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A303653 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 3*(1+x)^n - A(x) )^n / 3^(n+1).

%I A303653 #6 Apr 28 2018 14:35:11
%S A303653 1,15,291,20868,2501535,406641390,82021892979,19576367780568,
%T A303653 5370958558206975,1661471768423203359,571522497313691705223,
%U A303653 216322544080204799422227,89344723486622904627485286,39989870323587920736747152457,19285197574525200774860259575856,9970552400727667627167081347333058,5502200681071110455003310691040648913
%N A303653 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 3*(1+x)^n - A(x) )^n / 3^(n+1).
%H A303653 Paul D. Hanna, <a href="/A303653/b303653.txt">Table of n, a(n) for n = 0..50</a>
%F A303653 G.f.: 1 = Sum_{n>=0} 3^n * (1+x)^(n^2) / (3 + (1+x)^n * A(x))^(n+1).
%e A303653 G.f.: A(x) = 1 + 15*x + 291*x^2 + 20868*x^3 + 2501535*x^4 + 406641390*x^5 + 82021892979*x^6 + 19576367780568*x^7 + 5370958558206975*x^8 + ...
%e A303653 such that
%e A303653 1 = 1/3  +  (3*(1+x) - A(x))/3^2  +  (3*(1+x)^2 - A(x))^2/3^3  +  (3*(1+x)^3 - A(x))^3/3^4  +  (3*(1+x)^4 - A(x))^4/3^5  +  (3*(1+x)^5 - A(x))^5/3^6 + ...
%e A303653 Also,
%e A303653 1 = 1/(3 + A(x))  +  3*(1+x)/(3 + (1+x)*A(x))^2  +  3^2*(1+x)^4/(3 + (1+x)^2*A(x))^3  +  3^3*(1+x)^9/(3 + (1+x)^3*A(x))^4  +  3^4*(1+x)^16/(3 + (1+x)^4*A(x))^5  +  3^5*(1+x)^25/(3 + (1+x)^5*A(x))^6  + ...
%Y A303653 Cf. A301436.
%K A303653 nonn
%O A303653 0,2
%A A303653 _Paul D. Hanna_, Apr 28 2018