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A221097 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^(3*n))^n/n!.

%I A221097 #6 Apr 12 2025 18:30:13
%S A221097 1,1,6,90,2328,84660,3972060,229176654,15712089120,1248343353216,
%T A221097 112832687750400,11437476445244520,1285433373363701760,
%U A221097 158682294244352658312,21349655111889802728576,3110218068324341815470000,487862693943123978219847680,81999755541558838752430348800
%N A221097 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^(3*n))^n/n!.
%F A221097 E.g.f. also satisfies:
%F A221097 (1) A(x) = Sum_{n>=0} binomial(A(x)^(3*n), n) * x^n.
%F A221097 (2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(x)^(3*n*k)/n!.
%e A221097 E.g.f.: A(x) = 1 + x + 6*x^2/2! + 90*x^3/3! + 2328*x^4/4! + 84660*x^5/5! +...
%e A221097 where A(x) satisfies:
%e A221097 A(x) = 1 + log(1 + x*A(x)^3) + log(1 + x*A(x)^6)^2/2! + log(1 + x*A(x)^9)^3/3! +...
%e A221097 The e.g.f. also satisfies:
%e A221097 A(x) = 1 + A(x)^3*x + A(x)^6*(A(x)^6-1)*x^2/2! + A(x)^9*(A(x)^9-1)*(A(x)^9-2)*x^3/3! + A(x)^12*(A(x)^12-1)*(A(x)^12-2)*(A(x)^12-3)*x^4/4! +...+ binomial(A(x)^(3*n), n)*x^n +...
%o A221097 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, log(1+x*(A+x*O(x^n))^(3*m))^m/m!)); n!*polcoeff(A, n)}
%o A221097 for(n=0,20,print1(a(n),", "))
%o A221097 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, binomial((A+x*O(x^n))^(3*m), m)*x^m)); n!*polcoeff(A, n)}
%o A221097 for(n=0,20,print1(a(n),", "))
%o A221097 (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
%o A221097 {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))^(3*m*k))*x^m/m!)); n!*polcoeff(A, n)}
%o A221097 for(n=0,20,print1(a(n),", "))
%Y A221097 Cf. A189981, A221096, A221098, A221099.
%K A221097 nonn
%O A221097 0,3
%A A221097 _Paul D. Hanna_, Jan 01 2013