A338964 -id:A338964 - cp's OEIS frontend

A338965 Number of unoriented 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, 92307499707443390526727850063504, 124792381938502167392338612231208163827413085862945471, 122697712831832245109951221276235414511846772206539032522116543043328

Offset: 1

Robert A. Russell, Dec 04 2020

Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual.
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 formula is (960*n^20 + 1440*n^30 + 960*n^40 + 1200*n^50 + 2064*n^60 + 1440*n^66 + 40*n^100 + 1600*n^104 + 1200*n^114 + 624*n^120 + 60*n^150 + 1800*n^152 + 40*n^200 + 400*n^208 + 61*n^300 + 450*n^302 + 60*n^330 + n^600) / 14400.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the formula is (960 n^24 + 1440 n^36 + 960 n^48 + 1200 n^60 + 336 n^72 + 1728 n^76 + 1440 n^84 + 1640 n^120 + 1200 n^132 + 336 n^144 + 288 n^152 + 60 n^180 + 1800 n^182 + 440 n^240 + 61 n^360 + 450 n^364 + 60 n^396 + n^720) / 14400.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the formula is (960*n^40 + 1440*n^60 + 960*n^80 + 1200*n^100 + 2064*n^120 + 1440*n^128 + 40*n^200 + 1600*n^202 + 1200*n^216 + 624*n^240 + 60*n^300 + 1800*n^302 + 40*n^400 + 400*n^404 + 61*n^600 + 450*n^604 + 60*n^640 + n^1200) / 14400.

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

Cf. A338964 (oriented), A338966 (chiral), A338967 (achiral), A338981 (exactly n colors), A000389 (5-cell), A128767 (8-cell vertices, 16-cell facets), A337957(16-cell vertices, 8-cell facets), A338949 (24-cell).

Table[(960n^4+1440n^6+960n^8+1200n^10+336n^12+288n^16+1440n^17+1440n^19+40n^20+400n^22+1200n^23+336n^24+1200n^27+60n^30+1800n^31+288n^32+40n^40+400n^44+n^60+60n^61+450n^62+60n^75+n^120)/14400,{n,10}]

a(n) = (960*n^4 + 1440*n^6 + 960*n^8 + 1200*n^10 + 336*n^12 + 288*n^16 + 1440*n^17 + 1440*n^19 + 40*n^20 + 400*n^22 + 1200*n^23 + 336*n^24 + 1200*n^27 + 60*n^30 + 1800*n^31 + 288*n^32 + 40*n^40 + 400*n^44 + n^60 + 60*n^61 + 450*n^62 + 60*n^75 +*n^120) / 14400.
a(n) = Sum_{j=1..Min(n,120)} A338981(n) * binomial(n,j).
a(n) = A338964(n) - A338966(n) =(A338964(n) + A338967(n)) / 2 = A338966(n) + A338967(n).

A338966 Number of chiral pairs of 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.

Original entry on oeis.org

92307499707128546879177569498768, 124792381938502167387269721273817892704188259502965515, 122697712831832245109951209382504597654581237223625701047064169830144

Offset: 2

Robert A. Russell, Dec 04 2020

Each member of a chiral pair is a reflection but not a rotation of the other. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual.
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 formula is (960*n^20 + 1440*n^30 + 960*n^40 + 1200*n^50 - 816*n^60 - 1440*n^66 + 40*n^100 - 800*n^104 - 1200*n^114 + 624*n^120 + 60*n^150 - 1800*n^152 + 40*n^200 + 400*n^208 - 59*n^300 + 450*n^302 - 60*n^330 + n^600) / 14400.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the formula is (960 n^24 + 1440 n^36 + 960 n^48 + 1200 n^60 + 336 n^72 - 1152 n^76 - 1440 n^84 - 760 n^120 - 1200 n^132 + 336 n^144 + 288 n^152 + 60 n^180 - 1800 n^182 + 440 n^240 - 59 n^360 + 450 n^364 - 60 n^396 + n^720) / 14400.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the formula is (960*n^40 + 1440*n^60 + 960*n^80 + 1200*n^100 - 816*n^120 - 1440*n^128 + 40*n^200 - 800*n^202 - 1200*n^216 + 624*n^240 + 60*n^300 - 1800*n^302 + 40*n^400 + 400*n^404 - 59*n^600 + 450*n^604 - 60*n^640 + n^1200) / 14400.

