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A221099 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^(5*n))^n/n!.

%I A221099 #7 Apr 12 2025 18:29:52
%S A221099 1,1,10,240,9720,556400,41153220,3737360130,402876727680,
%T A221099 50302825722720,7141958361129600,1136668023900846360,
%U A221099 200486825731741824000,38826473000115470677800,8192096172894406564646400,1870885111733841408594984000,459893703431651653070494156800
%N A221099 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^(5*n))^n/n!.
%F A221099 E.g.f. also satisfies:
%F A221099 (1) A(x) = Sum_{n>=0} binomial(A(x)^(5*n), n) * x^n.
%F A221099 (2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(x)^(5*n*k)/n!.
%e A221099 E.g.f.: A(x) = 1 + x + 10*x^2/2! + 240*x^3/3! + 9720*x^4/4! + 556400*x^5/5! +...
%e A221099 where A(x) satisfies:
%e A221099 A(x) = 1 + log(1 + x*A(x)^5) + log(1 + x*A(x)^10)^2/2! + log(1 + x*A(x)^15)^3/3! +...
%e A221099 The e.g.f. also satisfies:
%e A221099 A(x) = 1 + A(x)^5*x + A(x)^10*(A(x)^10-1)*x^2/2! + A(x)^15*(A(x)^15-1)*(A(x)^15-2)*x^3/3! + A(x)^20*(A(x)^20-1)*(A(x)^20-2)*(A(x)^20-3)*x^4/4! +...+ binomial(A(x)^(5*n), n)*x^n +...
%o A221099 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, log(1+x*(A+x*O(x^n))^(5*m))^m/m!)); n!*polcoeff(A, n)}
%o A221099 for(n=0,20,print1(a(n),", "))
%o A221099 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, binomial((A+x*O(x^n))^(5*m), m)*x^m)); n!*polcoeff(A, n)}
%o A221099 for(n=0,20,print1(a(n),", "))
%o A221099 (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
%o A221099 {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))^(5*m*k))*x^m/m!)); n!*polcoeff(A, n)}
%o A221099 for(n=0,20,print1(a(n),", "))
%Y A221099 Cf. A189981, A221096, A221097, A221098.
%K A221099 nonn
%O A221099 0,3
%A A221099 _Paul D. Hanna_, Jan 01 2013