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A385057 E.g.f. satisfies A(x) = exp( Sum_{n>=1} (Integral A(x)^n dx)^n / n ).

%I A385057 #37 Aug 20 2025 10:56:58
%S A385057 1,1,3,18,173,2368,43025,991070,28030227,950818494,37995695979,
%T A385057 1763496545502,93967776822477,5692538342703978,388772833646583213,
%U A385057 29711642817587338986,2524166742181661207511,236956380718244960455206,24446253183753019240769463,2757979540962272093582650734,338712272097534292284500861745
%N A385057 E.g.f. satisfies A(x) = exp( Sum_{n>=1} (Integral A(x)^n dx)^n / n ).
%H A385057 Paul D. Hanna, <a href="/A385057/b385057.txt">Table of n, a(n) for n = 0..201</a>
%F A385057 E.g.f.: A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A385057 (1) A(x) = exp( Sum_{n>=1} (Integral A(x)^n dx)^n / n ).
%F A385057 (2) A'(x) = Sum_{n>=1} A(x)^(n+1) * (Integral A(x)^n dx)^(n-1).
%F A385057 (3) A(x) = exp(B(x)), where B(x) is the e.g.f. of A268294.
%e A385057 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 18*x^3/3! + 173*x^4/4! + 2368*x^5/5! + 43025*x^6/6! + 991070*x^7/7! + 28030227*x^8/8! + 950818494*x^9/9! + 37995695979*x^10/10! + ...
%e A385057 where
%e A385057 A(x) = exp( (Integral A(x) dx) + (Integral A(x)^2 dx)^2/2 + (Integral A(x)^3 dx)^3/3 + (Integral A(x)^4 dx)^4/4 + ... ).
%e A385057 Also,
%e A385057 A'(x) = A(x)^2 + A(x)^3*(Integral A(x)^2 dx) + A(x)^4*(Integral A(x)^3 dx)^2 + A(x)^5*(Integral A(x)^4 dx)^3 + ...
%e A385057 RELATED SERIES.
%e A385057 log(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 104*x^4/4! + 1437*x^5/5! + 26642*x^6/6! + 629127*x^7/7! + ... + A268294(n)*x^n/n! + ...
%o A385057 (PARI) {a(n) = my(A = 1 + x +x*O(x^n)); for(i=0, n+1, A = exp( sum(m=1, n+1, intformal(A^m)^m/m ) ) ); n!*polcoeff(A, n)}
%o A385057 for(n=0, 20, print1(a(n), ", "))
%Y A385057 Cf. A268294 (log).
%K A385057 nonn,new
%O A385057 0,3
%A A385057 _Paul D. Hanna_, Aug 19 2025