A331356 -id:A331356 - cp's OEIS frontend

A331360 Number of chiral pairs of colorings of the edges of a tesseract with n available colors.

0, 11158298, 4825452718593, 48038354542204960, 60632976384183154375, 20725679827848535509690, 2876113731787568888218778, 206323339833986421110604288, 8941884948181243949620880070

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}. 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 triangular faces of a regular 4-dimensional orthoplex {3,3,4} with n available colors.

Index entries for linear recurrences with constant coefficients, signature (33, -528, 5456, -40920, 237336, -1107568, 4272048, -13884156, 38567100, -92561040, 193536720, -354817320, 573166440, -818809200, 1037158320, -1166803110, 1166803110, -1037158320, 818809200, -573166440, 354817320, -193536720, 92561040, -38567100, 13884156, -4272048, 1107568, -237336, 40920, -5456, 528, -33, 1).

Cf. A331358 (oriented), A331359 (unoriented), A331361 (achiral).

Cf. A331352 (simplex), A331356 (orthoplex), A338954 (24-cell), A338966 (120-cell, 600-cell).

Table[(48n^4 - 92n^8 + 32n^12 + 3n^16 + 24n^18 - 16n^20 + n^32)/384, {n, 1, 25}]

a(n) = (48*n^4 - 92*n^8 + 32*n^12 + 3*n^16 + 24*n^18 - 16*n^20 + n^32) / 384.
a(n) = 11158298*C(n,2) + 4825419243699*C(n,3) + 48019052798280376*C(n,4) + 60392832865887732525*C(n,5) + 20362602448352682660450*C(n,6) + 2732305584323178619545720*C(n,7) + 183891356930602707657018720*C(n,8) + 7186781660616776435004792900*C(n,9) + 179941570948806294173832581400*C(n,10) + 3092495918794375534919002047600*C(n,11) + 38355721930663201428803366004000*C(n,12) + 356388702642050543223746618030400*C(n,13) + 2552262270629803579790727658473600*C(n,14) + 14398742619650630430045069333120000*C(n,15) + 65081946248235477116326789514496000*C(n,16) + 238774230958640305192143667115328000*C(n,17) + 718111905257279415879360961204608000*C(n,18) + 1783226074397879200641306482407680000*C(n,19) + 3674025240535453233675992278371840000*C(n,20) + 6297428247692138525542940292326400000*C(n,21) + 8984640042458034573900227275929600000*C(n,22) + 10651431202956156039912718487654400000*C(n,23) + 10448264801973961157855568414105600000*C(n,24) + 8418935641672774875938561280000000000*C(n,25) + 5510766716064148076659382317056000000*C(n,26) + 2882400456553496466714071801856000000*C(n,27) + 1175640370514915165746352603136000000*C(n,28) + 360177463966855890088916582400000000*C(n,29) + 77945658076061560043023564800000000*C(n,30) + 10621166594979816972895518720000000*C(n,31) + 685236554514826901477130240000000*C(n,32), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = A331358(n) - A331359(n) = (A331358(n) - A331361(n)) / 2 = A331359(n) - A331361(n).

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)

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).

A331357 Number of achiral colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

Original entry on oeis.org

1, 8200, 9080559, 1503323520, 81461669375, 2146080958056, 34228350856910, 377534786525184, 3140004522270465, 20896479183085000, 116094911796177061, 555622588428635520, 2346039511676401359, 8903083257215729960

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}. An achiral coloring is identical to its reflection. Also the number of achiral colorings of the square faces of a tesseract {4,3,3} with n available colors.
There are 192 elements in the automorphism group of the 4-dimensional orthoplex 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_2^2x_4^5
  31         8      4x_3^4x_6^2 + 4x_6^4
  22         3      8x_1^2x_2^1x_4^5
  211        6      2x_1^2x_2^11 + 2x_1^6x_2^9 + 4x_2^2x_4^5
  1111       1      4x_1^12x_2^6 + 4x_2^12

G. Royle, Partitions and Permutations
Index entries for linear recurrences with constant coefficients, signature (19, -171, 969, -3876, 11628, -27132, 50388, -75582, 92378, -92378, 75582, -50388, 27132, -11628, 3876, -969, 171, -19, 1).

