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

%I A092985 #40 Apr 02 2025 09:53:59
%S A092985 1,1,3,28,585,22176,1339975,118514880,14454403425,2326680294400,
%T A092985 478015854767451,122087424094272000,37947924636264267625,
%U A092985 14105590169042424729600,6178966019176767549393375,3150334059785191453342744576,1849556085478041490537172810625
%N A092985 a(n) is the product of the first n terms of an arithmetic progression with the first term 1 and common difference n.
%C A092985 We have the triangle (chopped versions of A076110, A162609)
%C A092985   1;
%C A092985   1 3;
%C A092985   1 4  7;
%C A092985   1 5  9 13;
%C A092985   1 6 11 16 21;
%C A092985   1 7 13 19 25 31;
%C A092985 ...
%C A092985 Sequence contains the product of the terms of the rows.
%C A092985 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
%C A092985 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
%H A092985 Alois P. Heinz, <a href="/A092985/b092985.txt">Table of n, a(n) for n = 0..200</a>
%F A092985 a(n) = Product_{k=1..n} (1+(k-1)*n) = 1*(1+n)*(1+2n)*...*(n^2-n+1).
%F A092985 a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*n^(n-k). - _Vladeta Jovovic_, Jan 28 2005
%F A092985 a(n) = n! * [x^n] 1/(1 - n*x)^(1/n) for n > 0. - _Ilya Gutkovskiy_, Oct 05 2018
%F A092985 a(n) ~ sqrt(2*Pi) * n^(2*n - 3/2) / exp(n). - _Vaclav Kotesovec_, Oct 05 2018
%e A092985 a(5) = 1*6*11*16*21 = 22176.
%p A092985 a:= n-> mul(n*j+1, j=0..n-1):
%p A092985 seq(a(n), n=0..20);  # _Alois P. Heinz_, Nov 24 2015
%t A092985 Flatten[{1, Table[n^n * Pochhammer[1/n, n], {n, 1, 20}]}] (* _Vaclav Kotesovec_, Oct 05 2018 *)
%o A092985 (PARI) vector(21, n, my(m=n-1); prod(j=0,m-1, j*m+1)) \\ _G. C. Greubel_, Mar 04 2020
%o A092985 (Magma) [1] cat [ (&*[j*n+1: j in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Mar 04 2020
%o A092985 (Sage) [product(j*n+1 for j in (0..n-1)) for n in (0..20)] # _G. C. Greubel_, Mar 04 2020
%o A092985 (GAP) List([0..20], n-> Product([0..n-1], j-> j*n+1) ); # _G. C. Greubel_, Mar 04 2020
%Y A092985 Cf. A057237, A092987.
%Y A092985 Main diagonal of A256268.
%K A092985 easy,nonn
%O A092985 0,3
%A A092985 _Amarnath Murthy_, Mar 28 2004
%E A092985 More terms from _Erich Friedman_, Aug 08 2005
%E A092985 Offset corrected by _Alois P. Heinz_, Nov 24 2015