A395280 a(n) = Product_{k=1..n-2} (n*k + n - 1)! / (n*k + 1)!.

1, 1, 5, 4620, 6399679104, 54803251858747392000, 8358350073642819567827288064000000, 52632311069252193290838179018133211213037568000000000, 27453701472592984509237630691917200216593343493991841417624813568000000000000, 2147874070223278653793958679139553326245411209910799706636182402354754938369146903440396050759680000000000

Offset

1,3

Comments

Product of the entries not on the border of an n X n square array with elements 1..n^2 listed in increasing order by rows (or 1 where no such entries exist). [See example]

Links

Table of n, a(n) for n=1..10.

Formula

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

a(n) ~ n^(2*n^2 - 7*n + 13/2) / (2*Pi * exp(n^2 - 3*n - 1/2)). - Vaclav Kotesovec, Apr 18 2026

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)      (1)        5     6*7*10*11 = 4620     6399679104

Mathematica

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

Crossrefs

Cf. A088020, A382532, A382612, A382620.

Sequence in context: A276238 A357125 A086896 * A177503 A072021 A196627

Adjacent sequences: A395277 A395278 A395279 * A395281 A395282 A395283

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 18 2026

Status

approved