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%I #14 Oct 03 2020 03:30:09
%S 1,2,8,64,976,23072,808688,48448384,5085859456,787587828992,
%T 172251228685568,61567677411810304,37205957567375604736,
%U 32218626571889542694912,38411427146174647342235648,73187646662485142233440845824,231273043503438376340776532770816
%N E.g.f.: Sum_{n>=0} 2^(-n*(n+1)/2!) * Product_{k=0..n} tan(2^k*x).
%C Limit a(n)*a(n+2)/a(n+1)^2 appears to have 4 attractors near [1.33088873225, 1.28507876546, 1.49830439017, 1.56094802901]. [Extended by _Vaclav Kotesovec_, Oct 03 2020]
%C Limit ( a(n)*a(n+5)/(a(n+1)*a(n+4)) )^(1/4) appears to converge (1.41...?).
%H Vaclav Kotesovec, <a href="/A194456/b194456.txt">Table of n, a(n) for n = 1..106</a>
%F E.g.f.: Sum_{n>=0} sin(x)^(n+1) * Product_{k=0..n} cos(2^k*x)^(n-1-k).
%e E.g.f.: A(x) = x + 2*x^2/2! + 8*x^3/3! + 64*x^4/4! + 976*x^5/5! +...
%e where
%e A(x) = tan(x) + tan(x)*tan(2*x)/2 + tan(x)*tan(2*x)*tan(4*x)/2^3 + tan(x)*tan(2*x)*tan(4*x)*tan(8*x)/2^6 +...
%e A(x) = sin(x)/cos(x) + sin(x)^2/cos(2*x) + sin(x)^3*cos(x)/cos(4*x) + sin(x)^4*cos(x)^2*cos(2*x)/cos(8*x) + sin(x)^5*cos(x)^3*cos(2*x)^2*cos(4*x)/cos(16*x) + sin(x)^6*cos(x)^4*cos(2*x)^3*cos(4*x)^2*cos(8*x)/cos(32*x) +...
%o (PARI) {a(n)=local(A=sum(m=0,n,2^(-m*(m+1)/2!)*prod(k=0,m,tan(2^k*x+x*O(x^n)))));n!*polcoeff(A,n)}
%o (PARI) {a(n)=local(X=x+x*O(x^n),A=sum(m=0,n,sin(X)^(m+1)*prod(k=0,m,cos(2^k*X)^(m-1-k))));n!*polcoeff(A,n)}
%Y Cf. A194457, A194026, A194027.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 24 2011
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