A052849 -id:A052849 - cp's OEIS frontend

A126776 Basis orbits of n-dimensional cubes.

Original entry on oeis.org

1, 1, 4, 17, 237, 9892

Offset: 1

Jonathan Vos Post, Feb 18 2007

a(7) >= 1456318. [Hughes & Anderson]

David Bremner, Mathieu Dutour Sikiric and Achill Schuermann, Polyhedral representation conversion up to symmetries, arXiv:math/0702239 [math.MG], 2007. See Table 1; note that it shows the dimension n+1 of the polyhedral cone, not the dimension n of the cube itself.
Robert B. Hughes and Michael R. Anderson, Simplexity of the cube, Discrete Mathematics, 158 (1996), 99-150.

Cf. A052849, A276412, A105231.

Edited by Andrei Zabolotskii, Aug 28 2025

A137320 Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (mx + n - 1)p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, 0, 2, 4, 0, 4, 12, 8, 0, 12, 44, 48, 16, 0, 48, 200, 280, 160, 32, 0, 240, 1096, 1800, 1360, 480, 64, 0, 1440, 7056, 12992, 11760, 5600, 1344, 128, 0, 10080, 52272, 105056, 108304, 62720, 20608, 3584, 256, 0, 80640, 438336, 944992, 1076544, 718368, 290304, 69888, 9216, 512

Offset: 0

Roger L. Bagula, Apr 20 2008

Row sums are factorials.

Also the Bell transform of A052849 (with a(0)=2). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			[0] {1},
[1] {0, 2},
[2] {0, 2,     4},
[3] {0, 4,     12,     8},
[4] {0, 12,    44,     48,     16},
[5] {0, 48,    200,    280,    160,     32},
[6] {0, 240,   1096,   1800,   1360,    480,    64},
[7] {0, 1440,  7056,   12992,  11760,   5600,   1344,   128},
[8] {0, 10080, 52272,  105056, 108304,  62720,  20608,  3584,  256},
[9] {0, 80640, 438336, 944992, 1076544, 718368, 290304, 69888, 9216, 512}.

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 62-63

Apart from signs, same as A137312.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n<2,2,2*n!), 8); # Peter Luschny, Jan 27 2016
p := (n,x) -> (n + 2*x - 1)!/(2*x - 1)!:
seq(seq(coeff(expand(p(n,x)), x, k), k=0..n), n=0..9); # Peter Luschny, Feb 26 2019

m = 2; p[x, 0] = 1; p[x, -1] = 0; p[x, 1] = m*x;
p[x_, n_] := p[x, n] = (m*x + n - 1)*p[x, n - 1];
Table[CoefficientList[p[x, n], x], {n, 0, 9}] // Flatten
(* Second program: *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, If[n < 2, 2, 2*n!]], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

From Peter Luschny, Feb 26 2019: (Start)
p(n, x) = n!*Sum_{k=0..n} (-1)^n*binomial(-x, k)*binomial(-x, n-k).
p(n, x) = (n + 2*x - 1)!/(2*x - 1)!.
T(n, k) = [x^k] p(n,x). (End)

Edited and offset set to 0 by Peter Luschny, Feb 26 2019

A281291 Numbers n such that 2*n! is not a refactorable number.

Original entry on oeis.org

2, 4, 8, 16, 256, 65536

Offset: 1

Altug Alkan, Jan 23 2017

See Conjecture 47 and Theorem 51 in Zelinsky's paper for related points.
In Theorem 51 Zelinsky gives a technical result which almost implies that for all sufficiently large n, n! is a refactorable number. (Corrected by Joshua Zelinsky, May 15 2020)
Also note that Luca & Young paper gives a proof for n! is a refactorable number for all n > 5.
This sequence focuses on the 2 * n! and we cannot say that 2 * n! is refactorable for all sufficiently large n at the moment. This is because if 2^(2^k) + 1 is a Fermat prime (A019434), then 2^(2^k) is a term of this sequence and it is not known yet sequence of Fermat primes is finite or not.

			8 is a term since d(2*8!) = 2^2 * 3^3 does not divide 2 * 8! = 2^8 * 3^2 * 5 * 7.

Florian Luca and Paul Thomas Young, On the number of divisors of n! and of the Fibonacci numbers
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999.
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Cf. A019434, A033950, A052849, A281498.

isA033950(n) = n % numdiv(n) == 0;
is(n) = !isA033950(2*n!);

A367962 Triangle read by rows. T(n, k) = Sum_{j=0..k} (n!/j!).

