A221099 - OEIS

%I #4 Jan 01 2013 21:24:57

%S 1,1,10,240,9720,556400,41153220,3737360130,402876727680,

%T 50302825722720,7141958361129600,1136668023900846360,

%U 200486825731741824000,38826473000115470677800,8192096172894406564646400,1870885111733841408594984000,459893703431651653070494156800

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

%F  E.g.f. also satisfies:

%F (1) A(x) = Sum_{n>=0} binomial(A(x)^(5*n), n) * x^n.

%F (2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(x)^(5*n*k)/n!.

%e  E.g.f.: A(x) = 1 + x + 10*x^2/2! + 240*x^3/3! + 9720*x^4/4! + 556400*x^5/5! +...

%e where A(x) satisfies:

%e 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 The e.g.f. also satisfies:

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

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

%o (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}

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

%Y  Cf. A189981, A221096, A221097, A221098.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 01 2013