%I #2 Mar 30 2012 18:37:20
%S 1,3,0,9,0,0,0,0,0,2187,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T 282429536481,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N Least possible nonnegative coefficients of x^n in G(x)^(3^n), n>=0, such that G(x) is an integer series with G'(0)=G(0)=1; the G(x) that satisfies this condition equals the g.f. of A167003.
%F G.f.: A(x) = 1 + Sum_{n>=0} 3^(3^n-n)*x^(3^n).
%F G.f.: A(x) = Sum_{n>=0} log(G(3^n*x))^n/n! where G(x) = g.f. of A167003.
%F a(n) = [x^n] G(x)^(3^n) for n>=0 where G(x) = g.f. of A167003.
%e G.f.: A(x) = 1 + 3*x + 9*x^3 + 2187*x^9 + 282429536481*x^27 +...
%e A(x) = 1 + 3^(1-0)*x + 3^(3-1)*x^3 + 3^(9-2)*x^9 + 3^(27-3)*x^27 + 3^(81-4)*x^81 +...
%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 + log(G(3x)) + log(G(9x))^2/2! + log(G(27x))^3/3! + 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 and equal 3^(3^m-m) at positions n=3^m for m>=0, with zeros elsewhere (except for the initial '1').
%o (PARI) {a(n)=if(n==0,1,if(n==3^valuation(n,3),3^(n-valuation(n,3)),0))}
%o (PARI) /* A(x) = Sum_{n>=0} log(G(3^n*x))^n/n!, G(x) = g.f. of A167003: */ {a(n)=local(A=[1,3],G=[1,1]);for(i=1,n,G=concat(G,0); A=Vec(sum(m=0,#G,log(subst(Ser(G),x,3^m*x))^m/m!)); G[ #G]=-floor(A[ #G]/3^(#G-1)));A[n+1]}
%Y Cf. A167003, A167005, variant: A167001.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 14 2009