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

Author

Wesley Ivan Hurt, Mar 30 2025

Keywords

• nonn

Comments

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).

Examples

			                                                       [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
		

Crossrefs

Row products of A081493.

Cf. A006003, A370753.

Programs

Mathematica

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

Formula

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