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a(n) = [x^n] G(x)^( 2^(n+1) ) / 4^n, where G(x) is the g.f. of A134046 such that G(x) satisfies: [x^n] G(x)^(2^n) = 4^n for n>=0.
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%I #2 Mar 30 2012 18:37:05

%S 1,2,6,50,1586,182274,70856770,91955753218,404598404260610,

%T 6173936430806583298,333433524033498566071298,

%U 64757369178015130100982820866,45786845522362297626576735328694274

%N a(n) = [x^n] G(x)^( 2^(n+1) ) / 4^n, where G(x) is the g.f. of A134046 such that G(x) satisfies: [x^n] G(x)^(2^n) = 4^n for n>=0.

%e Let G(x) be the g.f. of A134046 where [x^n] G(x)^(2^n) = 4^n,

%e then G(x) put to powers 2^n, n=0..6, begin as follows:

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

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

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

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

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

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

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

%e where coefficients in parenthesis, divided by respective powers of 4,

%e form the initial terms of this sequence.

%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); Vec(Ser(A)^(2^(n+1)))[n+1]/4^n}

%Y Cf. A134046.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 25 2007