A337952 -id:A337952 - cp's OEIS frontend

A338964 Number of oriented colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using subsets of a set of n colors.

1, 184614999414571937405905419562272, 249584763877004334779608333505026056531601345365910986, 245395425663664490219902430658740012166428009430164733569180712873472

Offset: 1

Robert A. Russell, Dec 04 2020

Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual. There are 7200 elements in the rotation group of the 120-cell. They divide into 41 conjugacy classes. The first formula is obtained by averaging the vertex (or facet) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
       Count   Even Cycle Indices      Count   Even Cycle Indices
           1   x_1^120                   400   x_2^3x_6^19
         450   x_1^4x_2^58             20+20   x_6^20
           1   x_2^60                144+144   x_2^5x_10^11
         400   x_1^6x_3^38        4*12+2*144   x_10^12
       20+20   x_3^40                600+600   x_12^10
     144+144   x_1^10x_5^22            4*240   x_15^8
       30+30   x_4^30                  4*360   x_20^6
  4*12+2*144   x_5^24                  4*240   x_30^4
Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide formulas here.
For the 600 facets of the 600-cell (vertices of the 120-cell), the cycle indices are:
       Count   Even Cycle Indices      Count   Even Cycle Indices
           1   x_1^600                   400   x_2^6x_6^98
         450   x_1^4x_2^298            20+20   x_6^100
           1   x_2^300            4*12+4*144   x_10^60
         400   x_1^12x_3^196         600+600   x_12^50
       20+20   x_3^200                 4*240   x_15^40
       30+30   x_4^150                 4*360   x_20^30
  4*12+4*144   x_5^120                 4*240   x_30^20
The formula is (960*n^20 + 1440*n^30 + 960*n^40 + 1200*n^50 + 624*n^60 + 40*n^100 + 400*n^104 + 624*n^120 + 60*n^150 + 40*n^200 + 400*n^208 + n^300 + 450*n^302 + n^600) / 7200.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the cycle indices are:
       Count   Even Cycle Indices      Count   Even Cycle Indices
           1   x_1^720              2*20+400   x_6^120
         450   x_1^8x_2^356          144+144   x_2^5x_10^71
           1   x_2^360            4*12+2*144   x_10^72
    2*20+400   x_3^240               600+600   x_12^60
       30+30   x_4^180                 4*240   x_15^48
     144+144   x_1^10x_5^142           4*360   x_20^36
  4*12+2*144   x_5^144                 4*240   x_30^24
The formula is (960*n^24 + 1440*n^36 + 960*n^48 + 1200*n^60 + 336*n^72 + 288*n^76 + 440*n^120 + 336*n^144 + 288*n^152 + 60*n^180 + 440*n^240 + n^360 + 450*n^364 + n^720) / 7200.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the cycle indices are:
       Count   Even Cycle Indices      Count   Even Cycle Indices
           1   x_1^1200                  400   x_2^3x_6^199
         450   x_1^8x_2^596            20+20   x_6^200
           1   x_2^600            4*12+4*144   x_10^120
         400   x_1^6x_3^398          600+600   x_12^100
       20+20   x_3^400                 4*240   x_15^80
       30+30   x_4^300                 4*360   x_20^60
  4*12+4*144   x_5^240                 4*240   x_30^40
The formula is (960*n^40 + 1440*n^60 + 960*n^80 + 1200*n^100 + 624*n^120 + 40*n^200 + 400*n^202 + 624*n^240 + 60*n^300 + 40*n^400 + 400*n^404 + n^600 + 450*n^604 + n^1200) / 7200.

Robert A. Russell, Table of n, a(n) for n = 1..30
Index entries for linear recurrences with constant coefficients, order 121.

Cf. A338965 (unoriented), A338966 (chiral), A338967 (achiral), A338980 (exactly n colors), A337895 (5-cell), A337952 (8-cell vertices, 16-cell facets), A337956(16-cell vertices, 8-cell facets), A338948 (24-cell).

