cp's OEIS Frontend

A101192 G.f. defined as the limit: A(x) = lim_{n->oo} F(n)^(1/3^(n-1)) where F(n) is the n-th iteration of: F(0) = 1, F(n) = F(n-1)^3 + (3x)^((3^n-1)/2) for n >= 1.

%I A101192 #8 Apr 30 2023 16:27:53
%S A101192 1,3,0,0,27,-162,729,-2916,10206,-28431,39366,216513,-2506302,
%T A101192 16395939,-87687765,419838390,-1879883964,8098629399,-33997343652,
%U A101192 136405492911,-478000355922,987247848321,4754553381171,-85842565710012,782970953914944,-5641921802462517,34830591205459716
%N A101192 G.f. defined as the limit: A(x) = lim_{n->oo} F(n)^(1/3^(n-1)) where F(n) is the n-th iteration of: F(0) = 1, F(n) = F(n-1)^3 + (3x)^((3^n-1)/2) for n >= 1.
%C A101192 The Euler transform of the power series A(x) at x=1/3 converges to the constant: c = Sum_{n>=0} (Sum_{k=0..n} C(n,k)*a(k)/3^k)/2^(n+1) = 2.080400667750319352117745232... which is the limit of S(n)^(1/3^(n-1)) where S(0)=1, S(n+1) = S(n)^3 + 1.
%F A101192 G.f. begins: A(x) = (1+m*x) + m^m*x^(m+1)/(1+m*x)^(m-1) + ... at m=3.
%e A101192 The iteration begins:
%e A101192 F(0) = 1,
%e A101192 F(1) = 1 +  3*x,
%e A101192 F(2) = 1 +  9*x +  27*x^2 +   27*x^3 +    81*x^4,
%e A101192 F(3) = 1 + 27*x + 324*x^2 + 2268*x^3 + 10449*x^4 + ... + 1594323*x^13.
%e A101192 The 3^(n-1)-th roots of F(n) tend to the limit of A(x):
%e A101192 F(1)^(1/3^0) = 1 + 3*x
%e A101192 F(2)^(1/3^1) = 1 + 3*x + 27*x^4 - 162*x^5 + 729*x^6 - 2916*x^7 + ...
%e A101192 F(3)^(1/3^2) = 1 + 3*x + 27*x^4 - 162*x^5 + 729*x^6 - 2916*x^7 + ...
%o A101192 (PARI) {a(n)=local(F=1,A,L);if(n==0,A=1,L=ceil(log(n+1)/log(3)); for(k=1,L,F=F^3+(3*x)^((3^k-1)/2)); A=polcoeff((F+x*O(x^n))^(1/3^(L-1)),n));A}
%Y A101192 Cf. A101189, A101193, A101194.
%K A101192 sign
%O A101192 0,2
%A A101192 _Paul D. Hanna_, Dec 07 2004