A000389 -id:A000389 - cp's OEIS frontend

A101104 a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4.

1, 12, 23, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24

Offset: 1

Cecilia Rossiter, Dec 15 2004

Original name: The first summation of row 4 of Euler's triangle - a row that will recursively accumulate to the power of 4.

D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Dead link]
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883
Eric Weisstein, Link to section of MathWorld: Eulerian Number
Eric Weisstein, Link to section of MathWorld: Nexus number
Eric Weisstein, Link to section of MathWorld: Finite Differences
Index entries for linear recurrences with constant coefficients, signature (1).

For other sequences based upon MagicNKZ(n,k,z):
..... |  n = 1  |  n = 2  |  n = 3  |  n = 4  |  n = 5  |  n = 6  |  n = 7
---------------------------------------------------------------------------
z = 0 | A000007 | A019590 | .......MagicNKZ(n,k,0) = A008292(n,k+1) .......
z = 1 | A000012 | A040000 | A101101 | thisSeq | A101100 | ....... | .......
z = 2 | A000027 | A005408 | A008458 | A101103 | A101095 | ....... | .......
z = 3 | A000217 | A000290 | A003215 | A005914 | A101096 | ....... | .......
z = 4 | A000292 | A000330 | A000578 | A005917 | A101098 | ....... | .......
z = 5 | A000332 | A002415 | A000537 | A000583 | A022521 | ....... | A255181
z = 6 | A000389 | A005585 | A024166 | A000538 | A000584 | A022522 | A255177
z = 7 | A000579 | A040977 | A101094 | A101089 | A000539 | A001014 | A022523
z = 8 | A000580 | A050486 | A101097 | A101090 | A101092 | A000540 | A001015
z = 9 | A000581 | A053347 | A101102 | A101091 | A101099 | A101093 | A000541
Cf. A101095 for an expanded table and more about MagicNKZ.

MagicNKZ = Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 4, 4}, {z, 1, 1}, {k, 0, 34}]
Join[{1, 12, 23},LinearRecurrence[{1},{24},56]] (* Ray Chandler, Sep 23 2015 *)

a(k) = MagicNKZ(4,k,1) where MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n+1-z,j)*(k-j+1)^n (cf. A101095). That is, a(k) = Sum_{j=0..k+1} (-1)^j*binomial(4, j)*(k-j+1)^4.
a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4. - Joerg Arndt, Nov 30 2014
G.f.: x*(1+11*x+11*x^2+x^3)/(1-x). - Colin Barker, Apr 16 2012

New name from Joerg Arndt, Nov 30 2014

Original Formula edited and Crossrefs table added by Danny Rorabaugh, Apr 22 2015

A002299 Binomial coefficients C(2*n+5,5).

Original entry on oeis.org

1, 21, 126, 462, 1287, 3003, 6188, 11628, 20349, 33649, 53130, 80730, 118755, 169911, 237336, 324632, 435897, 575757, 749398, 962598, 1221759, 1533939, 1906884, 2349060, 2869685, 3478761, 4187106, 5006386, 5949147, 7028847, 8259888, 9657648, 11238513

Offset: 0

N. J. A. Sloane, Eric Lane

Number of standard tableaux of shape (2n+1,1^5). - Emeric Deutsch, May 30 2004

G. C. Greubel, Table of n, a(n) for n = 0..5000 (terms 0..200 from Vincenzo Librandi)
Milan Janjic, Two Enumerative Functions.
J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math., Vol. 201, No. 1 (2006), pp. 143-179. [Th. 7.2(i), case a=1]
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000389, A053127.

[Binomial(2*n+5,5): n in [0..30]]; // Vincenzo Librandi, Oct 07 2011

Table[Binomial[2*n + 5, 5], {n, 0, 50}] (* G. C. Greubel, Nov 23 2017 *)

a(n)=n*(8*n^4+60*n^3+170*n^2+225*n+137)/30+1 \\ Charles R Greathouse IV, Apr 18 2012

a(n) = A000389(2*n+5).
G.f.: (1+15*x+15*x^2+x^3)/(1-x)^6 = (1+x)*(x^2+14*x+1)/(1-x)^6.
E.g.f.: (30 + 600*x + 1275*x^2 + 730*x^3 + 140*x^4 + 8*x^5)*exp(x)/30. - G. C. Greubel, Nov 23 2017
Sum_{n>=0} (-1)^n/a(n) = 5*(10/3 - Pi). - Matthieu Pluntz, Oct 08 2019
Sum_{n>=0} 1/a(n) = 40*log(2) - 80/3. - Amiram Eldar, Jan 03 2022
From Peter Bala, Sep 03 2023: (Start)
a(n) = Sum_{0 <= i <= j <= n} (j+1)*(2*i+1)^2.
a(n) = (n+2)*(2*n+5)/(n*(2*n-1))*a(n-1) with a(0) := 1. (End)
a(n) = 2*A225007(n) - A006324(n+1). - Yasser Arath Chavez Reyes, Feb 27 2024

A054559 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 8 1-simplexes.

