MathDB

Let

a_{1}=1

a_{n}=\frac{1}{n} \sum_{k=1}^{n-1}a_{k}a_{n-k}

for

n\geq 2

. Show that i)

\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}<2^{-\frac{1}{2}}

; ii)

\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}\geq \frac{2}{3}

Problem Statement