A158887 a(n) = (n+1)^n * n! * binomial(n-1 + 1/(n+1), n).

1, 1, 4, 45, 1056, 43225, 2756160, 253586025, 31872332800, 5252921480961, 1099886703552000, 285322741626047125, 89844523369696972800, 33764841634845724313625, 14930493174337400252809216

Offset: 0

Author

Paul D. Hanna, May 01 2009

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 diagonal, n > 0 (see example). - Wesley Ivan Hurt, May 16 2025

Examples

			a(1) = 1, a(2) = 1*4, a(3) = 1*5*9, a(4) = 1*6*11*16, a(5) = 1*7*13*19*25.
From _Wesley Ivan Hurt_, May 16 2025: (Start)
                                                       [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        4         45         1056             43225
(End)
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..230

Programs

GAP

List([0..15], n-> Product([0..n-1], j-> j*(n+1)+1) ); # G. C. Greubel, Mar 04 2020

Magma

[1] cat [&*[j*(n+1)+1: j in [0..n-1]]: n in [1..15]]; // G. C. Greubel, Mar 04 2020

Maple

seq(mul(j*(n+1)+1, j=0..n-1), n = 0..15); # G. C. Greubel, Mar 04 2020

Mathematica

Table[(n+1)^n n!Binomial[n-1+1/(n+1),n],{n,0,20}] (* Harvey P. Dale, Oct 26 2011 *)
a[n_] := (1 + n)^n Gamma[n + 1/(1 + n)]/Gamma[1/(n + 1)] // FullSimplify
Table[a[n], {n, 0, 20}] (* Gerry Martens, May 30 2018 *)

PARI

a(n)=(n+1)^n*n!*polcoeff(1/(1-x+x*O(x^n))^(1/(n+1)),n)

PARI

a(n)=if(n==0,1,prod(k=0,n-1,k*(n+1)+1))

Sage

[product(j*(n+1)+1 for j in (0..n-1)) for n in (0..15)] # G. C. Greubel, Mar 04 2020

Formula

a(n) = Product_{k=0..n-1} (k*(n+1) + 1) for n>0 with a(0)=1.

a(n) = coefficient of x^n/(n!*(n+1)^n) in 1/(1-x)^(1/(n+1)).

a(n) ~ sqrt(2*Pi) * exp(1-n) * n^(2*n-3/2). - Vaclav Kotesovec, Jun 28 2015

a(n) = (1+n)^n * gamma(n+1/(n+1)) / gamma(1/(n+1)). - Gerry Martens, May 30 2018