A134046 - OEIS

%I #3 Mar 30 2012 18:37:05

%S 1,2,-2,-20,-394,-72756,-38636660,-62621451464,-320115036508314,

%T -5370757579794299764,-307243789286348665328060,

%U -61750332256061511777753774808,-44549932891827456895274618101297860,-117151486172958209318246658698308578710856

%N G.f. A(x) satisfies: [x^n] A(x)^(2^n) = 4^n for n>=0.

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

%F 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_, Jan 05 2008

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

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

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

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

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

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

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

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

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

%e where coefficients enclosed in parenthesis are successive powers of 4.

%o (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]}

%Y Cf. A134047.

%K sign

%O 0,2

%A _Paul D. Hanna_, Oct 25 2007