A088020 -id:A088020 - cp's OEIS frontend

A364206 a(n) is the number of n X n nonsingular matrices using all the integers from 1 to n^2.

1, 24, 360144, 20914499571840

Offset: 1

Stefano Spezia, Jul 13 2023

Right diagonal of A364203.
Cf. A085000 (maximal determinant), A350565 (minimal permanent), A350566 (maximal permanent).
Cf. A364227 (with prime numbers).
Cf. A088020, A136609, A221976.

a(n) = (n^2)! - A221976(n). - Vaclav Kotesovec, Jul 16 2023

a(4) from Vaclav Kotesovec, Jul 16 2023 (using A221976)

A364226 Triangle read by rows: T(n, k) is the number of n X n matrices of rank k using all the first n^2 prime numbers.

Original entry on oeis.org

1, 0, 24, 0, 216, 362664

Offset: 1

Stefano Spezia, Jul 15 2023

			The triangle begins:
  1;
  0,  24;
  0, 216, 362664;
  ...

Cf. A180128 (maximal determinant), A088020 (row sums), A350858 (minimal permanent), A350859 (maximal permanent), A364227 (right diagonal).

Cf. A364203 (with the integers in [n^2]).

A062782 a(n) = (n^2)! / (n^n).

Original entry on oeis.org

1, 1, 6, 13440, 81729648000, 4963587213865915514880, 7973107998754741458076119859200000000, 738615790595351500616546025309298211371540807680000000000

Offset: 0

Jason Earls, Jul 18 2001

Seiichi Manyama, Table of n, a(n) for n = 0..21

Cf. A000312, A088020.

Join[{1},Table[(n^2)!/n^n,{n,8}]] (* Harvey P. Dale, Aug 10 2021 *)

PARI
```
for(n=0, 8, print1((n^2)!/(n^n), ", "))
```

a(n) = A088020(n)/A000312(n). - Seiichi Manyama, Mar 26 2019

a(0)=1 prepended by Seiichi Manyama, Mar 26 2019

A076181 a(n) = (n^2)!/(n!)^(n+1)/(2n-1).

Original entry on oeis.org

1, 1, 56, 375375, 577185873264, 337143653615801672640, 112387296492020598351382561996800, 30035919199125011172228882156381540400453089375, 8612584407483201348696124216860807740603132996062773468500000000

Offset: 1

Benoit Cloitre, Nov 01 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..27

Cf. A088020.

a[n_] := (n^2)!/n!^(n+1)/(2*n-1); Array[a, 10] (* Amiram Eldar, May 05 2025 *)

A091711 Exponent of 2 in (n^2)!.

Original entry on oeis.org

0, 3, 7, 15, 22, 34, 46, 63, 78, 97, 116, 142, 165, 193, 221, 255, 286, 321, 356, 397, 435, 479, 526, 574, 620, 672, 723, 781, 836, 896, 956, 1023, 1086, 1153, 1220, 1293, 1363, 1439, 1514, 1597, 1676, 1758, 1842, 1931, 2017, 2113, 2205, 2302, 2396

Offset: 1

Ralf Stephan, Feb 09 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A007814(A088020(n)).

[Valuation(Factorial(n^2), 2): n in [1..50]]; // Vincenzo Librandi, May 16 2013

Table[IntegerExponent[(n^2)!,2],{n,50}] (* Harvey P. Dale, Nov 15 2018 *)

PARI
```
a(n)=valuation((n*n)!,2)
```

A256511 a(n) = (n^2 - 1)!.

Original entry on oeis.org

1, 6, 40320, 1307674368000, 620448401733239439360000, 10333147966386144929666651337523200000000, 12413915592536072670862289047373375038521486354677760000000000

Offset: 1

Nichole Suomi Smith, Mar 31 2015

nonn

Cf. A000142, A000290, A088020.

Maple
```
seq(factorial(i^2-1), i = 1 .. 8);
```

(Range[10]^2-1)! (* Harvey P. Dale, Oct 07 2023 *)

a(n) = (2*Pi)^(1/2 - n/2)*n^(n^2 - 1/2)*Product_{j=0..n-1} Gamma(n+j/n). - Karol A. Penson, Jan 11 2024

A382612 a(n) = n^3 * (n^2 - n + 1).

Original entry on oeis.org

0, 1, 24, 189, 832, 2625, 6696, 14749, 29184, 53217, 91000, 147741, 229824, 344929, 502152, 712125, 987136, 1341249, 1790424, 2352637, 3048000, 3898881, 4930024, 6168669, 7644672, 9390625, 11441976, 13837149, 16617664, 19828257, 23517000, 27735421, 32538624, 37985409, 44138392, 51064125, 58833216, 67520449, 77204904, 87970077, 99904000, 113099361, 127653624

Offset: 0

Wesley Ivan Hurt, Mar 31 2025

Product of the entries in the corners of an n X n square array with elements 1..n^2 listed in increasing order by rows (see example).

			                                                        [1   2  3  4  5]
                                        [1   2  3  4]   [6   7  8  9 10]
                              [1 2 3]   [5   6  7  8]   [11 12 13 14 15]
                     [1 2]    [4 5 6]   [9  10 11 12]   [16 17 18 19 20]
             [1]     [3 4]    [7 8 9]   [13 14 15 16]   [21 22 23 24 25]
  ------------------------------------------------------------------------
    n         1        2         3            4                 5
  ------------------------------------------------------------------------
              1     1*2*3*4   1*3*7*9     1*4*13*16         1*5*21*25
  ------------------------------------------------------------------------
    a(n)      1       24        189          832               2625

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A088020 (product of all entries).
Cf. A382532 (product along main antidiagonal).
Cf. A382620 (product along border).

