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Search: id:A101192
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| A101192 |
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G.f. defined as the limit: A(x) = limit_{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. |
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3
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| 1, 3, 0, 0, 27, -162, 729, -2916, 10206, -28431, 39366, 216513, -2506302, 16395939, -87687765, 419838390, -1879883964, 8098629399, -33997343652, 136405492911, -478000355922, 987247848321, 4754553381171, -85842565710012, 782970953914944, -5641921802462517, 34830591205459716
(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=1/3 converges to the constant: c = Sum_{n=0..infty} Sum_{k=0..n} C(n,k)*a(k)/3^k*1.0)/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.
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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=3.
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EXAMPLE
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The iteration begins:
F(0) = 1,
F(1) = 1 +3*x
F(2) = 1 +9*x +27*x^2 +27*x^3 + 81*x^4
F(3) = 1 +27*x +324*x^2 +2268*x^3 +10449*x^4 +... + 1594323*x^13.
The 3^(n-1)-th roots of F(n) tend to the limit of A(x):
F(1)^(1/3^0) = 1 +3*x
F(2)^(1/3^1) = 1 +3*x +27*x^4 -162*x^5 +729*x^6 -2916*x^7 +...
F(3)^(1/3^2) = 1 +3*x +27*x^4 -162*x^5 +729*x^6 -2916*x^7 +...
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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(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}
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CROSSREFS
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Cf. A101189, A101193, A101194.
Sequence in context: A104751 A123474 A105786 this_sequence A037288 A160537 A009133
Adjacent sequences: A101189 A101190 A101191 this_sequence A101193 A101194 A101195
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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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