Robert A. Russell, Table of n, a(n) for n = 2..30
Index entries for linear recurrences with constant coefficients, order 121.

Cf. A338964 (oriented), A338965 (unoriented), A338967 (achiral), A338982 (exactly n colors), A000389 (5-cell), A337954 (8-cell vertices, 16-cell facets), A234249 (16-cell vertices, 8-cell facets), A338950 (24-cell).

Table[(960n^4 +1440n^6 +960n^8 +1200n^10 +336n^12 +288n^16 -1440n^17 -1440n^19 +40n^20 +400n^22 -1200n^23 +336n^24 -1200n^27 +60n^30 -1800n^31 +288n^32 +40n^40 +400n^44 +n^60 -60n^61 +450n^62 -60n^75 +n^120)/14400, {n,2,10}]

a(n) = (960*n^4 + 1440*n^6 + 960*n^8 + 1200*n^10 + 336*n^12 + 288*n^16 - 1440*n^17 - 1440*n^19 + 40*n^20 + 400*n^22 - 1200*n^23 + 336*n^24 - 1200*n^27 + 60*n^30 - 1800*n^31 + 288*n^32 + 40*n^40 + 400*n^44 + n^60 - 60*n^61 + 450*n^62 - 60*n^75 + n^120) / 14400.
a(n) = Sum_{j=2..Min(n,120)} A338982(n) * binomial(n,j).
a(n) = A338964(n) - A338965(n) =(A338964(n) - A338967(n)) / 2 = A338965(n) - A338967(n).

A338967 Number of achiral 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.

Original entry on oeis.org

1, 314843647550280564736, 5068890957390271123224826359979956, 11893730816857265534982913331475052373213184, 220581496716947452381892465686737251285705566406250

Offset: 1

Robert A. Russell, Dec 04 2020

An achiral coloring is identical to its reflection. 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 automorphism group of the 120-cell that are not in its rotation group. They divide into 9 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   Odd Cycle Indices      Count   Odd Cycle Indices
     60   x_1^30x_2^45            1200   x_1^2x_2^2x_6^19
     60   x_1^2x_2^59          720+720   x_2^5x_5^6x_10^8
   1800   x_2^2x_4^29          720+720   x_1^2x_2^4x_10^11
   1200   x_2^3x_3^10x_6^14
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    Odd Cycle Indices        Count    Odd Cycle Indices
    60     x_1^60x_2^270             1200     x_2^6x_6^98
    60     x_2^300                720+720     x_5^12x_10^54
  1800     x_1^2x_2^1x_4^149      720+720     x_10^60
  1200     x_2^6x_3^20x_6^88
The formula is (24*n^60 + 24*n^66 + 20*n^104 + 20*n^114 + 30*n^152 + n^300 + n^330) / 120.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the cycle indices are:
  Count   Odd Cycle Indices      Count   Odd Cycle Indices
     60   x_1^72x_2^324           1200   x_6^120
     60   x_2^360              720+720   x_1^2x_2^4x_5^14x_10^64
   1800   x_2^4x_4^178         720+720   x_2^5x_10^71
   1200   x_3^24x_6^108
The formula is (24*n^76 + 24*n^84 + 20*n^120 + 20*n^132 + 30*n^182 + n^360 + n^396) / 120.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the cycle indices are:
  Count   Odd Cycle Indices          Count   Odd Cycle Indices
     60   x_1^80x_2^560               1200   x_2^3x_6^199
     60   x_2^600                  720+720   x_5^16x_10^112
   1800   x_2^4x_4^298             720+720   x_10^120
   1200   x_1^2x_2^2x_3^26x_6^186
The formula is (24*n^120 + 24*n^128 + 20*n^202 + 20*n^216 + 30*n^302 + n^600 + n^640) / 120.

Robert A. Russell, Table of n, a(n) for n = 1..30
Index entries for linear recurrences with constant coefficients, order 76.

Cf. A338964 (oriented), A338965 (unoriented), A338966 (chiral), A338983 (exactly n colors), A132366 (5-cell), A337955 (8-cell vertices, 16-cell facets), A337958(16-cell vertices, 8-cell facets), A338951 (24-cell).

