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EXAMPLE
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G.f.: A(x) = 1 + x - x^2 - 16*x^4 - 1767*x^5 - 493164*x^6 -...
log(A(x)) = x - 3*x^2/2 + 4*x^3/3 - 71*x^4/4 - 8744*x^5/5 - 2948592*x^5/5 -...
ILLUSTRATE THE SERIES DEFINITION:
1 + log(A(2x)) + log(A(4x))^2/2! + log(A(8x))^3/3! + log(A(16x))^4/4! +...
= 1 + 2*x + 2*x^2 + 4*x^4 + 32*x^8 + 4096*x^16 + 134217728*x^32 +...
= 1 + 2^(1-0)*x + 2^(2-1)*x^2 + 2^(4-2)*x^4 + 2^(8-3)*x^8 + 2^(16-4)*x^16 +...
ILLUSTRATE (2^n)-th POWERS OF G.F. A(x).
The coefficients in the expansion of A(x)^(2^n) for n>=0 begin:
[(1),1,-1,0,-16,-1767,-493164,-422963721,-1130568823448,...];
[1,(2),-1,-2,-31,-3566,-989830,-846910236,-2261982587754,...];
[1,4,(2),-8,-69,-7252,-1993858,-1697772536,-4527350821567,...];
[1,8,20,(0),-198,-15088,-4045944,-3411523840,-9068291678061,...];
[1,16,104,320,(4),-33344,-8341216,-6888386304,-18191329536118,...];
[1,32,464,3968,21064,(0),-17646208,-14050624512,-36604843747036];
[1,64,1952,37632,511376,5030400,(0),-29063442432,-74124859451768];
[1,128,8000,325120,9649952,222432256,4056470528,(0),...];
[1,256,32384,2698240,166530624,8117172224,325157844992,10872157339648, (32),...]; ...
where the coefficients along the diagonal (in parenthesis) begin:
[1,2,2,0,4,0,0,0,32,0,0,0,0,0,0,0,4096,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 134217728,...]
and equal 2^(2^m-m) at positions n=2^m for m>=0, with zeros elsewhere (except for the initial '1').
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