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A301436 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 2*(1+x)^n - A(x) )^n / 2^(n+1).

%I A301436 #8 Apr 28 2018 00:28:02
%S A301436 1,6,50,1582,82722,5842550,511261682,52903385886,6290859281538,
%T A301436 843328959011622,125706002934030898,20617322695573745742,
%U A301436 3689811206934015405474,715633021826704924420758,149544785675949258192968178,33502338836970792659941911358,8011296279710787237594088464898,2036927238948023349890031708437830,548778491694092921577420334962662962,156179940994829385561873698156273034606
%N A301436 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 2*(1+x)^n - A(x) )^n / 2^(n+1).
%H A301436 Paul D. Hanna, <a href="/A301436/b301436.txt">Table of n, a(n) for n = 0..50</a>
%F A301436 G.f.: 1 = Sum_{n>=0} 2^n * (1+x)^(n^2) / (2 + (1+x)^n * A(x))^(n+1).
%e A301436 G.f.: A(x) = 1 + 6*x + 50*x^2 + 1582*x^3 + 82722*x^4 + 5842550*x^5 + 511261682*x^6 + 52903385886*x^7 + 6290859281538*x^8 + ...
%e A301436 such that
%e A301436 1 = 1/2  +  (2*(1+x) - A(x))/2^2  +  (2*(1+x)^2 - A(x))^2/2^3  +  (2*(1+x)^3 - A(x))^3/2^4  +  (2*(1+x)^4 - A(x))^4/2^5  +  (2*(1+x)^5 - A(x))^5/2^6 + ...
%e A301436 Also,
%e A301436 1 = 1/(2 + A(x))  +  2*(1+x)/(2 + (1+x)*A(x))^2  +  2^2*(1+x)^4/(2 + (1+x)^2*A(x))^3  +  2^3*(1+x)^9/(2 + (1+x)^3*A(x))^4  +  2^4*(1+x)^16/(2 + (1+x)^4*A(x))^5  +  2^5*(1+x)^25/(2 + (1+x)^5*A(x))^6  + ...
%Y A301436 Cf. A303653, A301465.
%K A301436 nonn
%O A301436 0,2
%A A301436 _Paul D. Hanna_, Mar 24 2018