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 6, 12, 15, 16, 24, 48, 60, 64, 65, 120, 240, 300, 320, 325, 326, 720, 1440, 1800, 1920, 1950, 1956, 1957, 5040, 10080, 12600, 13440, 13650, 13692, 13699, 13700, 40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 109601

Offset: 0

Peter Luschny, Dec 06 2023

			  [0]   1;
  [1]   1,    2;
  [2]   2,    4,    5;
  [3]   6,   12,   15,   16;
  [4]  24,   48,   60,   64,   65;
  [5] 120,  240,  300,  320,  325,  326;
  [6] 720, 1440, 1800, 1920, 1950, 1956, 1957;

Cf. A094587, A000142 (T(n, 0)), A052849 (T(n, 1)), A000522 (T(n, n)), A007526 (T(n,n-1)), A038154 (T(n, n-2)), A355268 (T(n, n/2)), A367963(n) (T(2*n, n)/n!).

Cf. A001339 (row sums), A087208 (alternating row sums), A082030 (accumulated sums), A053482, A331689.

T := (n, k) -> add(n!/j!, j = 0..k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Module[{n=1},NestList[Append[n#,1+Last[#]n++]&,{1},10]] (* or *)
Table[Sum[n!/j!,{j,0,k}],{n,0,10},{k,0,n}] (* Paolo Xausa, Dec 07 2023 *)

from functools import cache
@cache
def a_row(n: int) -> list[int]:
    if n == 0: return [1]
    row = a_row(n - 1) + [0]
    for k in range(n): row[k] *= n
    row[n] = row[n - 1] + 1
    return row

def T(n, k): return sum(falling_factorial(n, n - j) for j in range(k + 1))
for n in range(9): print([T(n, k) for k in range(n + 1)])

T(n, k) = A094587(n, k) * A000522(k).
T(n, k) = e * (n! / k!) * Gamma(k + 1, 1).
Sum_{k=0..n} T(n, k) * 2^(n - k) = A053482(n).
Sum_{k=0..n} T(n, k) * binomial(n, k) = A331689(n).
Recurrence: T(n, n) = T(n, n-1) + 1 starting with T(0, 0) = 1.
For k <> n: T(n, k) = n * T(n-1, k).

A371740 Triangle read by rows: g.f. (1 - t)^(-x) * (1 + t)^(2-x).

Original entry on oeis.org

1, 2, 1, 1, 0, 4, 0, 3, 1, 0, 6, 6, 0, 5, 6, 1, 0, 16, 24, 8, 0, 14, 23, 10, 1, 0, 60, 110, 60, 10, 0, 54, 105, 65, 15, 1, 0, 288, 600, 420, 120, 12, 0, 264, 574, 435, 145, 21, 1, 0, 1680, 3836, 3150, 1190, 210, 14, 0, 1560, 3682, 3199, 1330, 280, 28, 1, 0, 11520, 28224, 25984, 11760, 2800, 336, 16

Offset: 0

Peter Bala, Apr 09 2024

			Triangle begins
 n\k |  0    1     2    3    4   5
 - - - - - - - - - - - - - - - - -
  0  |  1
  1  |  2
  2  |  1    1
  3  |  0    4
  4  |  0    3     1
  5  |  0    6     6
  6  |  0    5     6    1
  7  |  0   16    24    8
  8  |  0   14    23   10    1
  9  |  0   60   110   60   10
 10  |  0   54   105   65   15   1
 ...

Cf. A052849, A132393, A371741.

with(combinat):
T := proc (n, k); if irem(n, 2) = 0 then abs(Stirling1((1/2)*n, k)) + (n/2)*abs(Stirling1((n-2)/2, k)) else (n+1)*abs(Stirling1((n-1)/2, k)) end if; end proc:
seq(print(seq(T(n, k), k = 0..floor(n/2))), n = 0..12);

G.f.: (1 - t)^(-x)*(1 + t)^(2-x) = Sum_{n >= 0} R(n, x)*t^n/floor((n+1)/2)! = 1 + 2*t/1! + (1 + x)*t^2/1! + 4*x*t^3/2! + x*(3 + x)*t^4/2! + 6*x*(1 + x)*t^5/3! + x*(1 + x)*(5 + x)*t^6/3! + 8*x*(1 + x)*(2 + x)*t^7/3! + x*(1 + x)*(2 + x)*(7 + x)*t^8/4! + 10*x*(1 + x)*(2 + x)*(3 + x)*t^9/5! + ....
Row polynomials: R(2*n, x) =  (2*n - 1 + x) * Product_{i = 0..n-2} (x + i) for n >= 1.
R(2*n+1, x) = (2*n + 2) * Product_{i = 0..n-1} (x + i) for n >= 0.
T(2*n, k) = |Stirling1(n, k)| + n*|Stirling1(n-1, k)| = A132393(n, k) + n* A132393(n-1, k);
T(2*n+1, k) = (2*n + 2)*|Stirling1(n, k)| = (2*n + 2)*A132393(n, k).
n-th row sum equals 2 * floor((n+1)/2)! for n >= 1.

A384494 Triangle read by rows: T(n, k) = (-1)^k(k+1)(n+1-k)!, n >= 0, k = 0..n.

