OFFSET
0,3
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
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..106
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!.
EXAMPLE
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 +...
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))^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)}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, May 03 2011
STATUS
approved