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A300050 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (1 + x*A(x)^n)^n / 2^(n+1).

%I A300050 #16 Apr 06 2018 20:27:32
%S A300050 1,1,4,28,250,2558,28594,340486,4255344,55300776,742732646,
%T A300050 10267434138,145692400018,2118364506746,31529605958892,
%U A300050 480186833802260,7483472464151002,119397596900634238,1951747376021480874,32721008993895160926,563260078608381337148,9967709437187109520736,181544883799333028286098,3406426523158387599683478,65894531117591548919270114
%N A300050 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (1 + x*A(x)^n)^n / 2^(n+1).
%C A300050 Compare to: G(x) = Sum_{n>=0} (1 + x*G(x)^k)^n / 2^(n+1) holds when G(x) = 1 + x*G(x)^(k+1) for fixed k.
%H A300050 Paul D. Hanna, <a href="/A300050/b300050.txt">Table of n, a(n) for n = 0..100</a>
%F A300050 G.f. A(x) satisfies:
%F A300050 (1) A(x) = Sum_{n>=0} (1 + x*A(x)^n)^n / 2^(n+1).
%F A300050 (2) A(x) = Sum_{n>=0} x^n * A(x)^(n^2) / (2 - A(x)^n)^(n+1).
%e A300050 G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 250*x^4 + 2558*x^5 + 28594*x^6 + 340486*x^7 + 4255344*x^8 + 55300776*x^9 + 742732646*x^10 + ...
%e A300050 such that
%e A300050 A(x) = 1/2 + (1 + x*A(x))/2^2 + (1 + x*A(x)^2)^2/2^3 + (1 + x*A(x)^3)^3/2^4 + (1 + x*A(x)^4)^4/2^5 + (1 + x*A(x)^5)^5/2^6 + (1 + x*A(x)^6)^6/2^7 + ...
%e A300050 Also, due to a series identity,
%e A300050 A(x) = 1 + x*A(x)/(2 - A(x))^2 + x^2*A(x)^4/(2 - A(x)^2)^3 + x^3*A(x)^9/(2 - A(x)^3)^4 + x^4*A(x)^16/(2 - A(x)^4)^5 + x^5*A(x)^25/(2 - A(x)^5)^6 + x^6*A(x)^36/(2 - A(x)^6)^7 + ... + x^n * A(x)^(n^2) / (2 - A(x)^n)^(n+1) + ...
%o A300050 (PARI) {a(n) = my(A=1); for(i=0,n, A = sum(m=0,n, x^m * A^(m^2) / (2 - A^m + x*O(x^n))^(m+1) )); polcoeff(A,n)}
%o A300050 for(n=0,30,print1(a(n),", "))
%Y A300050 Cf. A302103, A302104, A302105, A300279.
%K A300050 nonn
%O A300050 0,3
%A A300050 _Paul D. Hanna_, Feb 27 2018