Table[(24n^17+24n^19+20n^23+20n^27+30n^31+n^61+n^75)/120,{n,10}]

a(n)=(24*n^17+24*n^19+20*n^23+20*n^27+30*n^31+n^61+n^75)/120 \\ Charles R Greathouse IV, Jul 05 2024

a(n) = (24*n^17 + 24*n^19 + 20*n^23 + 20*n^27 + 30*n^31 + n^61 + n^75) / 120.
a(n) = Sum_{j=1..Min(n,75)} A338983(n) * binomial(n,j).
a(n) = 2*A338965(n) - A338964(n) =(A338964(n) - 2*A338966(n)) / 2 = A338965(n) - A338966(n).

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).

A331358 Number of oriented colorings of the edges of a tesseract with n available colors.

Original entry on oeis.org

1, 22409620, 9651199594275, 96076801068337216, 121265960728368199375, 41451359960612034644436, 5752227470227262715982165, 412646679764073090531066880, 17883769897375781105874361581

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}. 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 triangular faces of a regular 4-dimensional orthoplex {3,3,4} with n available colors.
There are 192 elements in the rotation group of the tesseract. 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_8^4
  31         8      4x_1^2x_3^10 + 4x_2^1x_6^5
  22         3      4x_2^16 + 4x_4^8
  211        6      4x_1^4x_2^14 + 4x_4^8
  1111       1      x_1^32 + 7x_2^16

G. Royle, Partitions and Permutations
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. A331359 (unoriented), A331360 (chiral), A331361 (achiral).

Cf. A331350 (simplex), A331354 (orthoplex), A338952 (24-cell), A338964 (120-cell, 600-cell).

Table[(48n^4 + 32n^6 + 36n^8 + 32n^12 + 19n^16 + 24n^18 + n^32)/192, {n, 1, 25}]

a(n) = (48*n^4 + 32*n^6 + 36*n^8 + 32*n^12 + 19*n^16 + 24*n^18 + n^32) / 192.
a(n) = C(n,1) + 22409618*C(n,2) + 9651132365418*C(n,3) + 96038196404417832*C(n,4) + 120785673234798359850*C(n,5) + 40725205155234194765220*C(n,6) + 5464611173328028329053040*C(n,7) + 367782713912186945387883840*C(n,8) + 14373563321596798877701789800*C(n,9) + 359883141899402124632485810800*C(n,10) + 6184991837595074128351177096800*C(n,11) + 76711443861342809436413801659200*C(n,12) + 712777405284132776184971034460800*C(n,13) + 5104524541259652946568783959507200*C(n,14) + 28797485239301310151711610238720000*C(n,15) + 130163892496470993203014850790912000*C(n,16) + 477548461917280632356433595575936000*C(n,17) + 1436223810514558840121822575516416000*C(n,18) + 3566452148795758403208660387955200000*C(n,19) + 7348050481070906467554726390758400000*C(n,20) + 12594856495384277051085880584652800000*C(n,21) + 17969280084916069147800454551859200000*C(n,22) + 21302862405912312079825436975308800000*C(n,23) + 20896529603947922315711136828211200000*C(n,24) + 16837871283345549751877122560000000000*C(n,25) + 11021533432128296153318764634112000000*C(n,26) + 5764800913106992933428143603712000000*C(n,27) + 2351280741029830331492705206272000000*C(n,28) + 720354927933711780177833164800000000*C(n,29) + 155891316152123120086047129600000000*C(n,30) + 21242333189959633945791037440000000*C(n,31) + 1370473109029653802954260480000000*C(n,32), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = A331359(n) + A331360(n) = 2*A331359(n) - A331361(n) = 2*A331360(n) + A331361(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).

A331350 Number of oriented colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

Original entry on oeis.org

1, 40, 1197, 18592, 166885, 1019880, 4738153, 17962624, 58248153, 166920040, 432738229, 1032709536, 2298857821, 4822806184, 9613704465, 18329410048, 33605960689, 59516325288, 102196242685, 170682720160, 278019522837

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}. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.
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 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  Even Cycle Indices
  5          24     x_5^2
  311        20     x_1^1x_3^3
  221        15     x_1^2x_2^4
  11111      1      x_1^10

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

Cf. A063843 (unoriented), A331352 (chiral), A331353 (achiral).
Other polychora: A331358 (8-cell), A331354 (16-cell), A338952 (24-cell), A338964 (120-cell, 600-cell).
Row 4 of A327083 (simplex edges and facets) and A337883 (simplex faces and peaks).

