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A317355 E.g.f. satisfies: A(x) = Sum_{n>=0} ( exp(n*x) - A(x) )^n / (2 - exp(n*x)*A(x))^(n+1).

%I A317355 #14 Aug 10 2018 12:06:28
%S A317355 1,1,5,85,5261,549061,79707245,15531175045,3926159465261,
%T A317355 1249497583485061,488841071584907885,230674363972514998405,
%U A317355 129251110556658394610861,84870052450743141454787461,64574784437643167984687238125,56377769340759003121860283852165,55996026841326090728124344073814061
%N A317355 E.g.f. satisfies: A(x) = Sum_{n>=0} ( exp(n*x) - A(x) )^n  / (2 - exp(n*x)*A(x))^(n+1).
%C A317355 E.g.f. A(x) = G(exp(x) - 1), where G(x) is the g.f. of A317350.
%H A317355 Vaclav Kotesovec, <a href="/A317355/b317355.txt">Table of n, a(n) for n = 0..144</a>
%F A317355 E.g.f. A(x) satisfies:
%F A317355 (1) A(x) = Sum_{n>=0} ( exp(n*x) - A(x) )^n  / (2 - exp(n*x)*A(x))^(n+1),
%F A317355 (2) A(x) = Sum_{n>=0} ( exp(n*x) + A(x) )^n  / (2 + exp(n*x)*A(x))^(n+1).
%F A317355 a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A317904 = 3.9561842030261697545408... and c = 0.16545672527... - _Vaclav Kotesovec_, Aug 10 2018
%e A317355 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 85*x^3/3! + 5261*x^4/4! + 549061*x^5/5! + 79707245*x^6/6! + 15531175045*x^7/7! + 3926159465261*x^8/8! + 1249497583485061*x^9/9! + ...
%e A317355 such that A = A(x) satisfies
%e A317355 A(x) = 1/(2 - A)  +  (exp(x) - A)/(2 - exp(x)*A)^2  +  (exp(2*x) - A)^2/(2 - exp(2*x)*A)^3  +  (exp(3*x) - A)^3/(2 - exp(3*x)*A)^4  +  (exp(4*x) - A)^4/(2 - exp(4*x)*A)^5  +  (exp(5*x) - A)^5/(2 - exp(5*x)*A)^6 + ...
%e A317355 Also,
%e A317355 A(x) = 1/(2 + A)  +  (exp(x) + A)/(2 + exp(x)*A)^2  +  (exp(2*x) + A)^2/(2 + exp(2*x)*A)^3  +  (exp(3*x) + A)^3/(2 + exp(3*x)*A)^4  +  (exp(4*x) + A)^4/(2 + exp(4*x)*A)^5  +  (exp(5*x) + A)^5/(2 + exp(5*x)*A)^6 + ...
%o A317355 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( exp(m*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 A317355 for(n=0, 20, print1(a(n), ", "))
%Y A317355 Cf. A317356, A317350.
%K A317355 nonn
%O A317355 0,3
%A A317355 _Paul D. Hanna_, Aug 02 2018