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%I #32 Nov 19 2020 04:37:24
%S 1,1,4,144,331776,2751882854400,272622932796264897576960000,
%T 3641839910835401567626683591527643364677019238400000000
%N Related to number of elements in the free band (idempotent semigroup) on n generators.
%C Continued square root 2 = sqrt(1 + sqrt(1 + sqrt(4 + sqrt(144 + ...)))) = sqrt(1 + sqrt(1 + 2*sqrt(1 + 3*sqrt(1 + 4*sqrt(1 + ...)))) [S. Ramanujan]. - _Michael Somos_, Dec 03 2017
%D John M. Howie, Fundamentals of Semigroup Theory, Oxford University Press 1995, p. 123.
%H <a href="/index/Se#semigroups">Index entries for sequences related to semigroups</a>
%F Binomial transform is A005345. - _Michael Somos_, Oct 22 2006
%F a(n) = (n*a(n-1))^2 if n > 0. a(0)=1. - _Michael Somos_, Oct 22 2006
%F a(n) = Product_{i=1..n} (n-i+1)^(2^i).
%F Sum_{n>=1} 1/a(n) = A258621. - _Amiram Eldar_, Nov 19 2020
%t s=1;lst={};Do[AppendTo[lst,s*=s*=n],{n,9}];lst (* _Vladimir Joseph Stephan Orlovsky_, Oct 20 2009 *)
%t Fold[Append[#1, (#2 Last[#1])^2] &, {1}, Range@ 7] (* _Michael De Vlieger_, Dec 03 2017 *)
%o (PARI) {a(n) = if(n<0, 0, prod(i=1, n, (n-i+1)^2^i))}; /* _Michael Somos_, Oct 22 2006 */
%o (Sage)
%o def A030450(n) :
%o return prod((n-i+1)^(2^i) for i in (1..n))
%o [A030450(n) for n in (0..9)] # _Jani Melik_, Jun 06 2015
%Y Cf. A005345, A030449, A258621.
%Y A052129(n) = n*a(n-1) if n > 0.
%K nonn
%O 0,3
%A Marcel Jackson (marcel_j(AT)hilbert.maths.utas.edu.au)
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