A338965 -id:A338965 - cp's OEIS frontend

A000389 Binomial coefficients C(n,5).

0, 0, 0, 0, 0, 1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334, 33649, 42504, 53130, 65780, 80730, 98280, 118755, 142506, 169911, 201376, 237336, 278256, 324632, 376992, 435897, 501942, 575757, 658008, 749398

Offset: 0

N. J. A. Sloane

a(n+4) is the number of inequivalent ways of coloring the vertices of a regular 4-dimensional simplex with n colors, under the full symmetric group S_5 of order 120, with cycle index (x1^5 + 10*x1^3*x2 + 20*x1^2*x3 + 15*x1*x2^2 + 30*x1*x4 + 20*x2*x3 + 24*x5)/120.
Figurate numbers based on 5-dimensional regular simplex. According to Hyun Kwang Kim, it appears that every nonnegative integer can be represented as the sum of g = 10 of these 5-simplex(n) numbers (compared with g=3 for triangular numbers, g=5 for tetrahedral numbers and g=8 for pentatope numbers). - Jonathan Vos Post, Nov 28 2004
The convolution of the nonnegative integers (A001477) with the tetrahedral numbers (A000292), which are the convolution of the nonnegative integers with themselves (making appropriate allowances for offsets of all sequences). - Graeme McRae, Jun 07 2006
a(n) is the number of terms in the expansion of (a_1 + a_2 + a_3 + a_4 + a_5 + a_6)^n. - Sergio Falcon, Feb 12 2007
Product of five consecutive numbers divided by 120. - Artur Jasinski, Dec 02 2007
Equals binomial transform of [1, 5, 10, 10, 5, 1, 0, 0, 0, ...]. - Gary W. Adamson, Feb 02 2009
Equals INVERTi transform of A099242 (1, 7, 34, 153, 686, 3088, ...). - Gary W. Adamson, Feb 02 2009
For a team with n basketball players (n>=5), this sequence is the number of possible starting lineups of 5 players, without regard to the positions (center, forward, guard) of the players. - Mohammad K. Azarian, Sep 10 2009
a(n) is the number of different patterns, regardless of order, when throwing (n-5) 6-sided dice. For example, one die can display the 6 numbers 1, 2, ..., 6; two dice can display the 21 digit-pairs 11, 12, ..., 56, 66. - Ian Duff, Nov 16 2009
Sum of the first n pentatope numbers (1, 5, 15, 35, 70, 126, 210, ...), see A000332. - Paul Muljadi, Dec 16 2009
Sum_{n>=0} a(n)/n! = e/120. Sum_{n>=4} a(n)/(n-4)! = 501*e/120. See A067764 regarding the second ratio. - Richard R. Forberg, Dec 26 2013
For a set of integers {1,2,...,n}, a(n) is the sum of the 2 smallest elements of each subset with 4 elements, which is 3*C(n+1,5) (for n>=4), hence a(n) = 3*C(n+1,5) = 3*A000389(n+1). - Serhat Bulut, Mar 11 2015
a(n) = fallfac(n,5)/5! is also the number of independent components of an antisymmetric tensor of rank 5 and dimension n >= 1. Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
Number of compositions (ordered partitions) of n+1 into exactly 6 parts. - Juergen Will, Jan 02 2016
Number of weak compositions (ordered weak partitions) of n-5 into exactly 6 parts. - Juergen Will, Jan 02 2016
a(n+3) could be the general number of all geodetic graphs of diameter n>=2 homeomorphic to the Petersen Graph. - Carlos Enrique Frasser, May 24 2018
From Robert A. Russell, Dec 24 2020: (Start)
a(n) is the number of chiral pairs of colorings of the 5 tetrahedral facets (or vertices) of the regular 4-D simplex (5-cell, pentachoron, Schläfli symbol {3,3,3}) using subsets of a set of n colors. Each member of a chiral pair is a reflection but not a rotation of the other.
a(n+4) is the number of unoriented colorings of the 5 tetrahedral facets of the regular 4-D simplex (5-cell, pentachoron) using subsets of a set of n colors. Each chiral pair is counted as one when enumerating unoriented arrangements. (End)
For integer m and positive integer r >= 4, the polynomial a(n) + a(n + m) + a(n + 2*m) + ... + a(n + r*m) in n has its zeros on the vertical line Re(n) = (4 - r*m)/2 in the complex plane. - Peter Bala, Jun 02 2024

