A272094 -id:A272094 - cp's OEIS frontend

A014062 a(n) = binomial(n^2, n).

1, 1, 6, 84, 1820, 53130, 1947792, 85900584, 4426165368, 260887834350, 17310309456440, 1276749965026536, 103619293824707388, 9176358300744339432, 880530516383349192480, 91005567811177478095440, 10078751602022313874633200, 1190739044344491048895397910

Offset: 0

N. J. A. Sloane

nonn

Roberts states that Gupta and Khare show that a(n) > A002110(n) for 2 < n < 1794 and that a(n) < A002110(n) for n >= 1794, where A002110(n) is the product of the first n primes. - T. D. Noe, Oct 03 2007
This sequence describes the number of ways to arrange n objects in an n X n array (for example, stars in a flag's field pattern). - Tom Young (mcgreg265(AT)msn.com), Jun 17 2010
It appears that a(n) == n (mod n^3) only if n is 1, an odd prime, the square of an odd prime, or the cube of an odd prime. - Gary Detlefs, Aug 06 2013; corrected by Michel Marcus, May 29 2015

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 265.

T. D. Noe, Table of n, a(n) for n=0..100
Horst A. Alzer and Jozsef J. B. Šandor, On a binomial coefficient and a product of prime numbers, Appl. An. Disc. Math. 5 (2011) 87-92.
Harlan J. Brothers, Pascal's Prism: Supplementary Material.
Harlan J. Brothers, Pascal's triangle, Sidi polynomials, and powers of e, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.
Hansraj Gupta and S. P. Khare, On C(k^2,k) and the product of the first k primes, Publ. Fac. Electrotechn. Belgrade, Ser. Math. Phys. 25-29 (1977) 577-598.

Main diagonal of A060539.

Cf. A177234, A177456, A177784, A177788, A272094, A295773.

[Binomial(n^2,n): n in [0..30]]; // G. C. Greubel, Apr 29 2024

Table[Binomial[n^2,n],{n,0,22}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2011 *)
Table[SeriesCoefficient[(1+x)^(n^2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *)

{a(n) = sum(k=0, n, binomial(n, k)*binomial(n^2-n, k))}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Nov 18 2015

[binomial(n^2,n) for n in range(31)] # G. C. Greubel, Apr 29 2024

a(n) ~ 1/sqrt(2*Pi) * (e*n)^(n - 1/2). - Charles R Greathouse IV, Jul 07 2007
a(n) = Sum_{k=0..n} binomial(n, k) * binomial(n^2 - n, k). - Paul D. Hanna, Nov 18 2015
a(n) = (n+1)*A177234(n). - R. J. Mathar, Jan 25 2019
From G. C. Greubel, Apr 29 2024: (Start)
a(n) = n*(n+1)*A177784(n).
a(n) = (n+1)*A177456(n)/(n-1).
a(n) = (n+1)*A177788(n)/n. (End)
a(n) = [x^n] (1+x)^(n^2). - Vaclav Kotesovec, Aug 06 2025

A088020 a(n) = (n^2)!.

Original entry on oeis.org

1, 1, 24, 362880, 20922789888000, 15511210043330985984000000, 371993326789901217467999448150835200000000, 608281864034267560872252163321295376887552831379210240000000000

Offset: 0

Hugo Pfoertner, Sep 18 2003

a(n) is the number of ways in which is possible to fill an n X n square matrix with n^2 distinct elements. - Stefano Spezia, Sep 16 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..21
Index entries for sequences related to tic-tac-toe

Cf. A000142 (n!), A000290 (n^2).
Cf. A039622, A179268. - Reinhard Zumkeller, Jul 06 2010
Cf. A272094, A272095, A272164.

List([0..10],n->Factorial(n^2)); # Muniru A Asiru, Sep 17 2018

[Factorial(n^2): n in [0..10]]; // Vincenzo Librandi, May 31 2011

seq(factorial(n^2),n=0..10); # Muniru A Asiru, Sep 17 2018

Table[(n^2)!,{n,0,9}] (* Vladimir Joseph Stephan Orlovsky, May 19 2011 *)

for(n=0,10,print1((n^2)!,",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 22 2006

A255322 a(n) = Product_{k=0..n} (k^2)!.

Original entry on oeis.org

1, 1, 24, 8709120, 182219087869378560000, 2826438545846116156142906806150103040000000000, 1051416277636507481568264360276689674557030810000137484550133942059008000000000000000000

Offset: 0

Vaclav Kotesovec, Feb 21 2015

nonn

Partial products of A088020. - Michel Marcus, Jul 06 2019

Seiichi Manyama, Table of n, a(n) for n = 0..12

Cf. A000178, A255358, A255359, A255360.
Cf. A002109, A051675, A255321, A255323, A255344.
Cf. A000142, A255268, A255269, A255504.
Cf. A272094, A372240, A372241.

