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Search: id:A101193
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| A101193 |
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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. |
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3
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| 1, 4, 0, 0, 0, 256, -3072, 24576, -163840, 983040, -5603328, 32112640, -195035136, 1283457024, -8975810560, 64281903104, -458387095552, 3216662069248, -22225382014976, 152271623028736, -1043452104015872, 7199883459035136, -50175319780360192, 353054558068408320
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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.
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FORMULA
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G.f. begins: A(x) = (1+m*x) + m^m*x^(m+1)/(1+m*x)^(m-1) +... at m=4.
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EXAMPLE
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The iteration begins:
F(0) = 1,
F(1) = 1 +4*x
F(2) = 1 +16*x +96*x^2 +256*x^3 +256*x^4 +1024*x^5
F(3) = 1 +64*x +1920*x^2 +35840*x^3 +... + 4398046511104*x^21.
The 4^(n-1)-th roots of F(n) tend to the limit of A(x):
F(1)^(1/4^0) = 1 +4*x
F(2)^(1/4^1) = 1 +4*x +256*x^5 -3072*x^6 +24576*x^7 -163840*x^8 +...
F(3)^(1/4^2) = 1 +4*x +256*x^5 -3072*x^6 +24576*x^7 -163840*x^8 +...
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PROGRAM
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(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}
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CROSSREFS
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Cf. A101189, A101192, A101194.
Sequence in context: A126849 A057386 A099306 this_sequence A013334 A096623 A078669
Adjacent sequences: A101190 A101191 A101192 this_sequence A101194 A101195 A101196
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KEYWORD
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sign
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Dec 07 2004
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