			G.f. = x^5 + 6*x^6 + 21*x^7 + 56*x^8 + 126*x^9 + 252*x^10 + 462*x^11 + ...
For A={1,2,3,4}, the only subset with 4 elements is {1,2,3,4}; sum of 2 minimum elements of this subset: a(4) = 1+2 = 3 = 3*C(4+1,5).
For A={1,2,3,4,5}, the subsets with 4 elements are {1,2,3,4}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}; sum of 2 smallest elements of each subset: a(5) = (1+2)+(1+2)+(1+2)+(1+3)+(2+3) = 18 = 3*C(5+1,5). - _Serhat Bulut_, Mar 11 2015
a(6) = 6 from the six independent components of an antisymmetric tensor A of rank 5 and dimension 6: A(1,2,3,4,5), A(1,2,3,4,6), A(1,2,3,5,6), A(1,2,4,5,6), A(1,3,4,5,6), A(2,3,4,5,6). See the Dec 10 2015 comment. - _Wolfdieter Lang_, Dec 10 2015

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
Gupta, Hansraj; Partitions of j-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Serhat Bulut, Subset Sum Problem, 2015.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
C. E. Frasser and G. N. Vostrov, Geodetic Graphs Homeomorphic to a Given Geodetic Graph, arXiv:1611.01873 [cs.DM], 2016. [p. 27]
H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 255
H. K. Kim, On Regular Polytope Numbers, Proc. Amer.Math. Soc. 131 (2003), 65-75.
Iva Kodrnja and Helena Koncul, Polynomials vanishing on a basis of S_m(Gamma_0(N)), Glasnik Matematički (2024) Vol. 59, No. 79, 313-325. See p. 324.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901. - _Juergen Will_, Jan 02 2016
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Composition.
A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A002299, A053127, A000332, A000579, A000580, A000581, A000582.
Cf. A000217, A005583, A051747, A000292.
Cf. A099242. - Gary W. Adamson, Feb 02 2009
Cf. A242023. A104712 (fourth column, k=5).
Cf. A001477, A049310, A052787, A067764, A110555, A277935.
5-cell colorings: A337895 (oriented), A132366(n-1) (achiral).
Unoriented colorings: A063843 (5-cell edges, faces), A128767 (8-cell vertices, 16-cell facets), A337957 (16-cell vertices, 8-cell facets), A338949 (24-cell), A338965 (600-cell vertices, 120-cell facets).
Chiral colorings: A331352 (5-cell edges, faces), A337954 (8-cell vertices, 16-cell facets), A234249 (16-cell vertices, 8-cell facets), A338950 (24-cell), A338966 (600-cell vertices, 120-cell facets).

a000389 n = a000389_list !! n
a000389_list = 0 : 0 : f [] a000217_list where
   f xs (t:ts) = (sum $ zipWith (*) xs a000217_list) : f (t:xs) ts
-- Reinhard Zumkeller, Mar 03 2015, Apr 13 2012

[Binomial(n, 5): n in [0..40]]; // Vincenzo Librandi, Mar 12 2015

f:=n->(1/120)*(n^5-10*n^4+35*n^3-50*n^2+24*n): seq(f(n), n=0..60);
ZL := [S, {S=Prod(B,B,B,B,B,B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n+1), n=0..42); # Zerinvary Lajos, Mar 13 2007
A000389:=1/(z-1)**6; # Simon Plouffe, 1992 dissertation

Table[Binomial[n, 5], {n, 5, 50}] (* Stefan Steinerberger, Apr 02 2006 *)
CoefficientList[Series[x^5 / (1 - x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 12 2015 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,0,0,0,0,1},50] (* Harvey P. Dale, Jul 17 2016 *)