Cf. A331354 (oriented), A331355 (unoriented), A331356 (chiral).
Other polychora: A331353 (5-cell), A331361 (8-cell), A338955 (24-cell), A338967 (120-cell, 600-cell).
Row 4 of A337414 (orthoplex edges, orthotope ridges) and A337890 (orthotope faces, orthoplex peaks).

Table[(8n^4 + 8n^6 + 18n^7 + 6n^8 + n^12 + 3n^13 + 3n^15 + n^18)/48, {n, 1, 25}]

a(n) = (8*n^4 + 8*n^6 + 18*n^7 + 6*n^8 + n^12 + 3*n^13 + 3*n^15 + n^18) / 48.
a(n) = C(n,1) + 8198*C(n,2) + 9055962*C(n,3) + 1467050480*C(n,4) + 74035775370*C(n,5) + 1679679306420*C(n,6) + 20864180531565*C(n,7) + 159341117375160*C(n,8) + 804216787965360*C(n,9) + 2808560520334800*C(n,10) + 6981656802951600*C(n,11) + 12540346820971200*C(n,12) + 16328843044113600*C(n,13) + 15272715797539200*C(n,14) + 10003790644848000*C(n,15) + 4357170994176000*C(n,16) + 1133753677056000*C(n,17) + 133382785536000*C(n,18), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = 2*A331355(n) - A331354(n) = A331354(n) - 2*A331356(n) = A331355(n) - A331356(n).

A331352 Number of chiral pairs of colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

Original entry on oeis.org

0, 6, 405, 7904, 76880, 486522, 2300305, 8806336, 28725192, 82626270, 214744629, 513368064, 1144198952, 2402617490, 4792612545, 9142333696, 16768783408, 29707141878, 51023629173, 85234690080, 138859666848

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}. The chiral colorings of its edges come in pairs, each the reflection of the other.

Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A331350 (oriented), A063843 (unoriented), A331353 (achiral).
Other polychora: A331360 (8-cell), A331356 (16-cell), A338954 (24-cell), A338966 (120-cell, 600-cell).
Row 4 of A327085 (simplex edges and ridges) and A337885 (simplex faces and peaks).

Table[(24n^2 - 50n^3 + 20n^4 + 15n^6 - 10n^7 + n^10)/120, {n, 1, 25}]

a(n) = (24*n^2 - 50*n^3 + 20*n^4 + 15*n^6 - 10*n^7 + n^10) / 120.
a(n) = 6*C(n,2) + 387*C(n,3) + 6320*C(n,4) + 41350*C(n,5) + 135792*C(n,6) + 246540*C(n,7) + 252000*C(n,8) + 136080*C(n,9) + 30240*C(n,10), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = A331350(n) - A063843(n) = (A331350(n) - A331353(n)) / 2 = A063843(n) - A331353(n).

A337413 Array read by descending antidiagonals: T(n,k) is the number of chiral pairs of colorings of the edges of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 3, 74, 0, 0, 15, 10704, 40927, 0, 0, 45, 345640, 731279799, 280317324, 0, 0, 105, 5062600, 732272925320, 3163614120031068, 24869435516248, 0, 0, 210, 45246810, 155180061396500, 314800331906964016128, 919853357924272852197243, 29931599129719666392, 0

Offset: 1

Robert A. Russell, Aug 26 2020

Each member of a chiral pair is a reflection, but not a rotation, of the other. 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 chiral pairs of colorings of the regular (n-2)-dimensional orthotopes (hypercubes) in a regular n-dimensional orthotope.

			Table begins with T(1,1):
0  0     0      0       0        0         0          0          0 ...
0  0     3     15      45      105       210        378        630 ...
0 74 10704 345640 5062600 45246810 288005144 1430618784 5881281480 ...
For T(2,3)=3, the chiral arrangements are AABC-AACB, ABBC-ACBB, and ABCC-ACCB.

