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E.g.f. satisfies: A(x) = Sum_{n>=0} 1/n! * Product_{k=1..n} log(1 + x*A(x)^k).
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%I #9 Aug 22 2024 16:44:07

%S 1,1,2,9,66,650,8250,127519,2318876,48626556,1154334060,30589513350,

%T 895415799960,28693464851688,999009599484624,37554576369815400,

%U 1516080931559327280,65418533528228549744,3004726893339734134128,146370356574519380115240

%N E.g.f. satisfies: A(x) = Sum_{n>=0} 1/n! * Product_{k=1..n} log(1 + x*A(x)^k).

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 66*x^4/4! + 650*x^5/5! +...

%e where

%e A(x) = 1 + log(1+x*A(x)) + log(1+x*A(x))*log(1+x*A(x)^2)/2! + log(1+x*A(x))*log(1+x*A(x)^2)*log(1+x*A(x)^3)/3! + log(1+x*A(x))*log(1+x*A(x)^2)*log(1+x*A(x)^3)*log(1+x*A(x)^4)/4! +...

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,prod(k=1,m,log(1+x*A^k+x*O(x^n)))/m!));n!*polcoeff(A,n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A189981, A033917.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 09 2013