(conv(u,v)=local(w); w=vector(length(u),i,sum(j=1,i,u[j]*v[i+1-j])); w);
(t(n)=n*(n+1)/2); u=vector(10,i,t(i)); conv(u,u)

G.f.: x^5/(1-x)^6.
a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)/120.
a(n) = (n^5-10*n^4+35*n^3-50*n^2+24*n)/120. (Replace all x_i's in the cycle index with n.)
a(n+2) = Sum_{i+j+k=n} i*j*k. - Benoit Cloitre, Nov 01 2002
Convolution of triangular numbers (A000217) with themselves.
Partial sums of A000332. - Alexander Adamchuk, Dec 19 2004
a(n) = -A110555(n+1,5). - Reinhard Zumkeller, Jul 27 2005
a(n+3) = (1/2!)*(d^2/dx^2)S(n,x)|A049310.%20-%20_Wolfdieter%20Lang">{x=2}, n>=2, one half of second derivative of Chebyshev S-polynomials evaluated at x=2. See A049310. - _Wolfdieter Lang, Apr 04 2007
a(n) = A052787(n+5)/120. - Zerinvary Lajos, Apr 26 2007
Sum_{n>=5} 1/a(n) = 5/4. - R. J. Mathar, Jan 27 2009
For n>4, a(n) = 1/(Integral_{x=0..Pi/2} 10*(sin(x))^(2*n-9)*(cos(x))^9). - Francesco Daddi, Aug 02 2011
Sum_{n>=5} (-1)^(n + 1)/a(n) = 80*log(2) - 655/12 = 0.8684411114... - Richard R. Forberg, Aug 11 2014
a(n) = -a(4-n) for all n in Z. - Michael Somos, Oct 07 2014
0 = a(n)*(+a(n+1) + 4*a(n+2)) + a(n+1)*(-6*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Oct 07 2014
a(n) = 3*C(n+1, 5) = 3*A000389(n+1). - Serhat Bulut, Mar 11 2015
From Ilya Gutkovskiy, Jul 23 2016: (Start)
E.g.f.: x^5*exp(x)/120.
Inverse binomial transform of A054849. (End)
From Robert A. Russell, Dec 24 2020: (Start)
a(n) = A337895(n) - a(n+4) = (A337895(n) - A132366(n-1)) / 2 = a(n+4) - A132366(n-1).
a(n+4) = A337895(n) - a(n) = (A337895(n) + A132366(n-1)) / 2 = a(n) + A132366(n-1).
a(n+4) = 1*C(n,1) + 4*C(n,2) + 6*C(n,3) + 4*C(n,4) + 1*C(n,5), where the coefficient of C(n,k) is the number of unoriented pentachoron colorings using exactly k colors. (End)

Corrected formulas that had been based on other offsets. - R. J. Mathar, Jun 16 2009

I changed the offset to 0. This will require some further adjustments to the formulas. - N. J. A. Sloane, Aug 01 2010

A063843 Number of n-multigraphs on 5 nodes.

Original entry on oeis.org

0, 1, 34, 792, 10688, 90005, 533358, 2437848, 9156288, 29522961, 84293770, 217993600, 519341472, 1154658869, 2420188694, 4821091920, 9187076352, 16837177281, 29809183410, 51172613512, 85448030080, 139159855989, 221554769150, 345523218536, 528767663040

Offset: 0

N. J. A. Sloane, Aug 25 2001

Equivalently, number of ways to color edges of complete graph on 5 nodes with n colors, under action of symmetric group S_5, of order 120, with cycle index on edges given by (1/120)*(24*x5^2 + 30*x2*x4^2 + 20*x3^3*x1 + 20*x3*x6*x1 + 15*x1^2*x2^4 + 10*x1^4*x2^3 + x1^10). Setting all x_i = n gives the sequence.
Number of vertex colorings of the Petersen graph. Marko Riedel, Mar 24 2016
Number of unoriented colorings of the 10 triangular edges or triangular faces of a pentachoron, Schläfli symbol {3,3,3}, using n or fewer colors. Also called a 5-cell or 4-simplex. - Robert A. Russell, Oct 17 2020

