A189981 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^n)^n/n!.

1, 1, 2, 12, 120, 1600, 28500, 621138, 16017792, 480474720, 16390969920, 626786792280, 26584872779520, 1238524175509608, 62873918454756864, 3455537675553482400, 204449393824639488000, 12958008875933613962880

Offset: 0

Author

Paul D. Hanna, May 03 2011

Keywords

• sign

Comments

The definition of the e.g.f. A(x) is an application of the identity:

* Sum_{n>=0} log(1 + q^n*x)^n/n! = Sum_{n>=0} binomial(q^n, n)*x^n at q = A(x).

Consider the function G(x) such that G(x) = 1 + x*G(x)^p, then

* G(x) = Sum_{n>=0} log(1 + x*G(x)^p)^n/n! (trivially), and

* G(x) = Sum_{n>=0} binomial(p*n+1,n)*x^n/(p*n+1) for fixed p;

does an analogous expression exist for the e.g.f. of this sequence?

Note that terms a(70)-a(83) are negative. - Vaclav Kotesovec, Jul 13 2014

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1600*x^5/5! +...
where A(x) satisfies:
A(x) = 1 + log(1 + x*A(x)) + log(1 + x*A(x)^2)^2/2! + log(1 + x*A(x)^3)^3/3! +...
The e.g.f. also satisfies:
A(x) = 1 + A(x)*x + A(x)^2*(A(x)^2-1)*x^2/2! + A(x)^3*(A(x)^3-1)*(A(x)^3-2)*x^3/3! + A(x)^4*(A(x)^4-1)*(A(x)^4-2)*(A(x)^4-3)*x^4/4! +...+ binomial(A(x)^n, n)*x^n +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..106

Crossrefs

Cf. A014070, A221101, A224797.

Programs

PARI

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,log(1+x*(A+x*O(x^n))^m)^m/m!));n!*polcoeff(A,n)}

PARI

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,binomial((A+x*O(x^n))^m,m)*x^m));n!*polcoeff(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))^(m*k))*x^m/m!));n!*polcoeff(A,n)}

Formula

E.g.f. also satisfies:

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

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