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A230714 a(n) = lcm(1,2,...,n)*a(floor(n/2))^2 for n > 0, a(0) = 1.

%I A230714 #15 Mar 04 2018 23:13:54
%S A230714 1,1,2,6,48,240,2160,15120,1935360,5806080,145152000,1596672000,
%T A230714 129330432000,1681295616000,82383485184000,82383485184000,
%U A230714 2699542042509312000,45892214722658304000,413029932503924736000,7847568717574569984000,4904730448484106240000000
%N A230714 a(n) = lcm(1,2,...,n)*a(floor(n/2))^2 for n > 0, a(0) = 1.
%C A230714 This sequence essentially documents the identity sqrt(A000142(n)*A205958(n)*A056040(n)*A180000(n)) = A003418(n)*a(floor(n/2)). Some interest derives from the fact that A056040 and A180000 can be computed by a structurally identical algorithm which can be used to compute A205958 and A000142, the latter being the fastest algorithm presently known to compute the factorial numbers. a(n) relates these numbers with lcm(1,2,3,...,n).
%H A230714 Michael De Vlieger, <a href="/A230714/b230714.txt">Table of n, a(n) for n = 0..351</a>
%F A230714 a(n) = A205958(n)*n!.
%p A230714 a := n -> `if`(n = 0, 1, lcm(seq(i,i=1..n))*a(floor(n/2))^2): seq(a(n),n=0..20);
%t A230714 Fold[Append[#1, Apply[LCM, Range@ #2] #1[[Floor[#2/2] + 1]]^2 ] &, {1}, Range@ 20] (* _Michael De Vlieger_, Mar 04 2018 *)
%o A230714 (Sage)
%o A230714 def A230714(n):
%o A230714     return factorial(n)*A205958(n)
%o A230714 [A230714(n) for n in (0..20)]
%o A230714 (PARI) a(n)=lcm(vector(n,i,i))*if(n>3,a(n\2)^2,1) \\ _Charles R Greathouse IV_, Oct 31 2013
%Y A230714 Cf. A003418, A180000.
%K A230714 nonn
%O A230714 0,3
%A A230714 _Peter Luschny_, Oct 28 2013