0,2

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

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 (pauldhanna(AT)juno.com), Jan 05 2008

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.

(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]}

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

Sequence in context: A067700 A037010 A114695 this_sequence A100247 A011342 A084046

Adjacent sequences: A134081 A134082 A134083 this_sequence A134085 A134086 A134087

sign

Paul D. Hanna (pauldhanna(AT)juno.com), Oct 25 2007