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Search: id:A101189
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| A101189 |
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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. |
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6
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| 1, 2, 0, 4, -8, 16, -40, 144, -512, 1696, -5696, 19840, -70048, 247744, -880128, 3152768, -11386624, 41389568, -151273728, 555794944, -2052141056, 7610274816, -28331018240, 105833345024, -396594444800, 1490425179136, -5615651143680, 21209004267520, -80276663808000
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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).
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FORMULA
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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.
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EXAMPLE
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The iteration begins:
F(0) = 1,
F(1) = F(0)^2 + (2*x)^(2^1-1)
= 1 +2*x,
F(2) = F(1)^2 + (2*x)^(2^2-1)
= 1 +4*x +4*x^2 +8*x^3,
F(3) = F(2)^2 + (2*x)^(2^3-1)
= 1 +8*x +24*x^2 +48*x^3 +80*x^4 +64*x^5 +64*x^6 +128*x^7.
The 2^(n-1)-th roots of F(n) tend to the limit of A(x):
F(1)^(1/2^0) = 1 +2*x
F(2)^(1/2^1) = 1 +2*x +4*x^3 -8*x^4 +16*x^5 -40*x^6 + ...
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 +...
The limit of this process is the g.f. A(x).
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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(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}
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CROSSREFS
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Cf. A101190, A101191, A005187, A004134, A003095.
Sequence in context: A120554 A120710 A115780 this_sequence A070015 A021492 A077119
Adjacent sequences: A101186 A101187 A101188 this_sequence A101190 A101191 A101192
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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 03 2004
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