%I #2 Mar 30 2012 18:37:20
%S 1,9,243,59076,111615732,1491228760410,142100883744107238,
%T 100726960061072884260690,551358313898624152526339325936,
%U 23916527760781825204570536862624479480
%N G.f.: A(x) = Sum_{n>=0} 3^n*log(G(3^n*x))^n/n! where G(x) = g.f. of A167003.
%C The g.f. of A167003, G(x), satisfies:
%C Sum_{n>=0} log(G(3^n*x))^n/n! = 1 + Sum_{n>=0} 3^(3^n-n)*x^(3^n).
%F a(n) = [x^n] G(x)^(3^(n+1)) for n>=0 where G(x) = g.f. of A167003.
%e G.f.: A(x) = 1 + 9*x + 243*x^2 + 59076*x^3 + 111615732*x^4 +...
%e Let G(x) equal the g.f. of A167003:
%e G(x) = 1 + x - 4*x^2 - 4*x^3 - 8220*x^4 - 16910960*x^5 - 220513689396*x^6 +...
%e then the g.f. A(x) of this sequence equals the series:
%e A(x) = 1 + 3*log(G(3x)) + 9*log(G(9x))^2/2! + 27*log(G(27x))^3/3! + 81*log(G(81x))^4/4! +...
%e ILLUSTRATE (3^n)-th POWERS OF G.F. G(x) OF A167003.
%e The coefficients in the expansion of G(x)^(3^n), n>=0, begin:
%e G^1: [1, 1, -4, -4, -8220, -16910960, -220513689396,...];
%e G^3: [(1), 3, -9, -35, -24648, -50782068, -661642361248,...];
%e G^9: [1, (9), 0, -240, -74574, -152788194, -1985840486856,...];
%e G^27: [1, 27, (243), 9, -236682, -462449898, -5965789971726,...];
%e G^81: [1, 81, 2916, (59076), 0, -1420876404, -17973134801100,...];
%e G^243: [1, 243, 28431, 2125845, (111615732), 0, -54490964413644,...];
%e G^729: [1, 729, 262440, 62178840, 10895760846, (1491228760410), 0,...]; ...
%e where the coefficients along the diagonal (shown in parenthesis) form the initial terms of this sequence.
%o (PARI) {a(n)=local(A=[1,9],B=[1,3],G=[1,1]);for(i=1,n,G=concat(G,0); B=Vec(sum(m=0,#G,log(subst(Ser(G),x,3^m*x))^m/m!)); G[ #G]=-floor(B[ #G]/3^(#G-1))); A=Vec(sum(m=0,#G,3^m*log(subst(Ser(G),x,3^m*x))^m/m!)); A[n+1]}
%Y Cf. A167003, A167004.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 14 2009