%I #6 Jun 19 2018 11:24:44
%S 2,0,68,4200,737432,376015248,596428719840,3087194714985696,
%T 54006009465104195072,3279501822205862690687680,
%U 705652071337358165341710594240,546562572350657623752732853465340160,1542500297504013104202649292306514734730496,16012038105195898099786632602589693961733953444608,615959454912822396904406370593148614749014922437494816768
%N L.g.f. A(x) satisfies: Sum_{n>=0} (log(1 + 2^n*x) - A(x))^n / n! = 1, where A(x) = Sum_{n>=1} a(n)*x^n/n.
%e L.g.f. A(x) = 2*x + 68*x^3/3 + 4200*x^4/4 + 737432*x^5/5 + 376015248*x^6/6 + 596428719840*x^7/7 + 3087194714985696*x^8/8 + 54006009465104195072*x^9/9 + ...
%e such that Sum_{n>=0} (log(1 + 2^n*x) - A(x))^n / n! = 1.
%e RELATED SERIES.
%e exp(A(x)) = 1 + 2*x + 2*x^2 + 24*x^3 + 1096*x^4 + 149632*x^5 + 62966568*x^6 + 85329761952*x^7 + 386069877057040*x^8 + 6001439689098721408*x^9 + 327962184329946415530336*x^10 + ... + A306062(n)*x^n + ...
%o (PARI) {a(n) = my(A=[2]); for(i=1,n, A=concat(A,0); A[#A] = Vec(sum(n=0,#A+1, (log(1 + 2^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. A306062.
%K nonn
%O 1,1
%A _Paul D. Hanna_, Jun 18 2018