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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 (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 = 12; f[x_] := Sum[a[k]*x^k, {k, 0, max}]; a[0] = a[1] = 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, 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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