A382532 -id:A382532 - cp's OEIS frontend

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

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

A382620 a(n) = n^(2n-4) (n!)^2 * (n^2)! * Pochhammer(1+1/n, n-1) / ((n^2-n+1) * (n^2-n)!).

Original entry on oeis.org

1, 24, 72576, 4528742400, 2423748096000000, 6787796602812825600000, 72775351435975459999580160000, 2410818176289650624878632291532800000, 211160088068074747246458003999015567360000000, 43450506124990177923906533235556142284800000000000000, 19145311724106592586650799558102522667408683773722624000000000

Offset: 1

Wesley Ivan Hurt, Apr 01 2025

nonn

Product of the entries on the border of an n X n square array with elements 1..n^2 listed in increasing order by rows.

			                                                        [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
  ------------------------------------------------------------------------
    a(n)      1        24      72576      4528742400    2423748096000000

Cf. A382532, A382612.

Table[n^(2n - 4)*(n!)^2*(n^2)!*Pochhammer[1 + 1/n, n - 1]/((n^2 - n + 1)*(n^2 - n)!), {n, 12}]

a(n) ~ 2^(3/2) * Pi^(3/2) * n^(7*n - 11/2) / exp(3*n + 1/2). - Vaclav Kotesovec, Apr 01 2025

A382597 a(n) = Product_{i=1..n} 1 - i + n(n - i + 1) - (n - 2i + 1)*((n - i + 1) mod 2).

Original entry on oeis.org

1, 1, 8, 105, 3840, 181545, 15814656, 1635491025, 261144576000, 47396578806225, 12046266925056000, 3390530144534798265, 1256223498048110592000, 506594307608708171477625, 257699484814807738928332800, 140934799049120316306629726625, 94240120920042785192632469422080

Offset: 0

Stefano Spezia, Mar 31 2025

nonn

a(n) is the product of the elements of the main antidiagonal of the n X n square matrix M(n) formed by writing the numbers 1, ..., n^2 successively forward and backward along the rows in zig-zag pattern (see A317614).

Except for n = 0, 2, and 6, a(n) has trailing zeros iff n is even.

			a(4) = 3840:
   1,  2,  3,  4;
   8,  7,  6,  5;
   9, 10, 11, 12;
  16, 15, 14, 13.

Cf. A317614, A370753, A382532.

a[n_]:=Product[1-i+n(n-i+1)-(n-2i+1)Mod[n-i+1,2],{i,n}]; Array[a,17,0]

cp's OEIS Frontend

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

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A382620 a(n) = n^(2n-4) (n!)^2 * (n^2)! * Pochhammer(1+1/n, n-1) / ((n^2-n+1) * (n^2-n)!).

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Formula

A382597 a(n) = Product_{i=1..n} 1 - i + n(n - i + 1) - (n - 2i + 1)*((n - i + 1) mod 2).

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A382620 a(n) = n^(2*n-4) * (n!)^2 * (n^2)! * Pochhammer(1+1/n, n-1) / ((n^2-n+1) * (n^2-n)!).

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A382597 a(n) = Product_{i=1..n} 1 - i + n*(n - i + 1) - (n - 2*i + 1)*((n - i + 1) mod 2).

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A382620 a(n) = n^(2n-4) (n!)^2 * (n^2)! * Pochhammer(1+1/n, n-1) / ((n^2-n+1) * (n^2-n)!).

A382597 a(n) = Product_{i=1..n} 1 - i + n(n - i + 1) - (n - 2i + 1)*((n - i + 1) mod 2).