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A092985
a(n) is the product of first n terms of an arithmetic progression with the first term 1 and common difference n.
8
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
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
Main diagonal of A256268.
Sequence in context: A367391 A056066 A174483 * A331196 A181588 A084880
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