A092985 - OEIS

%I #38 Mar 30 2023 16:23:16

%S 1,1,3,28,585,22176,1339975,118514880,14454403425,2326680294400,

%T 478015854767451,122087424094272000,37947924636264267625,

%U 14105590169042424729600,6178966019176767549393375,3150334059785191453342744576,1849556085478041490537172810625

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

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

%C   1;

%C   1 3;

%C   1 4  7;

%C   1 5  9 13;

%C   1 6 11 16 21;

%C   1 7 13 19 25 31;

%C ...

%C Sequence contains the product of the terms of the rows.

%C 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

%H Alois P. Heinz, <a href="/A092985/b092985.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = 1*(1+n)*(1+2n)*...*(n^2-n+1).

%F a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*n^(n-k). - _Vladeta Jovovic_, Jan 28 2005

%F a(n) = n! * [x^n] 1/(1 - n*x)^(1/n) for n > 0. - _Ilya Gutkovskiy_, Oct 05 2018

%F a(n) ~ sqrt(2*Pi) * n^(2*n - 3/2) / exp(n). - _Vaclav Kotesovec_, Oct 05 2018

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

%p a:= n-> mul(n*j+1, j=0..n-1):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Nov 24 2015

%t Flatten[{1, Table[n^n * Pochhammer[1/n, n], {n, 1, 20}]}] (* _Vaclav Kotesovec_, Oct 05 2018 *)

%o (PARI) vector(21, n, my(m=n-1); prod(j=0,m-1, j*m+1)) \\ _G. C. Greubel_, Mar 04 2020

%o (Magma) [1] cat [ (&*[j*n+1: j in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Mar 04 2020

%o (Sage) [product(j*n+1 for j in (0..n-1)) for n in (0..20)] # _G. C. Greubel_, Mar 04 2020

%o (GAP) List([0..20], n-> Product([0..n-1], j-> j*n+1) ); # _G. C. Greubel_, Mar 04 2020

%Y Cf. A057237, A092987.

%Y Main diagonal of A256268.

%K easy,nonn

%O 0,3

%A _Amarnath Murthy_, Mar 28 2004

%E More terms from _Erich Friedman_, Aug 08 2005

%E Offset corrected by _Alois P. Heinz_, Nov 24 2015