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

1, 1, 6, 90, 2328, 84660, 3972060, 229176654, 15712089120, 1248343353216, 112832687750400, 11437476445244520, 1285433373363701760, 158682294244352658312, 21349655111889802728576, 3110218068324341815470000, 487862693943123978219847680, 81999755541558838752430348800

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Formula

E.g.f. also satisfies:

(1) A(x) = Sum_{n>=0} binomial(A(x)^(3*n), n) * x^n.

(2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(x)^(3*n*k)/n!.

Example

 E.g.f.: A(x) = 1 + x + 6*x^2/2! + 90*x^3/3! + 2328*x^4/4! + 84660*x^5/5! +...
where A(x) satisfies:
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! +...
The e.g.f. also satisfies:
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 +...

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))^(3*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))^(3*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))^(3*m*k))*x^m/m!)); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A189981, A221096, A221098, A221099.

Sequence in context: A004996 A001499 A147630 * A177584 A177558 A177580

Adjacent sequences: A221094 A221095 A221096 * A221098 A221099 A221100

Keyword

nonn

Author

Paul D. Hanna, Jan 01 2013

Status

approved