%I #6 Jun 30 2018 19:22:11

%S 3,27,5967,5697567,31847802183,1195671270431187,326058737699333461707,

%T 675917435446065515610996255,10962564428448588841282872538419771,

%U 1418440155472251470046024633146709425948667,1484885879650092405217931878354260186060716460431319,12712226189522682755929156185294269966327457982317234267691359

%N L.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n satisfies: Sum_{n>=0} (log(1 + 3^n*x) - A(x))^n / n! = 1.

%e L.g.f. A(x) = 3*x + 27*x^2/2 + 5967*x^3/3 + 5697567*x^4/4 + 31847802183*x^5/5 + 1195671270431187*x^6/6 + 326058737699333461707*x^7/7 + 675917435446065515610996255*x^8/8 + ...

%e such that

%e 1 = 1 + (log(1 + 3*x) - A(x)) + (log(1 + 3^2*x) - A(x))^2/2! + (log(1 + 3^3*x) - A(x))^3/3! + (log(1 + 3^4*x) - A(x))^4/4! + (log(1 + 3^5*x) - A(x))^5/5! + ... + (log(1 + 3^n*x) - A(x))^n / n! + ...

%e RELATED SERIES.

%e exp(A(x)) = 1 + 3*x + 18*x^2 + 2034*x^3 + 1430514*x^4 + 6373869750*x^5 + 199297681460658*x^6 + 46580417624524112586*x^7 + ... + A316369(n)*x^n + ...

%o (PARI) {a(n) = my(A=[3]); for(i=1,n, A=concat(A,0); A[#A] = Vec(sum(n=0,#A+1, (log(1 + 3^n*x +x*O(x^#A) ) - x*Ser(A))^n/n! ))[#A+1]); n*A[n]}

%o for(n=1,20,print1(a(n),", "))

%Y Cf. A316369, A306061.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Jun 30 2018