T. D. Noe, Table of n, a(n) for n = 0..1000
Vladeta Jovovic, Formulae for the number T(n,k) of n-multigraphs on k nodes
Marko Riedel, Vertex colorings of the Petersen graph, Math StackExchange
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A063842. A row of A063841.
Cf. A331350 (oriented), A331352 (chiral), A331353 (achiral), A000389(n+4) (vertices and facets)
Other polychora: A331359 (8-cell), A331355 (16-cell), A338953 (24-cell), A338965 (120-cell, 600-cell).
Row 4 of A327084 (simplex edges and ridges) and A337884 (simplex faces and peaks).

f:=n-> 1/120*(24*n^2+50*n^3+20*n^4+15*n^6+10*n^7+n^10);

Table[(24n^2+50n^3+20n^4+15n^6+10n^7+n^10)/120,{n,0,30}] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,1,34,792,10688,90005,533358,2437848,9156288,29522961,84293770},30] (* Harvey P. Dale, Oct 20 2012 *)

a(n)=n^2*(n^8+10*n^5+15*n^4+20*n^2+50*n+24)/120 \\ Charles R Greathouse IV, Jan 20 2012

a(n) = (1/120)*(24*n^2+50*n^3+20*n^4+15*n^6+10*n^7+n^10).
a(n+1) = (1/5!)*(n^10 + 10*n^9 + 45*n^8 + 130*n^7 + 295*n^6 + 552*n^5 + 805*n^4 + 900*n^3 + 774*n^2 + 448*n + 120).
G.f. = (1 + 23*x + 473*x^2 + 3681*x^3 + 10717*x^4 + 11221*x^5 + 3779*x^6 + 339*x^7 + 6*x^8)/(1-x)^11.  - M. F. Hasler, Jan 19 2012
a(0)=0, a(1)=1, a(2)=34, a(3)=792, a(4)=10688, a(5)=90005, a(6)=533358, a(7)=2437848, a(8)=9156288, a(9)=29522961, a(10)=84293770, a(n)= 11*a(n-1)- 55*a(n-2)+165*a(n-3)-330*a(n-4)+462*a(n-5)-462*a(n-6)+ 330*a(n-7)- 165*a(n-8)+55*a(n-9)-11*a(n-10)+a(n-11). - Harvey P. Dale, Oct 20 2012
From Robert A. Russell, Oct 17 2020: (Start)
a(n) = A331350(n) - A331352(n) = (A331350(n) + A331353(n)) / 2 = A331352(n) + A331353(n).
a(n) = 1*C(n,1) + 32*C(n,2) + 693*C(n,3) + 7720*C(n,4) + 44150*C(n,5) + 138312*C(n,6) + 247380*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 unoriented colorings using exactly k colors. (End)

More terms from Vladeta Jovovic, Sep 02 2001

A331359 Number of unoriented colorings of the edges of a tesseract with n available colors.

Original entry on oeis.org

1, 11251322, 4825746875682, 48038446526132256, 60632984344185045000, 20725680132763499134746, 2876113738439693827763387, 206323339930086669420462592, 8941884949194537156253481511

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 unoriented colorings are the same if congruent; chiral pairs are counted as one. Also the number of unoriented 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), A331360 (chiral), A331361 (achiral).

Cf. A063843 (simplex), A331355 (orthoplex), A338953 (24-cell), A338965 (120-cell, 600-cell).

