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A202999 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 1)^n * x^n/n!.

%I A202999 #12 Mar 20 2023 15:05:17
%S A202999 1,2,8,80,1392,34352,1108576,44340704,2119928320,118111781888,
%T A202999 7524579815424,540141484897280,43182173208678400,3808622859938226176,
%U A202999 367715812648914460672,38610662734158029938688,4384921058923036753723392,536091721631513000647393280
%N A202999 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 1)^n * x^n/n!.
%F A202999 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following.
%F A202999 (1) A(x) = Sum_{n>=0} (A(x)^n + 1)^n * x^n/n!.
%F A202999 (2) A(x) = Sum_{n>=0} A(x)^(n^2) * exp(x*A(x)^n) * x^n/n!.
%F A202999 a(n) = Sum_{k=0..n} A361540(n,k). - _Paul D. Hanna_, Mar 20 2023
%e A202999 E.g.f.: A(x) = 1 + 2*x + 8*x^2/2! + 80*x^3/3! + 1392*x^4/4! + 34352*x^5/5! +...
%e A202999 where the e.g.f. satisfies following series identity:
%e A202999 A(x) = 1 + (A(x)+1)*x + (A(x)^2+1)^2*x^2/2! + (A(x)^3+1)^3*x^3/3! + (A(x)^4+1)^4*x^4/4! +...
%e A202999 A(x) = exp(x) + A(x)*exp(x*A(x))*x + A(x)^4*exp(x*A(x)^2)*x^2/2! + A(x)^9*exp(x*A(x)^3)*x^3/3! + A(x)^16*exp(x*A(x)^4)*x^4/4! +...
%o A202999 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0, n, (A^k+1+x*O(x^n))^k*x^k/k!));n!*polcoeff(A, n)}
%o A202999 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0, n, A^(k^2)*exp(A^k*x+x*O(x^n))*x^k/k!));n!*polcoeff(A, n)}
%Y A202999 Cf. A361053, A361054, A361055, A361057, A203013.
%Y A202999 Cf. A361540.
%K A202999 nonn
%O A202999 0,2
%A A202999 _Paul D. Hanna_, Dec 27 2011