%I
%S 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,
%T 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,
%U 0,0,288230376151711744,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N 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.
%F G.f.: A(x) = 1 + Sum_{n>=0} 2^(2^nn)*x^(2^n).
%F G.f.: A(x) = Sum_{n>=0} log(G(2^n*x))^n/n! where G(x) = g.f. of A167000.
%F a(n) = [x^n] G(x)^(2^n) for n>=0 where G(x) = g.f. of A167000.
%e G.f.: A(x) = 1 + 2*x + 2*x^2 + 4*x^4 + 32*x^8 + 4096*x^16 + 134217728*x^32 +...
%e A(x) = 1 + 2^(10)*x + 2^(21)*x^2 + 2^(42)*x^4 + 2^(83)*x^8 + 2^(164)*x^16 +...
%e Let G(x) equal the g.f. of A167000:
%e G(x) = 1 + x  x^2  16*x^4  1767*x^5  493164*x^6  422963721*x^7 +...
%e then the g.f. A(x) of this sequence equals the series:
%e A(x) = 1 + log(G(2x)) + log(G(4x))^2/2! + log(G(8x))^3/3! + log(G(16x))^4/4! +...
%e ILLUSTRATE (2^n)th POWERS OF G.F. G(x) OF A167000.
%e The coefficients in the expansion of G(x)^(2^n), n>=0, begin:
%e G^1: [(1),1,1,0,16,1767,493164,422963721,1130568823448,...];
%e G^2: [1,(2),1,2,31,3566,989830,846910236,...];
%e G^4: [1,4,(2),8,69,7252,1993858,1697772536,...];
%e G^8: [1,8,20,(0),198,15088,4045944,3411523840,...];
%e G^16: [1,16,104,320,(4),33344,8341216,6888386304,...];
%e G^32: [1,32,464,3968,21064,(0),17646208,14050624512,...];
%e G^64: [1,64,1952,37632,511376,5030400,(0),29063442432,...];
%e G^128: [1,128,8000,325120,9649952,222432256,4056470528,(0),...]; ...
%e where the coefficients along the diagonal (shown in parenthesis) form the initial terms of this sequence and equal 2^(2^mm) at positions n=2^m for m>=0, with zeros elsewhere (except for the initial '1').
%o (PARI) {a(n)=if(n==0,1,if(n==2^valuation(n,2),2^(nvaluation(n,2)),0))}
%o (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^(#G1)));A[n+1]}
%Y Cf. A167000, A167002, variant: A167004.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 14 2009
