This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A108552 #10 Dec 11 2020 18:02:01
%S A108552 1,1,8,180,1120,8064,604800,68428800,830269440,10897286400,
%T A108552 2324754432000,640237370572800,11585247657984000,221172909834240000,
%U A108552 93666727314800640000,2068161339110798131200,47726800133326110720000,1148978521728221184000000,28806532937614688256000000
%N A108552 Integer values of (1*2*...*k)/(1+2+...+k) = k!/T(k) = A000142(k)/A000217(k), k>=1.
%C A108552 A000142(n)/A000217(n) = n!/(n*(n+1)/2) = 2*(n-1)!/(n+1) is an integer iff n = 1 or n + 1 is composite; i.e., iff n is a term of A060462.
%F A108552 a(m) = 2*(A060462(m)-1)!/(A060462(m)+1) = A000142(A060462(m))/A000217(A060462(m)).
%p A108552 select(x-> denom(x)=1, [k!/(k*(k+1)/2)$k=1..30])[]; # _Alois P. Heinz_, Dec 11 2020
%t A108552 Select[Table[(n - 1)!/((n (n - 1))/2), {n, 2, 50}], IntegerQ[#] &] (* _Geoffrey Critzer_, May 02 2015 *)
%o A108552 (PARI) for(n=1,50, r=2*(n-1)!/(n+1); if(denominator(r)==1, print1(r,",")))
%Y A108552 Cf. A060462 (corresponding k), A000142 (factorials), A000217 (triangular numbers).
%K A108552 nonn
%O A108552 1,3
%A A108552 _Rick L. Shepherd_, Jun 09 2005
%E A108552 Offset corrected by _Alois P. Heinz_, Dec 11 2020