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A162659 E.g.f. A(x) satisfies A(x) = exp(x*A(x*A(x))).

%I A162659 #20 Mar 24 2025 22:33:48
%S A162659 1,1,3,22,281,5396,142297,4865806,207407489,10710044776,655655874641,
%T A162659 46789973764634,3840103504940881,358443042637767868,
%U A162659 37700333788138306937,4432826052558222878206,578707468284010393533953,83384676375176176768112720,13190759232920144271864441505
%N A162659 E.g.f. A(x) satisfies A(x) = exp(x*A(x*A(x))).
%F A162659 Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! with a(0,m)=1, then
%F A162659 a(n,m) = Sum_{k=0..n} C(n,k) * m*(n-k+m)^(k-1) * a(n-k,k).
%F A162659 ...
%F A162659 Let log(A(x)) = x*A(x*A(x)) = Sum_{n>=1} L(n)*x^n/n!, then
%F A162659 L(n) = Sum_{k=1..n} C(n,k) * (n-k)^(k-1) * a(n-k,k).
%F A162659 ...
%F A162659 E.g.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n/n! * [D^(n-1) A(x)^n] where operator D F(x) = d/dx x*F(x). - _Paul D. Hanna_, Mar 05 2013
%e A162659 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 281*x^4/4! + 5396*x^5/5! +...
%e A162659 A(x*A(x)) = 1 + x + 5*x^2/2! + 49*x^3/3! + 777*x^4/4! + 17581*x^5/5! +...
%e A162659 log(A(x)) = x + 2*x^2/2! + 15*x^3/3! + 196*x^4/4! + 3885*x^5/5! + 105486*x^6/6! +...
%t A162659 terms = 19; A[_] = 0; Do[A[x_] = Exp[x*A[x*A[x]]] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* _Stefano Spezia_, Mar 24 2025 *)
%o A162659 (PARI) {a(n,m=1)=if(n==0,1,if(m==0,0^n,sum(k=0,n,binomial(n,k)*m*(n-k+m)^(k-1)*a(n-k,k))))}
%o A162659 (PARI) /* Log(A(x)) = x*A(x*A(x)) = Sum_{n>=1} L(n)*x^n/n! where: */
%o A162659 {L(n)=if(n<1,0,sum(k=1,n,binomial(n,k)*(n-k)^(k-1)*a(n-k,k)))}
%Y A162659 Cf. A140049.
%K A162659 nonn
%O A162659 0,3
%A A162659 _Paul D. Hanna_, Jul 09 2009