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A167000 G.f. A(x) satisfies: Sum_{n>=0} log( A(2^n*x) )^n / n! = 1 + Sum_{n>=0} 2^(2^n-n) * x^(2^n). 4
1, 1, -1, 0, -16, -1767, -493164, -422963721, -1130568823448, -9811523398109059, -287512372919585565730, -29365896347484186250530846, -10704256920972727382240940549099, -14165930844739651162632827455464483815, -68918096446337603401330634164181238008617534 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..50

FORMULA

The coefficient of x^(2^n) in A(x)^(2^(2^n)) equals 2^(2^n-n):

[x^(2^n)] A(x)^(2^(2^n)) = 2^(2^n-n); while

[x^n] A(x)^(2^n) = 0 when n>0 is not a power of 2, with A(0)=1.

EXAMPLE

G.f.: A(x) = 1 + x - x^2 - 16*x^4 - 1767*x^5 - 493164*x^6 -...

log(A(x)) = x - 3*x^2/2 + 4*x^3/3 - 71*x^4/4 - 8744*x^5/5 - 2948592*x^5/5 -...

ILLUSTRATE THE SERIES DEFINITION:

1 + log(A(2x)) + log(A(4x))^2/2! + log(A(8x))^3/3! + log(A(16x))^4/4! +...

= 1 + 2*x + 2*x^2 + 4*x^4 + 32*x^8 + 4096*x^16 + 134217728*x^32 +...

= 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 +...

ILLUSTRATE (2^n)-th POWERS OF G.F. A(x).

The coefficients in the expansion of A(x)^(2^n) for n>=0 begin:

[(1),1,-1,0,-16,-1767,-493164,-422963721,-1130568823448,...];

[1,(2),-1,-2,-31,-3566,-989830,-846910236,-2261982587754,...];

[1,4,(2),-8,-69,-7252,-1993858,-1697772536,-4527350821567,...];

[1,8,20,(0),-198,-15088,-4045944,-3411523840,-9068291678061,...];

[1,16,104,320,(4),-33344,-8341216,-6888386304,-18191329536118,...];

[1,32,464,3968,21064,(0),-17646208,-14050624512,-36604843747036];

[1,64,1952,37632,511376,5030400,(0),-29063442432,-74124859451768];

[1,128,8000,325120,9649952,222432256,4056470528,(0),...];

[1,256,32384,2698240,166530624,8117172224,325157844992,10872157339648, (32),...]; ...

where the coefficients along the diagonal (in parenthesis) begin:

[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,...]

and equal 2^(2^m-m) at positions n=2^m for m>=0, with zeros elsewhere (except for the initial '1').

MATHEMATICA

max = 25; f[x_] := Sum[a[k]*x^k, {k, 0, max}]; init (* to speed-up computation *) = {1, 1, -1, 0}; skip = Length[init]; a[n_ /; n < skip] := init[[n+1]]; coes = CoefficientList[Series[Sum[Log[f[2^n*x]]^n/n!, {n, 0, max}] - 1 - Sum[2^(2^n-n)*x^2^n, {n, 0, Log[2, max]//Floor}], {x, 0, max} ], x]; Do[coes = coes /. (sol[k-1] = Solve[coes[[k]] == 0][[1, 1]]), {k, skip+1, Length[coes]}]; Join[init, Table[a[k] /. sol[k], {k, skip, max}]] (* Jean-Fran├žois Alcover, Mar 06 2013, updated Sep 04 2017 *)

PROG

(PARI) {a(n) = my(A=[1, 1], B=[1, 2]); for(i=1, n, A=concat(A, 0); B=Vec( sum(m=0, #A, log( subst(Ser(A), x, 2^m*x) )^m/m!) ); A[#A] = -floor( B[#A]/2^(#A-1) )); A[n+1]}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A167001, A167002, A167003.

Sequence in context: A263387 A264199 A160306 * A075413 A265213 A176886

Adjacent sequences:  A166997 A166998 A166999 * A167001 A167002 A167003

KEYWORD

nice,sign

AUTHOR

Paul D. Hanna, Nov 13 2009

EXTENSIONS

Typos in examples fixed by Paul D. Hanna, Nov 15 2009

STATUS

approved

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Last modified June 18 03:16 EDT 2021. Contains 345098 sequences. (Running on oeis4.)