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 A101189 G.f. defined as the limit: A(x) = limit_{n->oo} F(n)^(1/2^(n-1)) where F(n) is the n-th iteration of: F(0) = 1, F(n) = F(n-1)^2 + (2x)^(2^n-1) for n>=1. 5

%I

%S 1,2,0,4,-8,16,-40,144,-512,1696,-5696,19840,-70048,247744,-880128,

%T 3152768,-11386624,41389568,-151273728,555794944,-2052141056,

%U 7610274816,-28331018240,105833345024,-396594444800,1490425179136,-5615651143680,21209004267520,-80276663808000

%N G.f. defined as the limit: A(x) = limit_{n->oo} F(n)^(1/2^(n-1)) where F(n) is the n-th iteration of: F(0) = 1, F(n) = F(n-1)^2 + (2x)^(2^n-1) for n>=1.

%C The coefficients of x^n in A(x/2)^(1/2) equals A101190(n)/2^A005187(n). The coefficients of x^n in A(x/2)^(1/4) equals A101191(n)/2^A004134(n). A101190 and A101191 are related to doubly exponential numbers A003095 and to Catalan numbers (A000108).

%F G.f. A(x) = [Sum_{n>=0} A101190(n)/2^A005187(n)*(2x)^n]^2. G.f. A(x) = [Sum_{n>=0} A101191(n)/2^A004134(n)*(2x)^n]^4.

%e The iteration begins:

%e F(0) = 1,

%e F(1) = F(0)^2 + (2*x)^(2^1-1)

%e = 1 +2*x,

%e F(2) = F(1)^2 + (2*x)^(2^2-1)

%e = 1 +4*x +4*x^2 +8*x^3,

%e F(3) = F(2)^2 + (2*x)^(2^3-1)

%e = 1 +8*x +24*x^2 +48*x^3 +80*x^4 +64*x^5 +64*x^6 +128*x^7.

%e The 2^(n-1)-th roots of F(n) tend to the limit of A(x):

%e F(1)^(1/2^0) = 1 +2*x

%e F(2)^(1/2^1) = 1 +2*x +4*x^3 -8*x^4 +16*x^5 -40*x^6 + ...

%e F(3)^(1/2^2) = 1 +2*x +4*x^3 -8*x^4 +16*x^5 -40*x^6 +144*x^7 -512*x^8 +...

%e The limit of this process is the g.f. A(x).

%o (PARI) {a(n)=local(F=1,A,L);if(n==0,A=1,L=ceil(log(n+1)/log(2)); for(k=1,L,F=F^2+(2*x)^(2^k-1));A=polcoeff(F^(1/(2^(L-1)))+x*O(x^n),n));A}

%Y Cf. A101190, A101191, A005187, A004134, A003095.

%K sign

%O 0,2

%A _Paul D. Hanna_, Dec 03 2004

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Last modified September 20 08:51 EDT 2021. Contains 347579 sequences. (Running on oeis4.)