Table[(960n^4+1440n^6+960n^8+1200n^10+336n^12+288n^16+40n^20+400n^22+336n^24+60n^30+288n^32+40n^40+400n^44 +n^60+450n^62 +n^120)/7200,{n,10}]

a(n)=(960*n^4+1440*n^6+960*n^8+1200*n^10+336*n^12+288*n^16+40*n^20+400*n^22+336*n^24+60*n^30+288*n^32+40*n^40+400*n^44+n^60+450*n^62+n^120)/7200 \\ Charles R Greathouse IV, Jul 05 2024

a(n) = (960*n^4 + 1440*n^6 + 960*n^8 + 1200*n^10 + 336*n^12 + 288*n^16 + 40*n^20 + 400*n^22 + 336*n^24 + 60* n^30 + 288*n^32 + 40*n^40 + 400*n^44 + n^60 + 450*n^62 + n^120) / 7200.
a(n) = Sum_{j=1..Min(n,120)} A338980(n) * binomial(n,j).
a(n) = A338965(n) + A338966(n) = 2*A338965(n) - A338967(n) = 2*A338966(n) + A338967(n).

A338948 Number of oriented colorings of the 24 octahedral facets (or 24 vertices) of the 4-D 24-cell using subsets of a set of n colors.

Original entry on oeis.org

1, 30968, 490710246, 488689596200, 103480643539150, 8226360697111116, 332606338581801018, 8198553131754111456, 138483409168412322525, 1736111115543474313600, 17100230356306262961356, 138015359782116886130568

Offset: 1

Robert A. Russell, Nov 17 2020

Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbol of the 24-cell is {3,4,3}. It is self-dual. There are 576 elements in the rotation group of the 24-cell. They divide into 20 conjugacy classes. The first formula is obtained by averaging the vertex (or facet) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
   Count   Even Cycle Indices      Count   Even Cycle Indices
       1   x_1^24                     36   x_2^2x_4^5
      18   x_1^4x_2^10                32   x_2^3x_6^3
      72   x_1^2x_2^11               6+6   x_4^6
       1   x_2^12                 8+8+32   x_6^4
      32   x_1^6x_3^6              72+72   x_8^3
      36   x_1^4x_4^5              48+48   x_12^2
  8+8+32   x_3^8

Robert A. Russell, Table of n, a(n) for n = 1..30
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. A338949 (unoriented), A338950 (chiral), A338951 (achiral), A338952 (edges, faces), A337895 (5-cell), A337952 (8-cell vertices, 16-cell facets), A337956 (16-cell vertices, 8-cell facets), A338964 (120-cell, 600-cell).

Table[(96n^2+144n^3+48n^4+44n^6+36n^7+48n^8+36n^9+33n^12+72n^13+18n^14+n^24)/576,{n,15}]

a(n) = (96*n^2 + 144*n^3 + 48*n^4 + 44*n^6 + 36*n^7 + 48*n^8 + 36*n^9 + 33*n^12 + 72*n^13 + 18*n^14 + n^24) / 576.
a(n) = 1*C(n,1) + 30966*C(n,2) + 490617345*C(n,3) + 486726941020*C(n,4) + 101042102350935*C(n,5) + 7612797366078810*C(n,6) + 277177820254686645*C(n,7) + 5762279787373449480*C(n,8) + 75992221900428179850*C(n,9) + 682000715348622816300*C(n,10) + 4372841482811937689400*C(n,11) + 20731958137729666674000*C(n,12) + 74473828855001644068000*C(n,13) + 206154110634594043521600*C(n,14) + 444564429725793817440000*C(n,15) + 751083930907369899840000*C(n,16) + 994782360855398955840000*C(n,17) + 1027991414661948696960000*C(n,18) + 819571017352669021440000*C(n,19) + 494068244672052610560000*C(n,20) + 217722453472796912640000*C(n,21) + 66156028946382735360000*C(n,22) + 12387424687382384640000*C(n,23) + 1077167364120207360000*C(n,24), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
a(n) = A338949(n) + A338950(n) = 2*A338949(n) - A338951(n) = 2*A338950(n) + A338951(n).

