A370753 -id:A370753 - cp's OEIS frontend

A382532 a(n) = Product_{k=1..n} (n*k-k+1).

1, 1, 6, 105, 3640, 208845, 17873856, 2131900225, 337767408000, 68586144251625, 17361688356812800, 5359035747797893161, 1980990543353657472000, 863884504344556052483125, 438824910158909833337856000, 256841080519120696725634418625, 171586094900260237697765926076416

Offset: 0

Wesley Ivan Hurt, Mar 30 2025

nonn

Fill an n X n square array with the numbers 1..n^2 in increasing order by rows. a(n) is the product of the numbers along the main antidiagonal (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
  ------------------------------------------------------------------------
   a(n)      1        6         105         3640             208845

Row products of A081493.

Cf. A006003, A370753.

Table[Product[n*k - k + 1, {k, n}], {n, 0, 20}]

a(n) = (n-1)^n * Pochhammer(n/(n-1), n) for n>=2.
a(n) = Product_{k=1..n} A081493(n,k).
a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / exp(n+1). - 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

A382532 a(n) = Product_{k=1..n} (n*k-k+1).

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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

A382532 a(n) = Product_{k=1..n} (n*k-k+1).

Views

Author

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Comments

Examples

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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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Mathematica

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