Table[Product[(k^2)!, {k, 0, n}], {n, 0, 10}]
FoldList[Times,(Range[0,6]^2)!] (* Harvey P. Dale, Jan 30 2022 *)
Table[(n^2)!^(n+1) / Product[j^(Ceiling[Sqrt[j]]), {j, 1, n^2}], {n, 0, 6}] (* Vaclav Kotesovec, Apr 23 2024 *)
Table[(n^2)!^n * (n!)^2 / Product[j^(Floor[Sqrt[j]]), {j, 1, n^2}], {n, 0, 6}] (* Vaclav Kotesovec, Apr 23 2024 *)

{a(n) = prod(k=1, n, (k^2)!)} \\ Seiichi Manyama, Jul 06 2019

a(n) ~ c * n^((2*n + 1)*(2*n^2 + 2*n + 3)/6) * (2*Pi)^(n/2) / exp(5*n^3/9 + n^2/2 + n), where c = A255504 = 3.048330306522348566911920417337613015885313475... .
From Vaclav Kotesovec, Apr 23 2024: (Start)
a(n) = Product_{j=1..n^2} j^(n - ceiling(sqrt(j)) + 1).
a(n) = (n^2)!^n * (n!)^2 / Product_{j=1..n^2} j^(floor(sqrt(j))). (End)

A272095 a(n) = Product_{k=0..n} binomial(n^2,k).

Original entry on oeis.org

1, 1, 24, 27216, 1956864000, 11593630125000000, 7004354761049263478784000, 515246658615545697034849051407876096, 5368556637668593177532650186945239827409750982656, 9038577429104951379916309583338181472480254559457860096000000000

Offset: 0

Vaclav Kotesovec, Apr 20 2016

Cf. A272093, A272094, A272241.

Cf. A000178, A098694, A268196, A262261.

Table[Product[Binomial[n^2, k], {k, 0, n}], {n, 0, 10}]
Table[((n^2)!)^(n+1) * BarnesG[n^2 - n + 1] / (BarnesG[n^2 + 2] * BarnesG[n+2]), {n, 0, 10}]

a(n) = ((n^2)!)^(n+1) / (A272164(n) * A000178(n)).

a(n) ~ A * exp(3*n^2/4 + 5*n/6 - 1/8) * n^(n^2/2 - 5/12) / (2*Pi)^((n+1)/2), where A = A074962 is the Glaisher-Kinkelin constant.

A272093 a(n) = Product_{k=0..n} binomial(k*n,k).

Original entry on oeis.org

1, 1, 12, 3780, 44844800, 26352845268750, 953083353075475894272, 2537540586421634737033298208000, 579150777545101402084349505293757972480000, 12933741941622730846344367593442776825612980847409218750, 31768605393074559234133528464091374346848946682424165820313600000000000

Offset: 0

Vaclav Kotesovec, Apr 20 2016

Cf. A272094, A272094, A272095.

Cf. A000178, A098694, A268196, A262261.

Table[Product[Binomial[k*n, k], {k, 0, n}], {n, 0, 10}]

a(n) = A272096(n) / (A272166(n) * A000178(n)).

a(n) ~ A^2 * exp(n^2/2 + 3*n/4 + 1/12) * n^(n^2/2 - 1/3) / (2*Pi)^((n+1)/2), where A = A074962 is the Glaisher-Kinkelin constant.

A272168 a(n) = Product_{k=0..n} (k^2-k)!.

Original entry on oeis.org

1, 1, 2, 1440, 689762304000, 1678124094566146045378560000000, 445127215203413988036981576746329306509322538188800000000000000

Offset: 0

Vaclav Kotesovec, Apr 21 2016

The next term has 114 digits.

Cf. A272094, A272167.

Table[Product[(k^2-k)!, {k, 0, n}], {n, 0, 8}]

a(n) ~ c * n^(n*(2*n^2 + 1)/3) * (2*Pi)^(n/2) / exp(5*n^3/9 + n/2 - Zeta(3) / (2*Pi^2)), where c = Product_{k>=2} (k*(k-1))!/stirling(k*(k-1)) = 1.086533635964823338078329042... and stirling(n) = sqrt(2*Pi*n) * n^n / exp(n) is the Stirling approximation of n!.

A362187 a(n) = (n^2 - n)!.

Original entry on oeis.org

1, 1, 2, 720, 479001600, 2432902008176640000, 265252859812191058636308480000000, 1405006117752879898543142606244511569936384000000000, 710998587804863451854045647463724949736497978881168458687447040000000000000

Offset: 0

Stefano Spezia, Apr 10 2023

The next term has 104 digits.

For n > 0, a(n) is the number of n X n matrices using all the integers from 1 to n^2 and having the main diagonal given.

Winston de Greef, Table of n, a(n) for n = 0..21

Cf. A000142, A000290, A002378, A088020.
Cf. A039622, A179268, A272094, A272095, A272164.
Cf. A362209.

Mathematica
```
a[n_]:=(n^2-n)!; Array[a,9,0]
```

a(n) = (n^2 - n)*a(n-1) for n > 1.

a(n) = A000142(A002378(n-1)) for n > 0.

cp's OEIS Frontend

A014062 a(n) = binomial(n^2, n).

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A088020 a(n) = (n^2)!.

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A255322 a(n) = Product_{k=0..n} (k^2)!.

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A272095 a(n) = Product_{k=0..n} binomial(n^2,k).

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A272093 a(n) = Product_{k=0..n} binomial(k*n,k).

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A272168 a(n) = Product_{k=0..n} (k^2-k)!.

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A362187 a(n) = (n^2 - n)!.

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