A086687 -id:A086687 - cp's OEIS frontend

A036740 a(n) = (n!)^n.

1, 1, 4, 216, 331776, 24883200000, 139314069504000000, 82606411253903523840000000, 6984964247141514123629140377600000000, 109110688415571316480344899355894085582848000000000, 395940866122425193243875570782668457763038822400000000000000000000

Offset: 0

N. J. A. Sloane

(-1)^n*a(n) is the determinant of the n X n matrix m_{i,j} = T(n+i,j), 1 <= i,j <= n, where T(n,k) are the signed Stirling numbers of the first kind (A008275). Derived from methods given in Krattenthaler link. - Benoit Cloitre, Sep 17 2005
a(n) is also the number of binary operations on an n-element set which are right (or left) cancellative. These are also called right (left) cancellative magma or groupoids. The multiplication table of a right (left) cancellative magma is an n X n matrix with entries from an n element set such that the elements in each column (or row) are distinct. - W. Edwin Clark, Apr 09 2009
This sequence is mentioned in "Experimentation in Mathematics" as a sum-of-powers determinant. - John M. Campbell, May 07 2011
Determinant of the n X n matrix M_n = [m_n(i,j)] with m_n(i,j) = Stirling2(n+i,j) for 1<=i,j<=n. - Alois P. Heinz, Jul 26 2013

Jonathan Borwein, David Bailey and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Ltd., 2004, p. 207.

Alois P. Heinz, Table of n, a(n) for n = 0..30
Christian Krattenthaler, Advanced determinant calculus, in: D. Foata and G. N. Han (eds.), The Andrews Festschrift, Springer, Berlin, Heidelberg, 2001, pp. 349-426; arXiv preprint, arXiv:math/9902004 [math.CO], 1999.

Cf. A000142, A000169, A000312.
Cf. A086687, A225764, A261114, A261490.
Main diagonal of A225816.

a:= n-> n!^n:
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 25 2013

Table[(n!)^n,{n,0,10}] (* Harvey P. Dale, Sep 29 2013 *)

makelist(n!^n,n,0,10); /* Martin Ettl, Jan 13 2013 */

PARI
```
a(n)=n!^n;
```

a(n) = a(n-1)*n^n*(n-1)! = a(n-1)*A000169(n)*A000142(n) = A036740(n-1) * A000312(n)*A000142(n-1). - Henry Bottomley, Dec 06 2001
From Benoit Cloitre, Sep 17 2005: (Start)
a(n) = Product_{k=1..n} (k-1)!*k^k;
a(n) = A000178(n-1)*A002109(n) for n >= 1. (End)
a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2-1/12). - Vaclav Kotesovec, Nov 14 2014
a(n) = Product_{k=1..n} k^n. - José de Jesús Camacho Medina, Jul 12 2016
Sum_{n>=0} 1/a(n) = A261114. - Amiram Eldar, Nov 16 2020

A105291 Triangle read by rows: T(m,n) = binomial(m!,n), m>=0, 0 <= n <= m!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 6, 15, 20, 15, 6, 1, 1, 24, 276, 2024, 10626, 42504, 134596, 346104, 735471, 1307504, 1961256, 2496144, 2704156, 2496144, 1961256, 1307504, 735471, 346104, 134596, 42504, 10626, 2024, 276, 24, 1, 1, 120, 7140, 280840, 8214570, 190578024

Offset: 0

N. J. A. Sloane, Sep 03 2010, following a suggestion from R. H. Hardin, Aug 31 2010

This is the number of nXm arrays with each row a permutation of 1..m, and rows in lexicographically strictly increasing order.

For row 0, remember that 0!=1.

			Triangle begins:
[1, 1],
[1, 1],
[1, 2, 1],
[1, 6, 15, 20, 15, 6, 1],
[1, 24, 276, 2024, 10626, 42504, 134596, 346104, 735471, 1307504, 1961256, 2496144, 2704156, 2496144, 1961256, 1307504, 735471, 346104, 134596, 42504, 10626, 2024, 276, 24, 1],
...

Alois P. Heinz, Rows n = 0..6, flattened

See A180397 for another version.

Cf. A007318 (Pascal's triangle), A086687, A109892.

Flatten[Table[Binomial[m!,n],{m,0,5},{n,0,m!}]] (* Harvey P. Dale, Apr 16 2013 *)

A180397 T(n,m) = binomial(m!,n).

Original entry on oeis.org

1, 2, 0, 6, 1, 0, 24, 15, 0, 0, 120, 276, 20, 0, 0, 720, 7140, 2024, 15, 0, 0, 5040, 258840, 280840, 10626, 6, 0, 0, 40320, 12698280, 61949040, 8214570, 42504, 1, 0, 0, 362880, 812831040, 21324644880, 11104365420, 190578024, 134596, 0, 0, 0, 3628800

Offset: 1

R. H. Hardin, Sep 01 2010

T(n,m) = number of n X m arrays with each row a permutation of 1..m and rows lexicographically in strictly increasing order.

