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

0,3

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).

Links

Table of n, a(n) for n=0..16.

Formula

a(n) = (n-1)^n * Pochhammer(n/(n-1), n) for n>=2.

a(n) = Product_{k=1..n} A081493(n,k). - Alois P. Heinz, Mar 31 2025

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

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

Mathematica

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

Crossrefs

Row products of A081493.

Cf. A006003, A088020, A370753, A395280.

Sequence in context: A110342 A380647 A227207 * A126467 A013294 A013300

Adjacent sequences: A382529 A382530 A382531 * A382533 A382534 A382535

Keyword

nonn

Author

Wesley Ivan Hurt, Mar 30 2025

Status

approved