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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A244129 A342987 A090657 * A108563 A138476 A131381

Adjacent sequences:  A166998 A166999 A167000 * A167002 A167003 A167004

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 14 2009

STATUS

approved

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Last modified June 18 06:02 EDT 2021. Contains 345098 sequences. (Running on oeis4.)