of unoriented colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

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A331356 Number of chiral pairs of colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

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A331361 Number of achiral colorings of the edges of a tesseract with n available colors.

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A337414 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the edges of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

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A338955 Number of achiral colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

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A337955 Number of achiral colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract.

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