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A221098 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^(4*n))^n/n!.

%I A221098 #6 Apr 12 2025 18:30:03
%S A221098 1,1,8,156,5184,243280,14742240,1097403552,97012667136,9936480419424,
%T A221098 1157549828855040,151193318253405120,21890302973632558080,
%U A221098 3480525852596442818688,603034041051994953483264,113109668528001746742489600,22839699845167989485088522240
%N A221098 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^(4*n))^n/n!.
%F A221098 E.g.f. also satisfies:
%F A221098 (1) A(x) = Sum_{n>=0} binomial(A(x)^(4*n), n) * x^n.
%F A221098 (2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(x)^(4*n*k)/n!.
%e A221098 E.g.f.: A(x) = 1 + x + 8*x^2/2! + 156*x^3/3! + 5184*x^4/4! + 243280*x^5/5! +...
%e A221098 where A(x) satisfies:
%e A221098 A(x) = 1 + log(1 + x*A(x)^4) + log(1 + x*A(x)^8)^2/2! + log(1 + x*A(x)^12)^3/3! +...
%e A221098 The e.g.f. also satisfies:
%e A221098 A(x) = 1 + A(x)^4*x + A(x)^8*(A(x)^8-1)*x^2/2! + A(x)^12*(A(x)^12-1)*(A(x)^12-2)*x^3/3! + A(x)^16*(A(x)^16-1)*(A(x)^16-2)*(A(x)^16-3)*x^4/4! +...+ binomial(A(x)^(4*n), n)*x^n +...
%o A221098 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, log(1+x*(A+x*O(x^n))^(4*m))^m/m!)); n!*polcoeff(A, n)}
%o A221098 for(n=0,20,print1(a(n),", "))
%o A221098 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, binomial((A+x*O(x^n))^(4*m), m)*x^m)); n!*polcoeff(A, n)}
%o A221098 for(n=0,20,print1(a(n),", "))
%o A221098 (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
%o A221098 {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)*(A+x*O(x^n))^(4*m*k))*x^m/m!)); n!*polcoeff(A, n)}
%o A221098 for(n=0,20,print1(a(n),", "))
%Y A221098 Cf. A189981, A221096, A221097, A221099.
%K A221098 nonn
%O A221098 0,3
%A A221098 _Paul D. Hanna_, Jan 01 2013