Original entry on oeis.org

1, 2, -2, 6, -4, 3, 24, -12, 6, -4, 120, -48, 18, -8, 5, 720, -240, 72, -24, 10, -6, 5040, -1440, 360, -96, 30, -12, 7, 40320, -10080, 2160, -480, 120, -36, 14, -8, 362880, -80640, 15120, -2880, 600, -144, 42, -16, 9, 3628800, -725760, 120960, -20160, 3600, -720, 168, -48, 18, -10

Offset: 0

Wolfdieter Lang, May 31 2025

This triangle, written as (infinite) square matrix MT with vanishing upper diagonals 0, together with the Riordan triangle A104698, written also as such a square matrix MR, appears in the double sum formula for the number of certain restricted permutations given in A086852(n), as diagonal sequence A086852(n+2) = (2*MR*MT^t)_{n,n}, for n >=0, where t indicates matrix transpositon.

			The triangle T begins:
  n\k        0        1       2       3     4     5   6   7  8    9 ...
  ---------------------------------------------------------------------
  0:         1
  1:         2       -2
  2:         6       -4       3
  3:        24      -12       6      -4
  4:       120      -48      18      -8     5
  5:       720     -240      72     -24    10    -6
  6:      5040    -1440     360     -96    30   -12   7
  7:     40320   -10080    2160    -480    12   -36  14  -8
  8:    362880   -80640   15120   -2880   600  -144  42 -16  9
  9:   3628800  -725760  120960  -20160  3600  -720 168 -48 18 -10
  ...

Cf. A104698, A086852, A086852.

Column sequences: A000142(n+1), -A052849, A052560(n-1), -A052578(n-2), A052648(n-3), -A298881(n-4), A062098(n-5), -A159038(n-6), ...

Table[(-1)^k * (k+1) * (n+1-k)!, {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 31 2025 *)

T(n, k) = (-1)^k*(k+1)*(n+1-k)!, for n >= 0 and k = 0, 1, ..., n.

O.g.f. of row polynomials P(n, y) := Sum_{k=0..n} T(n, k) y^k: G(x, y) = ((N(x) - 1)/x) * (1/(1 + y*x)^2), with N(x) = hypergeometric([1,1], [], x), the o.g.f. of {n!}_{n>=0} (see A000142).

A117826 First four terms of the sequence are doubled, then those numbers are tripled and then those numbers are quadrupled, etc.

Original entry on oeis.org

1, 2, 3, 4, 2, 4, 6, 8, 6, 12, 18, 24, 24, 48, 72, 96, 120, 240, 360, 480, 720, 1440, 2160, 2880, 5040, 10080, 15120, 20160, 40320, 80640, 120960, 161280, 362880, 725760, 1088640, 1451520, 3628800, 7257600, 10886400, 14515200, 39916800, 79833600

Offset: 1

Robert G. Wilson v, Apr 25 2006

nonn

Cf. A000142, A115425.

Table[n!{1, 2, 3, 4}, {n, 11}] // Flatten

a(4n-3) = A000142(n). a(4n-2)=2*A000142(n)=A052849(n), a(4n-1)=3*A000142(n)=A052560(n), a(4n)=4*A000142(n)=2*A052849(n)=A052578(n).

A260229 a(n) = floor(e^(n!)).

Original entry on oeis.org

2, 7, 403, 26489122129, 13041808783936322797338790280986488113446079415755132

Offset: 1

Ilya Gutkovskiy, Jul 20 2015

nonn

The exponential growth in the number of permutations of n elements.

Next term is too big to be included.

			a(1) = floor(e^(1!)) = floor(e) = 2.

Cf. A000149, A000142.

Cf. A050923, A100731.

Mathematica
```
Table[Floor[E^n!], {n, 1, 7}]
```

default(realprecision, 100); vector(5, n, floor(exp(n!))) \\ Michel Marcus, Aug 06 2015

a(n) = A000149(A000142(n)).

a(n) = floor(sqrt(e^A052849(n) - e^A000142(n) + sqrt(e^A052849(n) - e^A000142(n) + sqrt(e^A052849(n) - e^A000142(n) + ...)))).

cp's OEIS Frontend

A126776 Basis orbits of n-dimensional cubes.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A137320 Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A281291 Numbers n such that 2*n! is not a refactorable number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A367962 Triangle read by rows. T(n, k) = Sum_{j=0..k} (n!/j!).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Python

SageMath

Formula

A371740 Triangle read by rows: g.f. (1 - t)^(-x) * (1 + t)^(2-x).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

A384494 Triangle read by rows: T(n, k) = (-1)^k*(k+1)*(n+1-k)!, n >= 0, k = 0..n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A117826 First four terms of the sequence are doubled, then those numbers are tripled and then those numbers are quadrupled, etc.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A260229 a(n) = floor(e^(n!)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A137320 Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (mx + n - 1)p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.

A384494 Triangle read by rows: T(n, k) = (-1)^k(k+1)(n+1-k)!, n >= 0, k = 0..n.