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A134084 G.f. A(x) = G(2x) where G(x) satisfies [x^(n+1)] G(x)^(2^n) = [x^n] G(x)^(2^n) for n>=0. 5
1, 2, 2, -4, -106, -6948, -1623788, -1213437064, -2912047916698, -23264250235542100, -641982248042094828676, -62929856484660987275500088, -22331407793040258023249030997892, -29057717949243934527799656871001480808 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

G.f. A(x) satisfies: A(x/2)^2 is the g.f. of an integer sequence (A134085).

LINKS

Table of n, a(n) for n=0..13.

FORMULA

A134086(n) = [x^n] G(x)^(2^n) for n>=0. A134087(n) = [x^n] G(x)^(2^(n+1)) for n>=0. G.f. A(x) satisfies: [x^(n+1)] A(x)^(2^n) = 2 * [x^n] A(x)^(2^n) for n>=0.

G.f. A(x) satisfies: 1 = Sum_{n>=0} (1/2^n - x) * log( A(2^(n-1)*x) )^n / n! = (1-x) + (1/2-x)log(A(x)) + (1/4-x)log(A(2x))^2/2! + (1/8-x)log(A(4x))^3/3! +... - Paul D. Hanna, Jan 05 2008

EXAMPLE

G.f. A(x) = 1 + 2*x + 2*x^2 - 4*x^3 - 106*x^4 - 6948*x^5 - ...

Define G(x) = A(x/2); illustrate that

G(x) satisfies [x^(n+1)] G(x)^(2^n) = [x^n] G(x)^(2^n)

by listing powers G(x)^(2^n) as follows:

G(x)^1 = (1 + x) + 1/2*x^2 - 1/2*x^3 - 53/8*x^4 - 1737/8*x^5 -...;

G(x)^2 = 1+(2x + 2x^2) + 0x^3 - 14x^4 - 448x^5 - 51184x^6 -...;

G(x)^4 = 1 +4x +(8x^2 + 8x^3) - 24x^4 - 952x^5 - 104216x^6 -...;

G(x)^8 = 1 +8x +32x^2 +(80x^3 + 80x^4) - 1968x^5 - 216368x^6 -...;

G(x)^16 = 1 +16x +128x^2 +672x^3 +(2464x^4 + 2464x^5) -452704x^6 -...;

G(x)^32 = 1 +32x +512x^2+5440x^3 +42816x^4+(255808x^5 + 255808x^6) -...;

to show that the coefficients within the parenthesis are equal.

Note also that G(x)^2 consists entirely of integer coefficients.

PROG

(PARI) {a(n)=local(A=[1], B); for(i=1, n, A=concat(A, 0); B=Vec(Ser(A)^(2^(#A-2))); A[ #A]=(B[ #B-1]-B[ #B])/2^(#A-2)); 2^n*A[n+1]}

CROSSREFS

Cf. A134085, A134086, A134087, A134088, A134089.

Sequence in context: A037010 A294184 A114695 * A267346 A264933 A012858

Adjacent sequences:  A134081 A134082 A134083 * A134085 A134086 A134087

KEYWORD

sign

AUTHOR

Paul D. Hanna, Oct 25 2007

STATUS

approved

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Last modified April 11 13:59 EDT 2021. Contains 342886 sequences. (Running on oeis4.)