A128767 Number of inequivalent n-colorings of the 4D hypercube under the full orthogonal group of the cube (of order 2^4*4! = 384).

Original entry on oeis.org

1, 402, 132102, 11756666, 405385550, 7416923886, 86986719477, 735192450952, 4834517667381, 26073250910950, 119759687845446, 481750080584202, 1733588303252702, 5673534527793146, 17109303241791825, 48047227408513056

Offset: 1

Ricardo Perez-Aguila (ricardo.perez.aguila(AT)gmail.com), Apr 04 2007

I assume this refers to colorings of the vertices of the cube. - N. J. A. Sloane, Apr 06 2007

Number of unoriented colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract (4-D cube) using up to n colors. Each chiral pair is counted as one when enumerating unoriented arrangements. 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. - Robert A. Russell, Oct 03 2020

			a(2)=402 because there are 402 inequivalent 2-colorings of the 4D hypercube.

Banks, D. C.; Linton, S. A. & Stockmeyer, P. K. Counting Cases in Substitope Algorithms. IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 4, pp. 371-384. 2004.
Perez-Aguila, Ricardo. Enumerating the Configurations in the n-Dimensional Orthogonal Polytopes Through Polya's Counting and A Concise Representation. Proceedings of the 3rd International Conference on Electrical and Electronics Engineering and XII Conference on Electrical Engineering ICEEE and CIE 2006, pp. 63-66.
Polya, G. & Read R. C. Combinatorial Enumeration of Groups, Graphs and Chemical Compounds. Springer-Verlag, 1987.

Banks, D. C.; Linton, S. A. & Stockmeyer, P. K., Counting Cases in Substitope Algorithms, IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 4, pp. 371-384. 2004.
Perez-Aguila, Ricardo, Orthogonal Polytopes: Study and Application, PhD Thesis. Universidad de las Americas, Puebla. November, 2006.
Perez-Aguila, Ricardo, Enumerating the Configurations in the n-Dimensional Orthogonal Polytopes Through Polya's Counting and A Concise Representation, Proceedings of the 3rd International Conference on Electrical and Electronics Engineering and XII Conference on Electrical Engineering ICEEE and CIE 2006, pp. 63-66.
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), A337954 (chiral), A337955 (achiral).
Other elements: A331359 (tesseract edges, hyperoctahedron faces), A331355 (tesseract faces, hyperoctahedron edges), A337957 (tesseract facets, hyperoctahedron vertices).
Other polychora: A000389(n+4) (4-simplex facets/vertices), A338949 (24-cell), A338965 (120-cell, 600-cell).
Row 4 of A325013 (orthoplex facets, orthotope vertices).

Table[(1/384)*( 48*n^2 + 180*n^4 + 48*n^6 + 83*n^8 + 12*n^10 + 12*n^12 + n^16),{n,30}]

a(n) = (1/384)*(48*n^2 + 180*n^4 + 48*n^6 + 83*n^8 + 12*n^10 + 12*n^12 + n^16)
G.f.: -x*(x +1)*(x^14 +384*x^13 +125020*x^12 +9439904*x^11 +213777216*x^10 +1821620108*x^9 +6527222787*x^8 +10098845160*x^7 +6527222787*x^6 +1821620108*x^5 +213777216*x^4 +9439904*x^3 +125020*x^2 +384*x +1) / (x -1)^17. [Colin Barker, Dec 04 2012]
From Robert A. Russell, Oct 03 2020: (Start)
a(n) = 1*C(n,1) + 400*C(n,2) + 130899*C(n,3) + 11230666*C(n,4) + 347919225*C(n,5) + 5158324560*C(n,6) + 43174480650*C(n,7) + 225086553300*C(n,8) + 775894225050*C(n,9) + 1831178115900*C(n,10) + 3008073915000*C(n,11) + 3439243962000*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 unoriented colorings using exactly k colors.
a(n) = A337952(n) - A337954(n) = (A337952(n) + A337955(n)) / 2 = A337954(n) + A337955(n).
(End)

A337895 Number of oriented colorings of the tetrahedral facets (or vertices) of a regular 4-dimensional simplex using n or fewer colors.

