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

1, 1, 10, 240, 9720, 556400, 41153220, 3737360130, 402876727680, 50302825722720, 7141958361129600, 1136668023900846360, 200486825731741824000, 38826473000115470677800, 8192096172894406564646400, 1870885111733841408594984000, 459893703431651653070494156800

Offset

0,3

Links

Table of n, a(n) for n=0..16.

Formula

E.g.f. also satisfies:

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

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

Example

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

Prog

 (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)}
for(n=0, 20, print1(a(n), ", "))
(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)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{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)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A189981, A221096, A221097, A221098.

Sequence in context: A215835 A006423 A067423 * A171204 A156443 A211088

Adjacent sequences: A221096 A221097 A221098 * A221100 A221101 A221102

Keyword

nonn

Author

Paul D. Hanna, Jan 01 2013

Status

approved