A179268 -id:A179268 - cp's OEIS frontend

A027441 a(n) = (n^4 + n)/2 (Row sums of an n X n X n magic cube, when it exists).

0, 1, 9, 42, 130, 315, 651, 1204, 2052, 3285, 5005, 7326, 10374, 14287, 19215, 25320, 32776, 41769, 52497, 65170, 80010, 97251, 117139, 139932, 165900, 195325, 228501, 265734, 307342, 353655, 405015, 461776, 524304, 592977, 668185, 750330, 839826, 937099

Offset: 0

Eric W. Weisstein

Starting with offset 1 = binomial transform of (1, 8, 25, 30, 12, 0, 0, 0, ...). - Gary W. Adamson, May 20 2009

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000 (terms 0-680 by Vincenzo Librandi.)
Eric Weisstein's World of Mathematics, Magic Constant
Eric Weisstein's World of Mathematics, Magic Cube
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Subsequence of A057590.

Cf. A002061, A139562, A179268.

[(n^4+n)/2: n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

A027441:=n->(n^4+n)/2: seq(A027441(n), n=0..30); # Wesley Ivan Hurt, Aug 13 2014

Table[(n^4 + n)/2, {n, 0, 30}] (* Wesley Ivan Hurt, Aug 13 2014 *)
LinearRecurrence[{5,-10,10,-5,1},{0,1,9,42,130},40] (* Harvey P. Dale, Apr 09 2018 *)

a(n)=(n^4 + n)/2 \\ Charles R Greathouse IV, Jul 28 2015

O.g.f.: x*(1+4*x+7*x^2)/(1-x)^5. - R. J. Mathar, Feb 13 2008
a(n) = Sum_{k=n..n^2} k; for n>0: a(n) = A037270(n) - A000217(n-1). - Reinhard Zumkeller, Jul 06 2010
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Aug 13 2014
a(n) = A071232(n) / n^2, for n > 0. - Wesley Ivan Hurt, Aug 13 2014
a(n) = A002061(n)*A000217(n). - Anton Zakharov, Dec 16 2016
a(n) = (n+1)*(a(n-1)/(n-1) + n*(n-1)), a(0)=0, a(1)=1. - Vladimir Kruchinin, Oct 10 2018

More terms from Wesley Ivan Hurt, Aug 13 2014

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

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.

A369523 a(n) = n*(n^2 - 1)!.

Original entry on oeis.org

1, 12, 120960, 5230697472000, 3102242008666197196800000, 61998887798316869577999908025139200000000, 86897409147752508696036023331613625269650404482744320000000000, 15860866523235520512929173666895185100358189521818149024850236796901838028800000000000000

Offset: 1

Thomas Scheuerle, Jan 25 2024

a(n) is the number of ways to fill an n X n square matrix with n^2 distinct elements such that a chosen element is in a designated row (or alternatively a column).

Cf. A088020, A179268.

Maple
```
seq(n*factorial(n^2-1), n=1..8);
```
Mathematica
```
Table[n*(n^2-1)!, {n, 1, 8}]
```
PARI
```
a(n) = n*(n^2-1)!
```

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

a(n) = Integral_{x>=0} x^(n - 1)*exp(-x^(1/n)) dx.

cp's OEIS Frontend

A027441 a(n) = (n^4 + n)/2 (Row sums of an n X n X n magic cube, when it exists).

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

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

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A369523 a(n) = n*(n^2 - 1)!.

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