Original entry on oeis.org

1, 6, 21, 56, 127, 258, 483, 848, 1413, 2254, 3465, 5160, 7475, 10570, 14631, 19872, 26537, 34902, 45277, 58008, 73479, 92114, 114379, 140784, 171885, 208286, 250641, 299656, 356091, 420762, 494543, 578368, 673233

Offset: 1

Robert A. Russell, Sep 28 2020

nonn

Each chiral pair is counted as two when enumerating oriented arrangements. Also called a 5-cell or pentachoron. The Schläfli symbol is {3,3,3}, and it has 5 tetrahedral facets (vertices).
There are 60 elements in the rotation group of the 4-dimensional simplex. Each is an even permutation of the vertices and can be associated with a partition of 5 based on the conjugacy class 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  Even Cycle Indices
  5          24     x_5^1
  311        20     x_1^2x_3^1
  221        15     x_1^1x_2^2
  11111      1      x_1^5

			For a(2)=6, the colors are AAAAA, AAAAB, AAABB, AABBB, ABBBB, and BBBBB.

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

Cf. A000389(n+4) (unoriented), A000389(chiral), A132366(n-1) (achiral), A331350 (edges, faces), A337952 (8-cell vertices, 16-cell facets), A337956(16-cell vertices, 8-cell facets), A338948 (24-cell), A338964 (120-cell, 600-cell).

Row 4 of A324999 (oriented colorings of facets or vertices of an n-simplex).

Table[n (24 + 35 n^2 + n^4)/60, {n, 40}]

a(n) = n * (24 + 35*n^2 + n^4) / 60.
a(n) = binomial[4+n,5] + binomial[n,5].
a(n) = 1*C(n,1) + 4*C(n,2) + 6*C(n,3) + 4*C(n,4) + 2*C(n,5), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
a(n) = A000389(n+4) + A000389(n) = 2*A000389(n+4) - A132366(n-1) = 2*A000389(n) + A132366(n-1).

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).

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).

A337956 Number of oriented 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, 730, 3270, 11991, 37450, 102726, 253485, 573265, 1205556, 2384460, 4475926, 8031765, 13858860, 23106196, 37372545, 58837851, 90421570, 135971430, 200486286, 290376955, 413769126, 580852650, 804281725

Offset: 1

Robert A. Russell, Oct 03 2020

Each chiral pair is counted as two when enumerating oriented arrangements. 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 (9,-36,84,-126,126,-84,36,-9,1).

Cf. A337957 (unoriented), A234249(n+1) (chiral), A337958 (achiral).
Other elements: A331354 (hyperoctahedron edges, tesseract faces), A331358 (hyperoctahedron faces, tesseract edges), A337952 (hyperoctahedron facets, tesseract vertices).
Other polychora: A337895 (5-cell), A338948 (24-cell), A338964 (120-cell, 600-cell).
Row 4 of A325004 (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 * (n+1) * (n^6 - n^5 + 7*n^4 + 29*n^3 + 16*n^2 - 4*n + 48) / 192.
a(n) = 1*C(n,1) + 13*C(n,2) + 84*C(n,3) + 312*C(n,4) + 735*C(n,5) + 1020*C(n,6) + 735*C(n,7) + 210*C(n,8), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
a(n) = A337957(n) + A234249(n+1) = 2*A337957(n) - A337958(n) = 2*A234249(n+1) + A337958(n).
From Stefano Spezia, Oct 04 2020: (Start)
G.f.: x*(1 + 6*x + 27*x^2 + 52*x^3 + 102*x^4 + 21*x^5 + x^6)/(1 - x)^9.
a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-8) for n > 8.
(End)

A338980 Number of oriented colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using exactly n colors.