[n^3*(n^2 - n + 1) : n in [0..50]]; // Wesley Ivan Hurt, Apr 15 2025

Mathematica
```
Table[n^3 (n^2 - n + 1), {n, 0, 60}]
```

G.f.: x*(1+18*x+60*x^2+38*x^3+3*x^4)/(x-1)^6. - R. J. Mathar, Apr 02 2025

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Wesley Ivan Hurt, Apr 15 2025

A264040 Number of possible permutations of the n X n generalization of the sliding block 15-puzzle.

Original entry on oeis.org

1, 12, 181440, 10461394944000, 7755605021665492992000000, 185996663394950608733999724075417600000000, 304140932017133780436126081660647688443776415689605120000000000, 63443466092942082051716694667580740401432758087272596099400947187607352115200000000000000

Offset: 1

Ben Whitmore, Nov 01 2015

For n > 1, of the permutations that can be reached by disassembling the puzzle and replacing the tiles, exactly half of them can be reached by sliding the tiles.

			a(4) = 10461394944000 because the standard 4 X 4 version of the 15-puzzle has exactly 10461394944000 permutations that can be reached by sliding the tiles.

Eric Weisstein's World of Mathematics, 15 Puzzle

Cf. A087725, A090031, A088020.

Mathematica
```
a[n_] := If[n == 1, 1, (n^2)!/2]
```

a(1) = 1; a(n) = (n^2)!/2 for n > 1.

a(1) added by Franklin T. Adams-Watters, Nov 11 2015

A364527 Triangle read by rows giving the number of square arrays composed of the numbers from 1 to n^2, counted up to rotation and reflection, with heterogeneity k, i.e., number of k different sums of rows, columns or diagonals with 1 <= k <= 2*n+2 for n > 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 3, 0, 1, 22, 346, 2060, 7989, 17160, 14662, 3120, 880

Offset: 1

Martin Renner, Jul 27 2023

T(n,1) gives the number of magic squares A006052(n).
For n > 1, T(n,2*n+2) gives the number of squares with maximum heterogeneity, i.e., all sums are different (but do not necessarily form a sequence of consecutive integers), sometimes called (super)heterogeneous squares or antimagic squares.
Subsets of T(n,2) or T(n,3) with one or both of the diagonal sums not equal to the magic constant are sometimes called semimagic squares.
Sum_{k=1..2*n+2} T(n,k) = A086829(n) = (n^2)!/8 for n > 1.

			T(n,k) starts with
  n = 1: 1;
  n = 2: 0, 0, 0, 0, 3, 0;
  n = 3: 1, 22, 346, 2060, 7989, 17160, 14662, 3120;
etc.
For n = 2 there are only three square arrays up to rotation and reflection, all of heterogeneity k = 5, i.e.,
  [1 2] [1 2] [1 3]
  [3 4] [4 3] [4 2]
since there are always the five different sums of rows, columns and diagonals 3, 4, 5, 6 and 7.
For n = 3 the lexicographically first square arrays of heterogeneity 1 <= k <= 8 are
  [2 7 6] [1 2 6] [1 2 5] [1 2 3] [1 2 3] [1 2 3] [1 2 3] [1 2 3]
  [9 5 1] [5 9 4] [3 9 6] [5 6 4] [4 5 6] [4 5 7] [4 5 6] [4 5 8]
  [4 3 8] [3 7 8] [4 7 8] [9 7 8] [7 8 9] [6 9 8] [7 9 8] [6 9 7]
For k = 1 we have the famous Lo Shu square with magic sum (n^3+n)/2 = 15. The other sums for the given examples are (9, 18), (8, 18, 19), (6, 15, 18, 24), (6, 12, 15, 18, 24), (6, 11, 14 16, 18, 23), (6, 12, 14, 15, 16, 17, 24) and (6, 11, 13, 14, 16, 17, 18, 22). Note that there are different sets of sums, namely a total of 6 with two values, 61 with three, 348 with four, 1295 with five, 2880 with six, 3845 with seven and 1538 with eight.

Pierre Berloquin, Garten der Sphinx. 150 mathematische Denkspiele, München 1984, p. 20, nr. 15 (Heterogene Quadrate), p. 20, nr. 16 (Antimagie), p. 86, nr. 148 (Höhere Antimagie), pp. 99-100, 178 (Solutions).

Eric Weisstein's World of Mathematics, Antimagic Square.
Eric Weisstein's World of Mathematics, Magic Square.
Eric Weisstein's World of Mathematics, Semimagic Square.

Cf. A006003, A006052, A050257, A086829, A088020.

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

A364206 a(n) is the number of n X n nonsingular matrices using all the integers from 1 to n^2.

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A364226 Triangle read by rows: T(n, k) is the number of n X n matrices of rank k using all the first n^2 prime numbers.

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A062782 a(n) = (n^2)! / (n^n).

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Mathematica

PARI

Formula

Extensions

A076181 a(n) = (n^2)!/(n!)^(n+1)/(2n-1).

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Mathematica

A091711 Exponent of 2 in (n^2)!.

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Magma

Mathematica

PARI

A256511 a(n) = (n^2 - 1)!.

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Maple

Mathematica

Formula

A382612 a(n) = n^3 * (n^2 - n + 1).

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Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A264040 Number of possible permutations of the n X n generalization of the sliding block 15-puzzle.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A364527 Triangle read by rows giving the number of square arrays composed of the numbers from 1 to n^2, counted up to rotation and reflection, with heterogeneity k, i.e., number of k different sums of rows, columns or diagonals with 1 <= k <= 2*n+2 for n > 1.

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Author

Keywords