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A167001
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Least possible nonnegative coefficients of x^n in G(x)^(2^n), n>=0, for an integer series G(x) such that G'(0)=G(0)=1; the G(x) that satisfies this condition equals the g.f. of A167000.
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3
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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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 288230376151711744, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: A(x) = 1 + Sum_{n>=0} 2^(2^n-n)*x^(2^n).
G.f.: A(x) = Sum_{n>=0} log(G(2^n*x))^n/n! where G(x) = g.f. of A167000.
a(n) = [x^n] G(x)^(2^n) for n>=0 where G(x) = g.f. of A167000.
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 2*x^2 + 4*x^4 + 32*x^8 + 4096*x^16 + 134217728*x^32 +...
A(x) = 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 +...
Let G(x) equal the g.f. of A167000:
G(x) = 1 + x - x^2 - 16*x^4 - 1767*x^5 - 493164*x^6 - 422963721*x^7 +...
then the g.f. A(x) of this sequence equals the series:
A(x) = 1 + log(G(2x)) + log(G(4x))^2/2! + log(G(8x))^3/3! + log(G(16x))^4/4! +...
ILLUSTRATE (2^n)-th POWERS OF G.F. G(x) OF A167000.
The coefficients in the expansion of G(x)^(2^n), n>=0, begin:
G^1: [(1),1,-1,0,-16,-1767,-493164,-422963721,-1130568823448,...];
G^2: [1,(2),-1,-2,-31,-3566,-989830,-846910236,...];
G^4: [1,4,(2),-8,-69,-7252,-1993858,-1697772536,...];
G^8: [1,8,20,(0),-198,-15088,-4045944,-3411523840,...];
G^16: [1,16,104,320,(4),-33344,-8341216,-6888386304,...];
G^32: [1,32,464,3968,21064,(0),-17646208,-14050624512,...];
G^64: [1,64,1952,37632,511376,5030400,(0),-29063442432,...];
G^128: [1,128,8000,325120,9649952,222432256,4056470528,(0),...]; ...
where the coefficients along the diagonal (shown in parenthesis) form the initial terms of this sequence 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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PROG
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(PARI) {a(n)=if(n==0, 1, if(n==2^valuation(n, 2), 2^(n-valuation(n, 2)), 0))}
(PARI) /* A(x) = Sum_{n>=0} log(G(2^n*x))^n/n!, G(x) = g.f. of A167000: */ {a(n)=local(A=[1, 2], G=[1, 1]); for(i=1, n, G=concat(G, 0); A=Vec(sum(m=0, #G, log(subst(Ser(G), x, 2^m*x))^m/m!)); G[ #G]=-floor(A[ #G]/2^(#G-1))); A[n+1]}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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