login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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.
3

%I #2 Mar 30 2012 18:37:20

%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^n-n)*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^(1-0)*x + 2^(2-1)*x^2 + 2^(4-2)*x^4 + 2^(8-3)*x^8 + 2^(16-4)*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^m-m) 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^(n-valuation(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^(#G-1)));A[n+1]}

%Y Cf. A167000, A167002, variant: A167004.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 14 2009