A272095 -id:A272095 - cp's OEIS frontend

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

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

A272094 a(n) = Product_{k=0..n} binomial(k^2,k).

Original entry on oeis.org

1, 1, 6, 504, 917280, 48735086400, 94925811409228800, 8154182636726616909619200, 36091760791026276649159689107865600, 9415901310649088228943246038670339934863360000, 162992165498634702043940163611264755298214594247272038400000

Offset: 0

Vaclav Kotesovec, Apr 20 2016

Cf. A255322, A272093, A272095.

Cf. A000178, A098694, A268196, A262261.

Table[Product[Binomial[k^2, k], {k, 0, n}], {n, 0, 10}]

a(n) = A255322(n) / (A272168(n) * A000178(n)).

a(n) ~ c1/c2 * A * exp(-1/12 + n/2 + n^2/4) * n^(1/12 + n^2/2) / (2*Pi)^(n/2), where c1 = Product_{k>=1} (k^2)!/stirling(k^2) = 1.14426047263759216966268786..., c2 = Product_{k>=2} (k*(k-1))!/stirling(k*(k-1)) = 1.086533635964823338078329..., stirling(n) = sqrt(2*Pi*n) * n^n / exp(n) is the Stirling approximation of n!, and A = A074962 is the Glaisher-Kinkelin constant.

A272164 a(n) = Product_{k=0..n} (n^2-k)!.

Original entry on oeis.org

1, 1, 288, 53094139822080000, 7114507432973653690572666462301501337370624000000000000

Offset: 0

Vaclav Kotesovec, Apr 21 2016

The next term has 392 digits.

Cf. A272095, A272163, A272238, A272241.

Table[Product[(n^2-k)!, {k, 0, n}], {n, 0, 6}]
Table[BarnesG[n^2 + 2]/BarnesG[n^2 - n + 1], {n, 0, 6}]

a(n) = A272163(n) * ((n^2)!)^(n+1) / A272179(n)^n.

a(n) ~ exp(1/24 + n/6 - n^2 - n^3) * n^(1 + n^2 + 2*n^3) * (2*Pi)^((n+1)/2).

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.

A371603 a(n) = Product_{k=0..n} binomial(n^2, k^2).

Original entry on oeis.org

1, 1, 4, 1134, 333132800, 1319947441510156250, 876533819183888230348458418944000, 1185269534290897564185384010731432113450477770983533184

Offset: 0

Vaclav Kotesovec, Mar 30 2024

nonn

Cf. A001142, A255322, A272095, A296591, A362288, A371624, A371646.

Table[Product[Binomial[n^2, k^2], {k, 0, n}], {n, 0, 8}]

a(n) = (n^2)!^(n+1) / (A255322(n) * A371624(n)).

a(n) ~ c * exp(2*n*(2*n^2/3 + 1)) / (A^(2*n) * 2^(4*n*(n^2 + 1)/3) * Pi^(n/2) * n^(7*n/6 - 1/4)), where c = 0.6367427... and A is the Glaisher-Kinkelin constant A074962.

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

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

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A272094 a(n) = Product_{k=0..n} binomial(k^2,k).

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A272164 a(n) = Product_{k=0..n} (n^2-k)!.

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

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A371603 a(n) = Product_{k=0..n} binomial(n^2, k^2).

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

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