Original entry on oeis.org

30, 180, 630, 1680, 3780, 7560, 13860, 23760, 38610, 60060, 90090, 131040, 185640, 257040, 348840, 465120, 610470, 790020, 1009470, 1275120, 1593900, 1973400, 2421900, 2948400, 3562650, 4275180, 5097330, 6041280, 7120080, 8347680, 9738960, 11309760, 13076910

Offset: 5

Vladeta Jovovic, Apr 10 2000

Number of {T_1,T_2,...,T_k} where T_i,i=1..k are 3-subsets of an n-set such that {D | D is 2-subset of T_i for some i=1..k} has l elements; k=5,l=8.

Let H be the n X n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row 3 of B equals -a(n+2). - T. D. Noe, May 01 2011

V. Jovovic, On the number of two-dimensional simplicial complexes (in Russian), Metody i sistemy tekhnicheskoy diagnostiki, Vypusk 16, Mezhvuzovskiy zbornik nauchnykh trudov, Izdatelstvo Saratovskogo universiteta, 1991.

Vincenzo Librandi, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A054557.

Cf. A000389, A052787.

I:=[30, 180, 630, 1680, 3780, 7560]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..30]]; // Vincenzo Librandi, Apr 29 2012

Table[n*(n+1)*(n+2)*(n+3)*(n+4)/4, {n,1,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2009 *)
CoefficientList[Series[30/(1-x)^6,{x,0,30}],x] (* Vincenzo Librandi, Apr 29 2012 *)

x='x+O('x^30); Vec(serlaplace(x^5*exp(x)/4)) \\ G. C. Greubel, Nov 23 2017

a(n) = 30*C(n,5) = 30*A000389(n) = n*(n-1)*(n-2)*(n-3)*(n-4)/4.
G.f.: 30*x^5/(1-x)^6. - Colin Barker, Jan 19 2012
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Vincenzo Librandi, Apr 29 2012
E.g.f.: x^5*exp(x)/4. - G. C. Greubel, Nov 23 2017
From Amiram Eldar, Mar 08 2022: (Start)
Sum_{n>=5} 1/a(n) = 1/24.
Sum_{n>=5} (-1)^(n+1)/a(n) = 8*log(2)/3 - 131/72. (End)

A099242 (6n+5)-th terms of expansion of 1/(1 - x - x^6).

Original entry on oeis.org

1, 7, 34, 153, 686, 3088, 13917, 62721, 282646, 1273690, 5739647, 25864698, 116554700, 525233175, 2366870474, 10665883415, 48063918336, 216591552484, 976031547888, 4398313653120, 19820223058176, 89316331907533

Offset: 0

Paul Barry, Oct 08 2004

A row of A099239.

Equals INVERT transform of A000389, C(n,5). [Gary W. Adamson, Feb 02 2009]

G. C. Greubel, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 16.
Index entries for linear recurrences with constant coefficients, signature (7,-15,20,-15,6,-1).

Cf. A000389.

CoefficientList[Series[1/((1 - x)^6 - x), {x, 0, 50}], x] (* G. C. Greubel, Nov 24 2017 *)
LinearRecurrence[{7,-15,20,-15,6,-1},{1,7,34,153,686,3088},30] (* Harvey P. Dale, May 06 2018 *)

my(x='x+O('x^50)); Vec(1/((1-x)^6-x)) \\ G. C. Greubel, Nov 24 2017

G.f.: 1/((1-x)^6-x).
a(n) = 7*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = Sum_{k=0..n} binomial(6*n-5*(k-1), k).
a(n) = Sum_{k=0..n} binomial(n+5*(k+1), k+5*(k+1)).
a(n) = Sum_{k=0..n} binomial(n+5*(k+1), n-k).

A238801 Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).

