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A101190 G.f.: A(x) = Sum_{a(n)/2^A005187(n)*x^n} = limit_{n->oo} F(n)^(1/2^n) where F(n) is the n-th iteration of: F(0) = 1, F(n) = F(n-1)^2 + x^(2^n-1) for n>=1. 2
1, 1, -1, 5, -53, 127, -677, 2221, -61133, 205563, -1394207, 4852339, -68586849, 243751723, -1741612525, 6265913725, -363239625661, 1323861506899, -9699189175227, 35700526467479, -527987675255931, 1960112858076289, -14606721595781139, 54604708004873403 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..23.

FORMULA

G.f. A(x) = [Sum_{n>=0} A101191(n)/2^A004134(n)*x^n]^2. G.f. A(2*x)^2 = Sum_{n>=0} A101189(n)*(2x)^n.

EXAMPLE

The iteration begins:

F(0) = 1,

F(1) = F(0)^2 + x^(2^1-1) = 1 +x,

F(2) = F(1)^2 + x^(2^2-1) = 1 +2*x +x^2 +x^3,

F(3) = F(2)^2 + x^(2^3-1) = 1 +4*x +6*x^2 +6*x^3 +5*x^4 +2*x^5 +x^6 +x^7.

The 2^n-th roots of F(n) tend to the limit of the g.f.:

F(1)^(1/2^1) = 1 +1/2*x -1/8*x^2 +1/16*x^3 -5/128*x^4 +7/256*x^5 +...

F(2)^(1/2^2) = 1 +1/2*x -1/8*x^2 +5/16*x^3 -53/128*x^4 +127/256*x^5 +...

F(3)^(1/2^3) = 1 +1/2*x -1/8*x^2 +5/16*x^3 -53/128*x^4 +127/256*x^5 +...

The limit of this process is the g.f. A(x) of this sequence.

The coefficients of x^k in the 2^n powers of the g.f. A(x) begin:

A^(2^0)=[1,1/2,-1/8,5/16,-53/128,127/256,-677/1024,2221/2048,...],

A^(2^1)=[1,1,0,1/2,-1/2,1/2,-5/8,9/8,-2,53/16,-89/16,155/16,...],

A^(2^2)=[1,2,1,1,0,0,0,1/2,-1,3/2,-5/2,9/2,-8,14,-197/8,44,...],

A^(2^3)=[1,4,6,6,5,2,1,1,0,0,0,0,0,0,0,1/2,-2,5,...],

A^(2^4)=[1,8,28,60,94,116,114,94,69,44,26,14,5,2,1,1,0,0,...].

Note: the sum of the coefficients of x^k in F(n) equals A003095(n):

1, 2=1+1, 5=1+2+1+1, 26=1+4+6+6+5+2+1+1, ...

The last n coefficients in F(n) read backwards are Catalan numbers (A000108).

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+x^(2^k-1)); A=polcoeff(F^(1/2^L)+x*O(x^n), n)); numerator(A)}

CROSSREFS

Cf. A101189, A101191, A005187, A003095, A000108.

Sequence in context: A107004 A139869 A072156 * A201017 A106097 A163580

Adjacent sequences:  A101187 A101188 A101189 * A101191 A101192 A101193

KEYWORD

sign

AUTHOR

Paul D. Hanna, Dec 03 2004

STATUS

approved

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Last modified March 30 06:40 EDT 2020. Contains 333119 sequences. (Running on oeis4.)