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A101189 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. 5
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; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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).

LINKS

Table of n, a(n) for n=0..28.

FORMULA

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.

EXAMPLE

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).

PROG

(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}

CROSSREFS

Cf. A101190, A101191, A005187, A004134, A003095.

Sequence in context: A120554 A120710 A115780 * A295321 A001443 A195287

Adjacent sequences:  A101186 A101187 A101188 * A101190 A101191 A101192

KEYWORD

sign

AUTHOR

Paul D. Hanna, Dec 03 2004

STATUS

approved

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Last modified July 31 12:29 EDT 2021. Contains 346373 sequences. (Running on oeis4.)