A167001 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.

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

Offset

0,2

Links

Table of n, a(n) for n=0..83.

Formula

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.

Example

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').

Prog

(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]}

Crossrefs

Cf. A167000, A167002, variant: A167004.

Sequence in context: A363907 A342987 A090657 * A108563 A378982 A138476

Adjacent sequences: A166998 A166999 A167000 * A167002 A167003 A167004

Keyword

nonn

Author

Paul D. Hanna, Nov 14 2009

Status

approved