A092985 a(n) is the product of 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

0,3

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

Links

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

Formula

a(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

Example

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

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 *)

Prog

(PARI) vector(21, n, my(m=n-1); prod(j=0, m-1, j*m+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
(Sage) [product(j*n+1 for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Mar 04 2020
(GAP) List([0..20], n-> Product([0..n-1], j-> j*n+1) ); # G. C. Greubel, Mar 04 2020

Crossrefs

Cf. A057237, A092987.

Main diagonal of A256268.

Sequence in context: A367391 A056066 A174483 * A331196 A181588 A084880

Adjacent sequences: A092982 A092983 A092984 * A092986 A092987 A092988

Keyword

easy,nonn

Author

Amarnath Murthy, Mar 28 2004

Extensions

More terms from Erich Friedman, Aug 08 2005

Offset corrected by Alois P. Heinz, Nov 24 2015

Status

approved