%I A101193
%S A101193 1,4,0,0,0,256,3072,24576,163840,983040,5603328,32112640,195035136,1283457024,
%T A101193 8975810560,64281903104,458387095552,3216662069248,22225382014976,152271623028736,
%U A101193 1043452104015872,7199883459035136,50175319780360192,353054558068408320
%V A101193 1,4,0,0,0,256,-3072,24576,-163840,983040,-5603328,32112640,-195035136,
               1283457024,
%W A101193 -8975810560,64281903104,-458387095552,3216662069248,-22225382014976,152271623028736,
%X A101193 -1043452104015872,7199883459035136,-50175319780360192,353054558068408320
%N A101193 G.f. defined as the limit: A(x) = limit_{n->oo} F(n)^(1/4^(n-1)) where 
               F(n) is the n-th iteration of: F(0) = 1, F(n) = F(n-1)^4 + (4x)^((4^n-1)/
               3) for n>=1.
%C A101193 The Euler transform of the power series A(x) at x=1/4 converges to the 
               constant: c = Sum_{n=0..infty} Sum_{k=0..n} C(n,k)*a(k)/4^k)/2^(n+1)) 
               = 2.030544704345910171947313128... which is the limit of S(n)^(1/
               4^(n-1)) where S(0)=1, S(n+1) = S(n)^4 +1.
%F A101193 G.f. begins: A(x) = (1+m*x) + m^m*x^(m+1)/(1+m*x)^(m-1) +... at m=4.
%e A101193 The iteration begins:
%e A101193 F(0) = 1,
%e A101193 F(1) = 1 +4*x
%e A101193 F(2) = 1 +16*x +96*x^2 +256*x^3 +256*x^4 +1024*x^5
%e A101193 F(3) = 1 +64*x +1920*x^2 +35840*x^3 +... + 4398046511104*x^21.
%e A101193 The 4^(n-1)-th roots of F(n) tend to the limit of A(x):
%e A101193 F(1)^(1/4^0) = 1 +4*x
%e A101193 F(2)^(1/4^1) = 1 +4*x +256*x^5 -3072*x^6 +24576*x^7 -163840*x^8 +...
%e A101193 F(3)^(1/4^2) = 1 +4*x +256*x^5 -3072*x^6 +24576*x^7 -163840*x^8 +...
%o A101193 (PARI) {a(n)=local(F=1,A,L);if(n==0,A=1,L=ceil(log(n+1)/log(4)); for(k=1,
               L,F=F^4+(4*x)^((4^k-1)/3)); A=polcoeff((F+x*O(x^n))^(1/4^(L-1)),n));
               A}
%Y A101193 Cf. A101189, A101192, A101194.
%Y A101193 Sequence in context: A162296 A057386 A099306 this_sequence A013334 A156393 
               A096623
%Y A101193 Adjacent sequences: A101190 A101191 A101192 this_sequence A101194 A101195 
               A101196
%K A101193 sign
%O A101193 0,2
%A A101193 Paul D. Hanna (pauldhanna(AT)juno.com), Dec 07 2004