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%I #2 Mar 30 2012 18:37:20
%S 1,4,20,320,21064,5030400,4056470528,10872157339648,98162974155542592,
%T 3052890463194814939136,334052589949087491382968320,
%U 130858881562759880830581892710400
%N G.f.: A(x) = Sum_{n>=0} 2^n*log(G(2^n*x))^n/n! where G(x) = g.f. of A167000.
%C The g.f. of A167000, G(x), satisfies:
%C Sum_{n>=0} log(G(2^n*x))^n/n! = 1 + Sum_{n>=0} 2^(2^n-n)*x^(2^n).
%F a(n) = [x^n] G(x)^(2^(n+1)) for n>=0 where G(x) = g.f. of A167000.
%e G.f.: A(x) = 1 + 4*x + 20*x^2 + 320*x^3 + 21064*x^4 + 5030400*x^5 +...
%e Let G(x) equal the g.f. of A167000:
%e G(x) = 1 + x - x^2 - 16*x^4 - 1767*x^5 - 493164*x^6 - 422963721*x^7 +...
%e then the g.f. A(x) of this sequence equals the series:
%e A(x) = 1 + 2*log(G(2x)) + 4*log(G(4x))^2/2! + 8*log(G(8x))^3/3! + 16*log(G(16x))^4/4! +...
%e ILLUSTRATE (2^n)-th POWERS OF G.F. G(x) OF A167000.
%e The coefficients in the expansion of G(x)^(2^n), n>=0, begin:
%e G^1: [1,1,-1,0,-16,-1767,-493164,-422963721,-1130568823448,...];
%e G^2: [(1),2,-1,-2,-31,-3566,-989830,-846910236,...];
%e G^4: [1,(4),2,-8,-69,-7252,-1993858,-1697772536,...];
%e G^8: [1,8,(20),0,-198,-15088,-4045944,-3411523840,...];
%e G^16: [1,16,104,(320),4,-33344,-8341216,-6888386304,...];
%e G^32: [1,32,464,3968,(21064),0,-17646208,-14050624512,...];
%e G^64: [1,64,1952,37632,511376,(5030400),0,-29063442432,...];
%e G^128: [1,128,8000,325120,9649952,222432256,(4056470528),0,...]; ...
%e where the coefficients along the diagonal (shown in parenthesis) form the initial terms of this sequence.
%o (PARI) {a(n)=local(A=[1,4],B=[1,2],G=[1,1]);for(i=1,n,G=concat(G,0); B=Vec(sum(m=0,#G,log(subst(Ser(G),x,2^m*x))^m/m!)); G[ #G]=-floor(B[ #G]/2^(#G-1))); A=Vec(sum(m=0,#G,2^m*log(subst(Ser(G),x,2^m*x))^m/m!)); A[n+1]}
%Y Cf. A167000, A167001.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 14 2009
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