Original entry on oeis.org

1, 2, 0, 3, 0, 1, 4, 0, 4, 0, 5, 0, 10, 0, 1, 6, 0, 20, 0, 6, 0, 7, 0, 35, 0, 21, 0, 1, 8, 0, 56, 0, 56, 0, 8, 0, 9, 0, 84, 0, 126, 0, 36, 0, 1, 10, 0, 120, 0, 252, 0, 120, 0, 10, 0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1, 12, 0, 220, 0, 792, 0, 792, 0, 220, 0, 12, 0

Offset: 0

Philippe Deléham, Mar 05 2014

Row sums are powers of 2.

			Triangle begins:
1;
2, 0;
3, 0, 1;
4, 0, 4, 0;
5, 0, 10, 0, 1;
6, 0, 20, 0, 6, 0;
7, 0, 35, 0, 21, 0, 1;
8, 0, 56, 0, 56, 0, 8, 0;
9, 0, 84, 0, 126, 0, 36, 0, 1;
10, 0, 120, 0, 252, 0, 120, 0, 10, 0; etc.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. Columns: A000027, A000292, A000389, A000580, A000582, A001288, A010966, A010968, A010970, A010972, A010974, A010976, A010980, A010982, A010984, A010986, A010988, A010990, A010992, A010994, A010996, A010998, A011000, A017713, A017715, A017717, A017719, A017721, A017723, A017725, A017727, A017729, A017731, A017733, A017735, A017737, A017739, A017741, A017743, A017745, A017747, A017749, A017751, A017753, A017755, A017757, A017759, A017761, A017763.

Cf. A095704, A178616.

Table[Binomial[n + 1, k + 1]*(1 - Mod[k , 2]), {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Nov 22 2017 *)

T(n,k) = binomial(n+1, k+1)*(1-(k % 2));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Nov 23 2017

G.f.: 1/((1+(y-1)*x)*(1-(y+1)*x)).
T(n,k) = 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
Sum_{k=0..n} T(n,k)*x^k = A000027(n+1), A000079(n), A015518(n+1), A003683(n+1), A079773(n+1), A051958(n+1), A080920(n+1), A053455(n), A160958(n+1) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively.

A278325 T(n,k)=Number of nXk 0..1 arrays with rows and columns in lexicographic nondecreasing order but with exactly two mistakes.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 1, 20, 20, 1, 6, 94, 236, 94, 6, 21, 395, 1678, 1678, 395, 21, 56, 1492, 9714, 19854, 9714, 1492, 56, 126, 4991, 51229, 186987, 186987, 51229, 4991, 126, 252, 14848, 251892, 1548002, 3058990, 1548002, 251892, 14848, 252, 462, 39832, 1144205

Offset: 1

R. H. Hardin, Nov 18 2016

Table starts
...0.....0........0..........1............6..............21................56
...0.....3.......20.........94..........395............1492..............4991
...0....20......236.......1678.........9714...........51229............251892
...1....94.....1678......19854.......186987.........1548002..........11947350
...6...395.....9714.....186987......3058990........43798840.........570738085
..21..1492....51229....1548002.....43798840......1133034160.......26606146271
..56..4991...251892...11947350....570738085.....26606146271.....1159826452573
.126.14848..1144205...87562135...7010453565....577253086055....46588303100299
.252.39832..4762445..605431820..82490704797..11856994168275..1741053359883624
.462.97835.18164685.3905165468.927749763030.234252250004719.61687083048618477

			Some solutions for n=4 k=4
..0..1..1..1. .1..1..1..1. .0..0..1..1. .0..1..0..1. .0..0..0..0
..0..0..1..0. .0..1..0..0. .0..1..1..0. .1..0..0..0. .1..0..1..1
..0..1..0..0. .1..0..0..0. .1..1..0..1. .1..0..1..0. .0..0..0..1
..1..1..0..0. .1..0..0..0. .0..0..1..0. .0..0..1..1. .0..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..199

Column 1 is A000389(n+1).

