A167000 - OEIS

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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).

1, 1, -1, 0, -16, -1767, -493164, -422963721, -1130568823448, -9811523398109059, -287512372919585565730, -29365896347484186250530846, -10704256920972727382240940549099 (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 - 16x^4 - 1767x^5 - 493164x^6 -...` `log(A(x)) = x - 3x^2/2 + 4x^3/3 - 71x^4/4 - 8744x^5/5 - 2948592x^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 + 2x + 2x^2 + 4x^4 + 32x^8 + 4096x^16 + 134217728x^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 = 12; f[x_] := Sum[a[k]x^k, {k, 0, max}]; a[0] = a[1] = 1; coes = CoefficientList[ Series[ Sum[ Log[ f[2^nx]]^n/n!, {n, 0, max}] - 1 - Sum[ 2^(2^n - n)x^2^n, {n, 0, max}], {x, 0, max}], x]; Table[a[k] , {k, 0, max}] /. Solve[Thread[coes == 0]] // First ( Jean-François Alcover, Mar 06 2013 *)`
	PROG	`(PARI) {a(n)=local(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]}`
	CROSSREFS	`Cf. A167001, A167002, A167003.` `Sequence in context: A071900 A145406 A160306 * A075413 A176886 A159389` `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 May 25 08:17 EDT 2013. Contains 225646 sequences.