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A317356 E.g.f. satisfies: A(x) = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n / (2 - exp(n*x)*A(x))^(n+1).

%I A317356 #12 Aug 10 2018 12:06:22
%S A317356 1,2,14,134,4358,589622,102434534,21285122294,5530748479718,
%T A317356 1792785367579382,711595226383338854,339665400624638782454,
%U A317356 192071493764203628322278,127053485326157331378577142,97253813187878484942034153574,85330814329687863076988482842614,85104598195236153766017309663096038
%N A317356 E.g.f. satisfies: A(x) = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n  / (2 - exp(n*x)*A(x))^(n+1).
%C A317356 E.g.f. A(x) = G(exp(x) - 1), where G(x) is the g.f. of A317351.
%H A317356 Vaclav Kotesovec, <a href="/A317356/b317356.txt">Table of n, a(n) for n = 0..136</a>
%F A317356 E.g.f. A(x) satisfies:
%F A317356 (1) A(x) = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n  / (2 - exp(n*x)*A(x))^(n+1),
%F A317356 (2) A(x) = Sum_{n>=0} ( exp((n+1)*x) + A(x) )^n  / (2 + exp(n*x)*A(x))^(n+1).
%F A317356 a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A317904 = 3.9561842030261697545408... and c = 0.2625457134... - _Vaclav Kotesovec_, Aug 10 2018
%e A317356 E.g.f.: A(x) = 1 + 2*x + 14*x^2/2! + 134*x^3/3! + 4358*x^4/4! + 589622*x^5/5! + 102434534*x^6/6! + 21285122294*x^7/7! + 5530748479718*x^8/8! + 1792785367579382*x^9/9! + ...
%e A317356 such that A = A(x) satisfies
%e A317356 A(x) = 1/(2 - A)  +  (exp(2*x) - A)/(2 - exp(x)*A)^2  +  (exp(3*x) - A)^2/(2 - exp(2*x)*A)^3  +  (exp(4*x) - A)^3/(2 - exp(3*x)*A)^4  +  (exp(5*x) - A)^4/(2 - exp(4*x)*A)^5  +  (exp(6*x) - A)^5/(2 - exp(5*x)*A)^6 + ...
%e A317356 Also,
%e A317356 A(x) = 1/(2 + A)  +  (exp(2*x) + A)/(2 + exp(x)*A)^2  +  (exp(3*x) + A)^2/(2 + exp(2*x)*A)^3  +  (exp(4*x) + A)^3/(2 + exp(3*x)*A)^4  +  (exp(5*x) + A)^4/(2 + exp(4*x)*A)^5  +  (exp(6*x) + A)^5/(2 + exp(5*x)*A)^6 + ...
%o A317356 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( exp((m+1)*x +x*O(x^#A)) - Ser(A) )^m  / (2 - exp(m*x +x*O(x^#A))*Ser(A))^(m+1) ) ) ); n!*A[n+1]}
%o A317356 for(n=0, 20, print1(a(n), ", "))
%Y A317356 Cf. A317356, A317351.
%K A317356 nonn
%O A317356 0,2
%A A317356 _Paul D. Hanna_, Aug 02 2018