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A305134 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} ( 2*exp(n*x) - A(x) )^n / 2^(n+1).

%I A305134 #6 May 29 2018 20:53:14
%S A305134 1,6,106,9798,2042986,721198086,378754904746,274462194065478,
%T A305134 261211828432706026,315282684090141417606,470124979835875652863786,
%U A305134 848422945353825106452994758,1822526603267557240862350671466,4596139606368556055825161023870726,13448584326250762088160567798167642026,45199506338787031550197525974862852621638
%N A305134 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} ( 2*exp(n*x) - A(x) )^n / 2^(n+1).
%H A305134 Paul D. Hanna, <a href="/A305134/b305134.txt">Table of n, a(n) for n = 0..50</a>
%F A305134 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
%F A305134 (1) 1 = Sum_{n>=0} ( 2*exp(n*x) - A(x) )^n / 2^(n+1).
%F A305134 (2) 1 = Sum_{n>=0} 2^n * exp(n^2*x) / (2 + exp(n*x) * A(x))^(n+1).
%e A305134 E.g.f.: A(x) = 1 + 6*x + 106*x^2/2! + 9798*x^3/3! + 2042986*x^4/4! + 721198086*x^5/5! + 378754904746*x^6/6! + 274462194065478*x^7/7! + 261211828432706026*x^8/8! + 315282684090141417606*x^9/9! + 470124979835875652863786*x^10/10! + ...
%e A305134 such that
%e A305134 1 = 1/2  +  (2*exp(x) - A(x))/2^2  +  (2*exp(2*x) - A(x))^2/2^3  +  (2*exp(3*x) - A(x))^3/2^4  +  (2*exp(4*x) - A(x))^4/2^5  +  (2*exp(5*x) - A(x))^5/2^6 + ...
%e A305134 Also,
%e A305134 1 = 1/(2 + A(x))  +  2*exp(x)/(2 + exp(x)*A(x))^2  +  2^2*exp(4*x)/(2 + exp(2*x)*A(x))^3  +  2^3*exp(9*x)/(2 + exp(3*x)*A(x))^4  +  2^4*exp(16*x)/(2 + exp(4*x)*A(x))^5  +  2^5*exp(25*x)/(2 + exp(5*x)*A(x))^6  + ...
%e A305134 RELATED SERIES.
%e A305134 log(A(x)) = 6*x + 70*x^2/2! + 8322*x^3/3! + 1812142*x^4/4! + 657412530*x^5/5! + 351254035150*x^6/6! + 257586196964082*x^7/7! + 247297892785673422*x^8/8! + 300478711708843324530*x^9/9! + 450397140484880214948430*x^10/10! + ...
%Y A305134 Cf. A304640, A301436.
%K A305134 nonn
%O A305134 0,2
%A A305134 _Paul D. Hanna_, May 29 2018