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), A337414 (achiral).
Rows 2-4 are A050534, A337406, A331356.
Cf. A327085 (simplex edges), A337409 (orthotope edges), A325006 (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; 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,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) = A337411(n,k) - A337412(n,k) = (A337411(n,k) - A337414(n,k)) / 2 = A337412(n,k) - A337414(n,k).

A338954 Number of chiral pairs of 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

68774446614978208476646592, 5523164445430505077912054084256733211946217, 5448873034189827051926943172520863487560602391778344960, 10956401461402941741829554371669666304159415287557559324930859375

Offset: 2

Robert A. Russell, Nov 17 2020

Each member of a chiral pair is a reflection but not a rotation of the other. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual.

Robert A. Russell, Table of n, a(n) for n = 2..30
Index entries for linear recurrences with constant coefficients, order 97.

Cf. A338952 (oriented), A338953 (unoriented), A338955 (achiral), A338958 (exactly n colors), A338950 (vertices, facets), A331352 (5-cell), A331360 (8-cell edges, 16-cell faces), A331356 (16-cell edges, 8-cell faces), A338966 (120-cell, 600-cell).

Table[(96n^8+144n^12-48n^16-64n^18-192n^20-60n^24+48n^32+32n^36-5n^48+72n^50-12n^52-12n^60+n^96)/1152,{n,2,15}]

a(n) = (96*n^8 + 144*n^12 - 48*n^16 - 64*n^18 - 192*n^20 - 60*n^24 +
48*n^32 + 32*n^36 - 5*n^48 + 72*n^50 - 12*n^52 - 12*n^60 + n^96) / 1152.
a(n) = Sum_{j=2..Min(n,96)} A338958(n) * binomial(n,j).
a(n) = A338952(n) - A338953(n) = (A338952(n) - A338955(n)) / 2 = A338953(n) - A338955(n).

A337954 Number of chiral pairs of colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract.

Original entry on oeis.org

0, 94, 97974, 10700090, 390081800, 7280687610, 86121007714, 730895668104, 4816861200630, 26010740238450, 119563513291420, 481192778757834, 1732132086737234, 5669991002636870, 17101193825828700, 48029634770843680

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 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 (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Cf. A337952 (oriented), A128767 (unoriented), A337955 (achiral).
Other elements: A331360 (tesseract edges, hyperoctahedron faces), A331356 (tesseract faces, hyperoctahedron edges), A234249(n+1) (tesseract facets, hyperoctahedron vertices).
Other polychora: A000389 (4-simplex facets/vertices), A338950 (24-cell), A338966 (120-cell, 600-cell).
Row 4 of A325014 (orthoplex facets, orthotope vertices).

Table[(n^16-12n^12+12n^10+43n^8-48n^6-44n^4+48n^2)/384,{n, 30}]

a(n) = (n-1) * n^2 * (n+1) * (n^12 + n^10 - 11*n^8 + n^6 + 44 n^4 - 4 n^2 - 48) / 384.
a(n) = 94*C(n,2) + 97692*C(n,3) + 10308758*C(n,4) + 337560150*C(n,5) + 5098740090*C(n,6) + 42976836210*C(n,7) + 224685801060*C(n,8) + 775389028050*C(n,9) + 1830791421900*C(n,10) + 3007909258200*C(n,11) + 3439214024400*C(n,12) + 2685727044000*C(n,13) + 1366701336000*C(n,14) + 408648240000*C(n,15) + 54486432000*C(n,16), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
a(n) = A337952(n) - A128767(n) = (A337952(n) - A337955(n)) / 2 = A128767(n) - A337955(n).

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).

cp's OEIS Frontend

A331360 Number of chiral pairs of colorings of the edges of a tesseract with n available 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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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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A331357 Number of achiral colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

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A331352 Number of chiral pairs of colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

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A337413 Array read by descending antidiagonals: T(n,k) is the number of chiral pairs of colorings of the edges of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

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A338954 Number of chiral pairs of 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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