			Table starts
1 2 6 24 120 720 5040 40320 362880 3628800
0 1 15 276 7140 258840 12698280 812831040 65840765760
0 0 20 2024 280840 61949040 21324644880 10923907290240
0 0 15 10626 8214570 11104365420 26853059065140
0 0 6 42504 190578024 1590145128144
0 0 1 134596 3652745460
0 0 0 346104
0 0 0
0 0
0

R. H. Hardin, Table of n, a(n) for n = 1..1275

See A105291 for another version.
First row gives A000142.
Main diagonal gives A086687.

t[n_, m_] := Binomial[m!, n]; Table[t[m - n + 1, n], {m, 9}, {n, m, 1, -1}] // Flatten (* to display table in Comment *) Table[ t[m, n], {m, 10}, {n, 8}] // TableForm (* Robert G. Wilson v, Sep 02 2010 *)

T(n,m) = binomial(m!,n).

A067454 Binomial(a,b) where a = n! and b = sum of first n natural numbers.

Original entry on oeis.org

1, 1, 0, 1, 1961256, 4730523156632595024, 14714642085645788453359573419196194379200, 141671570521482692258866117076218807527442238234361038404229857860891009480

Offset: 0

Amarnath Murthy, Feb 05 2002

Terms from Robert G. Wilson v.

Cf. A067454, A086687.

Table[Binomial[n!, n(n + 1)/2], {n, 0, 8}] (* Robert G. Wilson v *)

From Vaclav Kotesovec, May 29 2025: (Start)
a(n) ~ n!^(n*(n+1)/2) / (n*(n+1)/2)!.
a(n) ~ 2^(3*n^2/4 + 3*n/4) * Pi^(n^2/4 + n/4 - 1/2) * n^(n^3/2 - n^2/4 - 3*n/4 - 1) / exp(n^3/2 - n/24 + 5/24). (End)

A107447 a(n) = binomial(n!, n^2).

Original entry on oeis.org

1, 0, 0, 735471, 41749257038001257014137504, 8072776113194557737391130747136885454937928869204466648295480, 34145180671088019813488798475366394184193477615213303683031012996650190080826664983024305988320017979711374101114480000

Offset: 1

Zak Seidov, May 26 2005

G. C. Greubel, Table of n, a(n) for n = 1..11

Cf. A014062 C(n^2, n), A086687 C(n!, n), A067454.

[Binomial(Factorial(n), n^2): n in [1..9]]; // G. C. Greubel, Mar 24 2024

Mathematica
```
a[n_]:= Binomial[n!, n^2]; Array[a, 10]
```

[binomial(factorial(n), n^2) for n in range(1,10)] # G. C. Greubel, Mar 24 2024

From Vaclav Kotesovec, May 28 2025: (Start)
a(n) ~ n!^(n^2) / (n^2)!.
a(n) ~ (2*Pi)^((n^2 - 1)/2) * n^(n^3 - 3*n^2/2 - 1) / exp(n^3 -n^2 - n/12). (End)

A187083 a(n) = binomial(n^n, n).

Original entry on oeis.org

1, 6, 2925, 174792640, 2475588476563125, 14320984850603177651837856, 50975600425441237253196072020826978589, 155681826868802708662507744652859497547627180714885120, 541851389452483826218851027234763464912884507272826833630475746754951097

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 03 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..27
Harlan J. Brothers, Pascal's triangle, Sidi polynomials, and powers of e, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.

Cf. A014062, A014070, A086687.

[Binomial(n^n, n): n in [1..15]]; // Vincenzo Librandi, Apr 22 2011

Mathematica
```
Table[Binomial[n^n,n],{n,12}]
```

a(n) ~ n^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016

A365628 a(n) = binomial(primorial(n), n).

Original entry on oeis.org

1, 2, 15, 4060, 78738660, 545754554499462, 1018081517447240182211275, 1793004475784081302284255717158418120, 1943305407393725342965469143054357602760779899437185, 3772316402417100592416011698371929155605067111502494326520988270728160

Offset: 0

Darío Clavijo, Sep 13 2023

nonn

Cf. A002110, A007318, A014062, A060604, A086687.

a(n) = binomial(vecprod(primes(n)), n); \\ Michel Marcus, Sep 14 2023

from sympy import binomial, primorial
a = lambda n: binomial(primorial(n), n)
print([a(n) for n in range(1,10)])

a(n) = binomial(A002110(n), n).

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A036740 a(n) = (n!)^n.

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A105291 Triangle read by rows: T(m,n) = binomial(m!,n), m>=0, 0 <= n <= m!.

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A180397 T(n,m) = binomial(m!,n).

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A067454 Binomial(a,b) where a = n! and b = sum of first n natural numbers.

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A107447 a(n) = binomial(n!, n^2).

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A187083 a(n) = binomial(n^n, n).

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Magma

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A365628 a(n) = binomial(primorial(n), n).

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