OFFSET
0,3
COMMENTS
Compare definition of e.g.f. A(x) to the trivial statement:
if F(x) = 1/(1-x) then F(x) = 1 + x*exp(Integral F(x) dx).
Here Integral F(x) dx does not include the constant of integration.
Limit n->infinity (a(n)/n!)^(1/n) = 2.792845... - Vaclav Kotesovec, Feb 28 2014
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..184
FORMULA
E.g.f. derivative: A'(x) = [1 + x*A(x)^4]*(A(x) - 1)/x.
EXAMPLE
E.g.f. A(x) = 1+ x + 2*x^2/2! + 12*x^3/3! + 88*x^4/4! + 860*x^5/5! +...
A(x)^4 = 1 + 4*x + 20*x^2/2! + 144*x^3/3! +1384*x^4/4! +16400*x^5/5!+...
Let L(x) = Integral A(x)^4 dx where A(x) = 1 + x*exp(L(x)), then
L(x) = x + 4*x^2/2! + 20*x^3/3! + 144*x^4/4! + 1384*x^5/5! +...
exp(L(x)) = 1 + x + 5*x^2/2! + 33*x^3/3! + 297*x^4/4! + 3385*x^5/5! +...
PROG
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*exp(intformal(A^4))); n!*polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 06 2008
STATUS
approved