Table[(48n^4 + 64n^6 + 164n^8 + 32n^12 + 35n^16 + 24n^18 + 16n^20 + n^32)/384, {n, 1, 25}]

a(n) = (48*n^4 + 64*n^6 + 164*n^8 + 32*n^12 + 35*n^16 + 24*n^18 + 16*n^20 + n^32) / 384.
a(n) = C(n,1) + 11251320*C(n,2) + 4825713121719*C(n,3) + 48019143606137456*C(n,4) + 60392840368910627325*C(n,5) + 20362602706881512104770*C(n,6) + 2732305589004849709507320*C(n,7) + 183891356981584237730865120*C(n,8) + 7186781660980022442696996900*C(n,9) + 179941570950595830458653229400*C(n,10) + 3092495918800698593432175049200*C(n,11) + 38355721930679608007610435655200*C(n,12) + 356388702642082232961224416430400*C(n,13) + 2552262270629849366778056301033600*C(n,14) + 14398742619650679721666540905600000*C(n,15) + 65081946248235516086688061276416000*C(n,16) + 238774230958640327164289928460608000*C(n,17) + 718111905257279424242461614311808000*C(n,18) + 1783226074397879202567353905547520000*C(n,19) + 3674025240535453233878734112386560000*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) - A331360(n) = (A331358(n) - A331361(n)) / 2 = A331360(n) + A331361(n).

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.

Original entry on oeis.org

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).

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).

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).

A338949 Number of unoriented 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, 18736, 249563343, 245072692820, 51780391393325, 4114243321427946, 166320182540310771, 4099464588809407728, 69243270244113372390, 868065984969662449300, 8550173137863803682016, 69007957379144017626756

Offset: 1

Robert A. Russell, Nov 17 2020

Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbol of the 24-cell is {3,4,3}. It is self-dual.

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. A338948 (oriented), A338950 (chiral), A338951 (achiral), A338953 (edges, faces), A000389 (5-cell), A128767 (8-cell vertices, 16-cell facets), A337957 (16-cell vertices, 8-cell facets), A338965 (120-cell, 600-cell).

Table[(96n^2+144n^3+144n^4+140n^6+300n^7+120n^8+36n^9+45n^12+84n^13+18n^14+12n^15+12n^18+n^24)/1152,{n,15}]
LinearRecurrence[{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},{1,18736,249563343,245072692820,51780391393325,4114243321427946,166320182540310771,4099464588809407728,69243270244113372390,868065984969662449300,8550173137863803682016,69007957379144017626756,471182396311499869193288,2790108355121570273031710,14612960014479438426745050,68774495831757984888966336,294660451484256436406752191,1161683435155207577365494648,4252399462403852518286044405,14563558286595288907896687700,46968928774940328123724865031,143447144215320073513164583826,416884377543198363455158598933,1157756823443195554136397711600,3083952997773835021725260467500},20] (* Harvey P. Dale, Mar 24 2024 *)

a(n) = (96*n^2 + 144*n^3 + 144*n^4 + 140*n^6 + 300*n^7 + 120*n^8 + 36*n^9 + 45*n^12 + 84*n^13 + 18*n^14 + 12*n^15 + 12*n^18 + n^24) / 1152.
a(n) = 1*C(n,1) + 18734*C(n,2) + 249507138*C(n,3) + 244074551860*C(n,4) + 50557523375300*C(n,5) + 3807232072474470*C(n,6) + 138599298699649830*C(n,7) + 2881219380682352640*C(n,8) + 37996512548398853085*C(n,9) + 341001760994302265550*C(n,10) + 2186424231002014796100*C(n,11) + 10365985337974980021000*C(n,12) + 37236922591331944681200*C(n,13) + 103077062953464218018400*C(n,14) + 222282219864764987928000*C(n,15) + 375541967632270447008000*C(n,16) + 497391180994576316448000*C(n,17) + 513995707397665741248000*C(n,18) + 409785508676334510720000*C(n,19) + 247034122336026305280000*C(n,20) + 108861226736398456320000*C(n,21) + 33078014473191367680000*C(n,22) + 6193712343691192320000*C(n,23) + 538583682060103680000*C(n,24), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors.
a(n) = A338948(n) - A338950(n) = (A338948(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)

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A000389 Binomial coefficients C(n,5).

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A063843 Number of n-multigraphs on 5 nodes.

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A331359 Number of unoriented colorings of the edges of a tesseract with n available colors.

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

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