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%I #5 Mar 30 2012 18:37:28
%S 1,1,2,2,-104,-3704,-113968,-2059408,362847616,113325214336,
%T 25903723142912,3643199606296832,-2713129993221899264,
%U -5529958919819578778624,-7506437411940645017540608,-6418592915828033819245078528,20775854326350479088039843168256
%N E.g.f.: Sum_{n>=0} sin(2^n*x)^n / 2^(n^2).
%C Signs of terms are of period 8: [+,+,+,+,-,-,-,-].
%F E.g.f.: Sum_{n>=0} sin(x)^n * Product_{k=0..n-1} cos(2^k*x)^n.
%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 2*x^3/3! - 104*x^4/4! - 3704*x^5/5! +...
%e where
%e A(x) = 1 + sin(2*x)/2 + sin(2^2*x)^2/2^4 + sin(2^3*x)^3/2^9 + sin(2^4*x)^4/2^16 +...
%e Also,
%e A(x) = 1 + sin(x)*cos(x) + sin(x)^2*cos(x)^2*cos(2*x)^2 + sin(x)^3*cos(x)^3*cos(2*x)^3*cos(4*x)^3 + sin(x)^4*cos(x)^4*cos(2*x)^4*cos(4*x)^4*cos(8*x)^4 +...
%o (PARI) {a(n)=local(X=x+x*O(x^n),A=sum(m=0,n,sin(2^m*X)^m/2^(m^2)));n!*polcoeff(A,n)}
%o (PARI) {a(n)=local(X=x+x*O(x^n),A=sum(m=0,n,sin(X)^m*prod(k=0,m-1,cos(2^k*X)^m)));n!*polcoeff(A,n)}
%Y Cf. A194026.
%K sign
%O 0,3
%A _Paul D. Hanna_, Aug 12 2011
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