0,3

Compare the definition of e.g.f. A(x) to this trivial statement:

if F(x) = 1/(1-2x) then F(x) = 1 + 2*x*exp(2*Integral F(x)dx).

E.g.f. satisfies: A'(x) = [1 + 2*x*A(x)]*(A(x) - 1)/x where A'(x) = d/dx A(x).

E.g.f. A(x) = 1 + x + 4*x^2/2! + 18*x^3/3! + 112*x^4/4! +...

CONVERGENCE AND ASYMPTOTICS.

Let r be the radius of convergence of the power series A(x), then:

a(n)/n! ~ (1/2)/r^(n+1) where

r=0.63923227138053689755467936951149007771973874430987288272658905276...

so that the power series A(x) diverges at x=r.

Note: A(-r) is evaluated as 1/(2r) since Integral A(x)dx is a

convergent alternating series at x=-r having the sum:

Sum_{n>=0} a(n)*(-r)^(n+1)/(n+1)! = log(r - 1/2)/2 - log(r);

however, as N approaches infinity, the N-th partial sum of A(x) at x=-r,

Sum_{n>=0..N} a(n)*(-r)^n/n!, oscillates between 1/(4r) and 3/(4r).

Thus the power series A(x) converges only for |x| < r.

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*exp(2*intformal(A)+x*O(x^n))); n!*polcoeff(A, n)}

Sequence in context: A144085 A003708 A000986 this_sequence A113356 A062805 A053483

Adjacent sequences: A143917 A143918 A143919 this_sequence A143921 A143922 A143923

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Sep 05 2008