Table[(24n^2 + 20n^4 + 15n^6 + n^10)/60, {n, 1, 25}]

a(n) = (24*n^2 + 20*n^4 + 15*n^6 + n^10) / 60.
a(n) = C(n,1) + 38*C(n,2) + 1080*C(n,3) + 14040*C(n,4) + 85500*C(n,5) + 274104*C(n,6) + 493920*C(n,7) + 504000*C(n,8) + 272160*C(n,9) + 60480*C(n,10), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = A063843(n) + A331352(n) = 2*A063843(n) - A331353(n) = 2*A331352(n) + A331353(n).

A338952 Number of oriented 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, 137548893254081168086800768, 11046328890861011039111168376671536861388643, 10897746068379654103881579020805286236644252743361724416

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 has 24 octahedral facets. 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 edge (or face) 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_1^96              6+6+36+36   x_4^24
      72   x_1^4x_2^46                32   x_2^3x_6^15
    1+18   x_2^48                 8+8+32   x_6^16
      32   x_1^6x_3^30             72+72   x_8^12
  8+8+32   x_3^32                  48+48   x_12^8

Robert A. Russell, Table of n, a(n) for n = 1..30
Index entries for linear recurrences with constant coefficients, order 97.

Cf. A338953 (unoriented), A338954 (chiral), A338955 (achiral), A338956 (exactly n colors), A338948 (vertices, facets), A331350 (5-cell), A331358 (8-cell edges, 16-cell faces), A331354 (16-cell edges, 8-cell faces), A338964 (120-cell, 600-cell).

Table[(96n^8+144n^12+48n^16+32n^18+84n^24+48n^32+32n^36+19n^48+72n^50+n^96)/576,{n,15}]

a(n) = (96*n^8 + 144*n^12 + 48*n^16 + 32*n^18 + 84*n^24 + 48*n^32 + 32*n^36 + 19*n^48 + 72*n^50 + n^96) / 576.
a(n) = Sum_{j=1..Min(n,96)} A338956(n) * binomial(n,j).
a(n) = A338953(n) + A338954(n) = 2*A338953(n) - A338955(n) = 2*A338954(n) + A338955(n).

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).

A337952 Number of oriented colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract.

Original entry on oeis.org

1, 496, 230076, 22456756, 795467350, 14697611496, 173107727191, 1466088119056, 9651378868011, 52083991149400, 239323201136866, 962942859342036, 3465720389989936, 11343525530430016, 34210497067620525

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. There are 192 elements in the rotation group of the tesseract. 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 cycle indices for each rotation by partition. 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  Even Cycle Indices
        6      8x_8^2
       8      4x_1^4x_3^4 + 4x_2^2x_6^2
       3      4x_1^4x_2^6 + 4x_4^4
      6      4x_2^8 + 4x_4^4
     1      x_1^16 + 7x_2^8

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. A128767 (unoriented), A337954 (chiral), A337955 (achiral).
Other elements: A331358 (tesseract edges, hyperoctahedron faces), A331354 (tesseract faces, hyperoctahedron edges), A337956 (tesseract facets, hyperoctahedron vertices).
Other polychora: A337895 (4-simplex facets/vertices), A338948 (24-cell), A338964 (120-cell, 600-cell).
Row 4 of A325012 (orthoplex facets, orthotope vertices).

Table[(n^16+12n^10+63n^8+68n^4+48n^2)/192,{n,30}]

a(n) = n^2 * (n^14 + 12*n^8 + 63*n^6 + 68*n^2 + 48) / 192.
a(n) = 1*C(n,1) + 494*C(n,2) + 228591*C(n,3) + 21539424*C(n,4) + 685479375*C(n,5) + 10257064650*C(n,6) + 86151316860*C(n,7) + 449772354360*C(n,8) + 1551283253100*C(n,9) + 3661969537800*C(n,10) + 6015983173200*C(n,11) + 6878457986400*C(n,12) + 5371454088000*C(n,13) + 2733402672000*C(n,14) + 817296480000*C(n,15) + 108972864000*C(n,16), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
a(n) = A128767(n) + A337954(n) = 2*A128767(n) - A337955(n) = 2*A337954(n) + A337955(n).

cp's OEIS Frontend

A338965 Number of unoriented 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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A338966 Number of chiral pairs of 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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A338967 Number of achiral 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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A331354 Number of oriented colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

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A331358 Number of oriented colorings of the edges of a tesseract with n available 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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A331350 Number of oriented colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

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A338952 Number of oriented 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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