0,2

A134047(n) = [x^n] A(x)^( 2^(n+1) ) / 4^n.

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

To illustrate the property [x^n] A(x)^(2^n) = 4^n,

put the g.f. A(x) to powers 2^n, n=0..6, as follows:

A(x)^1 = (1) + 2x - 2x^2 - 20x^3 - 394x^4 - 72756x^5 - 38636660x^6 +...;

A(x)^2 = 1 + (4)x + 0x^2 - 48x^3 - 864x^4 -147008x^5 - 77562368x^6 +...;

A(x)^4 = 1 + 8x +(16)x^2 - 96x^3 -2112x^4 -300928x^5 -156298496x^6 +...;

A(x)^8 = 1 +16x + 96x^2 +(64)x^3 -5504x^4 -638720x^5 -317470208x^6 +...;

A(x)^16= 1 +32x +448x^2 +3200x^3+(256)x^4-1441280x^5 -656432128x^6 +...;

A(x)^32= 1 +64x+1920x^2+35072x^3+406016x^4+(1024)x^5-1394636800x^6 +...;

A(x)^64= 1+128x+7936x^2+315904x^3+8987648x^4+186648576x^5+(4096)x^6+...;

where coefficients enclosed in parenthesis are successive powers of 4.

(PARI) {a(n)=local(A=[]); for(i=0, n, A=concat(A, 0); A[i+1]=(4^i - Vec(Ser(A)^(2^i))[i+1])/2^i); A[n+1]}

Cf. A134047.

Adjacent sequences: A134043 A134044 A134045 this_sequence A134047 A134048 A134049

Sequence in context: A009340 A053593 A002907 this_sequence A081687 A082811 A014353

sign

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