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A221096 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^(2*n))^n/n!.

%I A221096 #6 Apr 12 2025 18:30:23
%S A221096 1,1,4,42,768,19460,637200,25724916,1233957312,68591031120,
%T A221096 4338982958400,307907317681920,24229505587541760,2094548798610726432,
%U A221096 197370092438311892736,20140182770328963216000,2213078753956025271214080,260601290312643875434817280
%N A221096 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^(2*n))^n/n!.
%F A221096 E.g.f. also satisfies:
%F A221096 (1) A(x) = Sum_{n>=0} binomial(A(x)^(2*n), n) * x^n.
%F A221096 (2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(x)^(2*n*k)/n!.
%e A221096 E.g.f.: A(x) = 1 + x + 4*x^2/2! + 42*x^3/3! + 768*x^4/4! + 19460*x^5/5! +...
%e A221096 where A(x) satisfies:
%e A221096 A(x) = 1 + log(1 + x*A(x)^2) + log(1 + x*A(x)^4)^2/2! + log(1 + x*A(x)^6)^3/3! +...
%e A221096 The e.g.f. also satisfies:
%e A221096 A(x) = 1 + A(x)^2*x + A(x)^4*(A(x)^4-1)*x^2/2! + A(x)^6*(A(x)^6-1)*(A(x)^6-2)*x^3/3! + A(x)^8*(A(x)^8-1)*(A(x)^8-2)*(A(x)^8-3)*x^4/4! +...+ binomial(A(x)^(2*n), n)*x^n +...
%o A221096 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, log(1+x*(A+x*O(x^n))^(2*m))^m/m!)); n!*polcoeff(A, n)}
%o A221096 for(n=0,20,print1(a(n),", "))
%o A221096 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, binomial((A+x*O(x^n))^(2*m), m)*x^m)); n!*polcoeff(A, n)}
%o A221096 for(n=0,20,print1(a(n),", "))
%o A221096 (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
%o A221096 {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))^(2*m*k))*x^m/m!)); n!*polcoeff(A, n)}
%o A221096 for(n=0,20,print1(a(n),", "))
%Y A221096 Cf. A189981, A221097, A221098, A221099.
%K A221096 nonn
%O A221096 0,3
%A A221096 _Paul D. Hanna_, Jan 01 2013