OFFSET
1,2
COMMENTS
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]
Limit ( a(n)*a(n+5)/(a(n+1)*a(n+4)) )^(1/4) appears to converge (1.41...?).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..106
FORMULA
E.g.f.: Sum_{n>=0} sin(x)^(n+1) * Product_{k=0..n} cos(2^k*x)^(n-1-k).
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 8*x^3/3! + 64*x^4/4! + 976*x^5/5! +...
where
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 +...
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) +...
PROG
(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)}
(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)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 24 2011
STATUS
approved