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A203013 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (2*A(x)^n - 1)^n * x^n/n!.

%I A203013 #8 Mar 20 2023 15:12:32
%S A203013 1,1,5,55,993,24871,802873,31793035,1493163745,81186783535,
%T A203013 5018214016041,347636382949747,26685235607680081,2248760378885064487,
%U A203013 206430769607981879353,20507793044444903462251,2192507508237447321800385,251034864831917236610746207
%N A203013 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (2*A(x)^n - 1)^n * x^n/n!.
%F A203013 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following.
%F A203013 (1) A(x) = Sum_{n>=0} (2*A(x)^n - 1)^n * x^n/n!.
%F A203013 (2) A(x) = Sum_{n>=0} 2^n * A(x)^(n^2) * exp(-x*A(x)^n) * x^n/n!.
%F A203013 a(n) = Sum_{k=0..n} A361540(n,k) * 2^(n-k) * (-1)^k. - _Paul D. Hanna_, Mar 20 2023
%e A203013 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 993*x^4/4! + 24871*x^5/5! +...
%e A203013 where the e.g.f. satisfies following series identity:
%e A203013 A(x) = 1 + (2*A(x)-1)*x + (2*A(x)^2-1)^2*x^2/2! + (2*A(x)^3-1)^3*x^3/3! + (2*A(x)^4-1)^4*x^4/4! +...
%e A203013 is equal to
%e A203013 A(x) = exp(-x) + 2*A(x)*exp(-x*A(x))*x + 2^2*A(x)^4*exp(-x*A(x)^2)*x^2/2! + 2^3*A(x)^9*exp(-x*A(x)^3)*x^3/3! + 2^4*A(x)^16*exp(-x*A(x)^4)*x^4/4! +...
%o A203013 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0, n, (2*A^k-1+x*O(x^n))^k*x^k/k!));n!*polcoeff(A, n)}
%o A203013 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0, n, 2^k*A^(k^2)*exp(-A^k*x+x*O(x^n))*x^k/k!));n!*polcoeff(A, n)}
%Y A203013 Cf. A202999, A361053, A361054, A361055, A361056, A361057.
%Y A203013 Cf. A361540.
%K A203013 nonn
%O A203013 0,3
%A A203013 _Paul D. Hanna_, Dec 27 2011