A092985 a(n) is the product of the first n terms of an arithmetic progression with the first term 1 and common difference n.

1, 1, 3, 28, 585, 22176, 1339975, 118514880, 14454403425, 2326680294400, 478015854767451, 122087424094272000, 37947924636264267625, 14105590169042424729600, 6178966019176767549393375, 3150334059785191453342744576, 1849556085478041490537172810625

Offset: 0

Author

Amarnath Murthy, Mar 28 2004

Keywords

• easy
• nonn

Comments

We have the triangle (chopped versions of A076110, A162609)

1;

1 3;

1 4 7;

1 5 9 13;

1 6 11 16 21;

1 7 13 19 25 31;

...

Sequence contains the product of the terms of the rows.

a(n) = b(n-1) where b(n) = n^n*Gamma(n+1/n)/Gamma(1/n) and b(0) is limit n->0+ of b(n). - Gerald McGarvey, Nov 10 2007

Product of the entries in the first column of an n X n square array with elements 1..n^2 listed in increasing order by rows. - Wesley Ivan Hurt, Apr 02 2025

Examples

			a(5) = 1*6*11*16*21 = 22176.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A057237, A092987.

Main diagonal of A256268.

Programs

GAP

List([0..20], n-> Product([0..n-1], j-> j*n+1) ); # G. C. Greubel, Mar 04 2020

Magma

[1] cat [ (&*[j*n+1: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Mar 04 2020

Maple

a:= n-> mul(n*j+1, j=0..n-1):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 24 2015

Mathematica

Flatten[{1, Table[n^n * Pochhammer[1/n, n], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 05 2018 *)

PARI

vector(21, n, my(m=n-1); prod(j=0,m-1, j*m+1)) \\ G. C. Greubel, Mar 04 2020

Sage

[product(j*n+1 for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Mar 04 2020

Formula

a(n) = Product_{k=1..n} (1+(k-1)*n) = 1*(1+n)*(1+2n)*...*(n^2-n+1).

a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*n^(n-k). - Vladeta Jovovic, Jan 28 2005

a(n) = n! * [x^n] 1/(1 - n*x)^(1/n) for n > 0. - Ilya Gutkovskiy, Oct 05 2018

a(n) ~ sqrt(2*Pi) * n^(2*n - 3/2) / exp(n). - Vaclav Kotesovec, Oct 05 2018

Extensions

More terms from Erich Friedman, Aug 08 2005

Offset corrected by Alois P. Heinz, Nov 24 2015