Empirical for column k:
k=1: a(n) = (1/120)*n^5 - (1/24)*n^4 + (1/24)*n^3 + (1/24)*n^2 - (1/20)*n
k=2: [polynomial of degree 11]
k=3: [polynomial of degree 23]
k=4: [polynomial of degree 47]
k=5: [polynomial of degree 95]

A278778 T(n,k)=Number of nXk 0..1 arrays with rows in nondecreasing lexicographic order and columns in nonincreasing lexicographic order, but with exactly two mistakes.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 1, 20, 20, 1, 6, 117, 266, 117, 6, 21, 503, 1972, 1972, 503, 21, 56, 1750, 10784, 19750, 10784, 1750, 56, 126, 5209, 48501, 150085, 150085, 48501, 5209, 126, 252, 13751, 189595, 955347, 1673658, 955347, 189595, 13751, 252, 462, 33000

Offset: 1

R. H. Hardin, Nov 28 2016

Table starts
...0.....0.......0.........1...........6............21..............56
...0.....2......20.......117.........503..........1750............5209
...0....20.....266......1972.......10784.........48501..........189595
...1...117....1972.....19750......150085........955347.........5355983
...6...503...10784....150085.....1673658......16205001.......141166787
..21..1750...48501....955347....16205001.....251740932......3634987413
..56..5209..189595...5355983...141166787....3634987413.....90752836672
.126.13751..665212..27218249..1126917480...48847405083...2155380363189
.252.33000.2138149.127644118..8340736743..611199661843..48042054699217
.462.73282.6384894.559023840.57745890265.7140933364136.999491681597761

			Some solutions for n=4 k=4
..1..0..0..1. .1..1..0..0. .1..0..0..1. .1..1..1..1. .1..0..0..0
..1..1..0..1. .1..0..1..1. .1..1..0..0. .1..0..0..0. .0..1..0..1
..0..1..1..1. .0..1..1..1. .1..1..1..0. .0..1..1..0. .0..1..0..1
..0..1..1..1. .1..0..1..1. .0..0..1..0. .0..1..1..0. .1..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..219

Column 1 is A000389(n+1).

Empirical for column k:
k=1: a(n) = (1/120)*n^5 - (1/24)*n^4 + (1/24)*n^3 + (1/24)*n^2 - (1/20)*n
k=2: [polynomial of degree 10]
k=3: [polynomial of degree 19]
k=4: [polynomial of degree 36]
k=5: [polynomial of degree 69]
k=6: [polynomial of degree 134]

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, 780198, 900389, 1035000, 1185295, 1352610

Offset: 1

Robert A. Russell, Sep 28 2020

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}]

my(x='x+O('x^39)); Vec(x*(x^4+1)/(x-1)^6) \\ Elmo R. Oliveira, Aug 31 2025

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).
From Elmo R. Oliveira, Aug 31 2025: (Start)
G.f.: x*(1 + x^4)/(1 - x)^6.
E.g.f.: x*(60 + 120*x + 60*x^2 + 10*x^3 + x^4)*exp(x)/60.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 6. (End)

More terms from Elmo R. Oliveira, Aug 31 2025

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).

A337957 Number of unoriented 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, 715, 3060, 10626, 31465, 82251, 194580, 424270, 864501, 1663740, 3049501, 5359095, 9078630, 14891626, 23738715, 36890001, 56031760, 83369265, 121747626, 174792640, 247073751, 344291325, 473490550, 643304376

Offset: 1

Robert A. Russell, Oct 03 2020

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.

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A337956 (oriented), A234249(n+1) (chiral), A337958 (achiral).
Other elements: A331355 (hyperoctahedron edges, tesseract faces), A331359 (hyperoctahedron faces, tesseract edges), A128767 (hyperoctahedron facets, tesseract vertices).
Other polychora: A000389(n+4) (5-cell), A338949 (24-cell), A338965 (120-cell, 600-cell).
Row 4 of A325005 (orthotope facets, orthoplex vertices).

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

a(n) = binomial(binomial(n+1,2)+3,4).
a(n) = n * (n+1) * (n^2 + n + 2) * (n^2 + n + 4) * (n^2 + n + 6) / 384.
a(n) = 1*C(n,1) + 13*C(n,2) + 84*C(n,3) + 297*C(n,4) + 600*C(n,5) + 690*C(n,6) + 420*C(n,7) + 105*C(n,8), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors.
a(n) = A337956(n) - A234249(n+1) = (A337956(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 + 37*x^3 + 27*x^4 + 6*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)

cp's OEIS Frontend

A101104 a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4.

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A002299 Binomial coefficients C(2*n+5,5).

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A054559 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 8 1-simplexes.

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A099242 (6n+5)-th terms of expansion of 1/(1 - x - x^6).

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A238801 Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).

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A278325 T(n,k)=Number of nXk 0..1 arrays with rows and columns in lexicographic nondecreasing order but with exactly two mistakes.

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A278778 T(n,k)=Number of nXk 0..1 arrays with rows in nondecreasing lexicographic order and columns in nonincreasing lexicographic order, but with exactly two mistakes.

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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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