Original entry on oeis.org

0, 1, 184614999414571937405905419562270, 249584763877004334779054488506782340719383629107224173, 245395425663663491880846922641400894840783985813370231599231766603156

Offset: 0

Robert A. Russell, Dec 13 2020

Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual. For n>120, a(n) = 0.
Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide generating functions here using bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k.
For the 600 facets of the 600-cell (vertices of the 120-cell), the generating function is 2*bp(20)/15 + bp(30)/5 + 2*bp(40)/15 + bp(50)/6 + 13*bp(60)/150 + bp(100)/180 + bp(104)/18 + 13*bp(120)/150 + bp(150)/120 + bp(200)/180 + bp(208)/18 + bp(300)/7200 + bp(302)/16 + bp(600)/7200.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the generating function is 2*bp(24)/15 + bp(36)/5 + 2*bp(48)/15 + bp(60)/6 + 7*bp(72)/150 + bp(76)/25 + 11*bp(120)/180 + 7*bp(144)/150 + bp(152)/25 + bp(180)/120 + 11*bp(240)/180 + bp(360)/7200 + bp(364)/16 + bp(720)/7200.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the generating function is 2*bp(40)/15 + bp(60)/5 + 2*bp(80)/15 + bp(100)/6 + 13*bp(120)/150 + bp(200)/180 + bp(202)/18 + 13*bp(240)/150 + bp(300)/120 + bp(400)/180 + bp(404)/18 + bp(600)/7200 + bp(604)/16 + bp(1200)/7200.

Robert A. Russell, Table of n, a(n) for n = 0..120

Cf. A338981 (unoriented), A338982 (chiral), A338983 (achiral), A338964 (up to n colors), A337895 (5-cell), A337952 (8-cell vertices, 16-cell facets), A337956 (16-cell vertices, 8-cell facets), A338948 (24-cell).

bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, j}] (*binomial series*)
CoefficientList[2bp[4]/15+bp[6]/5+2bp[8]/15+bp[10]/6+7bp[12]/150+bp[16]/25+bp[20]/180+bp[22]/18+7bp[24]/150+bp[30]/120+bp[32]/25+bp[40]/180+bp[44]/18+bp[60]/7200+bp[62]/16+bp[120]/7200,x]

A338964(n) = Sum_{j=1..Min(n,120)} a(n) * binomial(n,j).
a(n) = A338981(n) + A338982(n) = 2*A338981(n) - A338983(n) = 2*A338982(n) + A338983(n).
G.f.: 2*bp(4)/15 + bp(6)/5 + 2*bp(8)/15 + bp(10)/6 + 7*bp(12)/150 + bp(16)/25 + bp(20)/180 + bp(22)/18 + 7*bp(24)/150 + bp(30)/120 + bp(32)/25 + bp(40)/180 + bp(44)/18 + bp(60)/7200 + bp(62)/16 + bp(120)/7200, where bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k and S2(j,k) is the Stirling subset number, A008277.

cp's OEIS Frontend

A338964 Number of oriented colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using subsets of a set of n colors.

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A338948 Number of oriented colorings of the 24 octahedral facets (or 24 vertices) of the 4-D 24-cell using subsets of a set of n colors.

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A128767 Number of inequivalent n-colorings of the 4D hypercube under the full orthogonal group of the cube (of order 2^4*4! = 384).

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A337895 Number of oriented colorings of the tetrahedral facets (or vertices) of a regular 4-dimensional simplex using n or fewer colors.

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

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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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A337956 Number of oriented colorings of the 8 cubic facets of a tesseract or of the 8 vertices of a hyperoctahedron.

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A338980 Number of oriented colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using exactly n colors.

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