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A101194 G.f. defined as the limit: A(x) = limit_{n->oo} F(n)^(1/5^(n-1)) where F(n) is the n-th iteration of: F(0) = 1, F(n) = F(n-1)^5 + (5x)^((5^n-1)/4) for n>=1. 2
1, 5, 0, 0, 0, 0, 3125, -62500, 781250, -7812500, 68359375, -546875000, 4082031250, -28417968750, 179443359375, -939941406250, 2685546875000, 23010253906250, -569122314453125, 7669982910156250, -84739685058593750, 836715698242187500, -7611751556396484375 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The Euler transform of the power series A(x) at x=1/5 converges to the constant: c = Sum_{n=0..infty} Sum_{k=0..n} C(n,k)*a(k)/5^k)/2^(n+1)) = 2.012346619142363112612326559... which is the limit of S(n)^(1/5^(n-1)) where S(0)=1, S(n+1) = S(n)^5 +1.

LINKS

Table of n, a(n) for n=0..22.

FORMULA

G.f. begins: A(x) = (1+m*x) + m^m*x^(m+1)/(1+m*x)^(m-1) +... at m=5.

EXAMPLE

The iteration begins:

F(0) = 1,

F(1) = 1 +5*x

F(2) = 1 +25*x +250*x^2 +1250*x^3 +3125*x^4 +3125*x^5 +15625*x^6

F(3) = 1 +125*x +7500*x^2 +287500*x^3 +... + 5^31*x^31.

The 5^(n-1)-th root of F(n) tend to the limit of A(x):

F(1)^(1/5^0) = 1 +5*x

F(2)^(1/5^1) = 1 +5*x +3125*x^6 -62500*x^7 +781250*x^8 +...

F(3)^(1/5^2) = 1 +5*x +3125*x^6 -62500*x^7 +781250*x^8 +...

PROG

(PARI) {a(n)=local(F=1, A, L); if(n==0, A=1, L=ceil(log(n+1)/log(5)); for(k=1, L, F=F^5+(5*x)^((5^k-1)/4)); A=polcoeff((F+x*O(x^n))^(1/5^(L-1)), n)); A}

CROSSREFS

Cf. A101189, A101192, A101193.

Sequence in context: A284104 A260911 A228631 * A196344 A106222 A090750

Adjacent sequences:  A101191 A101192 A101193 * A101195 A101196 A101197

KEYWORD

sign

AUTHOR

Paul D. Hanna, Dec 07 2004

STATUS

approved

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Last modified September 27 10:35 EDT 2020. Contains 337380 sequences. (Running on oeis4.)