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Write the natural numbers in groups: 1; 2,3; 4,5,6; 7,8,9,10; ... and multiply the members of each group.
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%I #16 Jul 13 2021 09:42:26

%S 1,6,120,5040,360360,39070080,5967561600,1220096908800,

%T 321570878428800,106137499051584000,42873948150095462400,

%U 20803502274492921984000,11938961126118491232768000,7998487694738166709923840000,6185983879158893706209144832000

%N Write the natural numbers in groups: 1; 2,3; 4,5,6; 7,8,9,10; ... and multiply the members of each group.

%C Each group begins with a triangular number + 1 and proceeds until the next triangular number.

%H Alois P. Heinz, <a href="/A057003/b057003.txt">Table of n, a(n) for n = 1..200</a>

%F (n (n + 1)/2)!/((n - 1) n /2)!.

%F a(n) ~ n^(2*n)/2^n. - _Vaclav Kotesovec_, Jul 13 2021

%p a:= n-> binomial(n*(n+1)/2, n) * n!:

%p seq(a(n), n=1..20); # _Alois P. Heinz_, Aug 17 2012

%t Table[(n (n + 1)/2)!/((n - 1) n /2)!, {n, 1, 15}]

%t With[{nn=15},Times@@@TakeList[Range[(nn(nn+1))/2],Range[nn]]] (* Requires Mathematica version 11 or later *) (* _Harvey P. Dale_, Sep 21 2017 *)

%Y Cf. A006003.

%K nonn

%O 1,2

%A _Robert G. Wilson v_, Sep 09 2000