Sequence A93637 of the OEIS (https://oeis.org/A093637) starting as $1,1,2,4,9,20,49,117,297,746,1947,\ldots$ is defined by the coefficients $a_0,a_1,\ldots$ of the unique formal power series defined by the equality $$A(x)=\prod_{n=0}^\infty \frac{1}{1-a_nx^{n+1}}=\sum_{n=0}^\infty a_n x^n\ .$$ Experimentally, $a_{n+1}/a_n$ seems to converge to some a limit roughly given by $2.96777$ suggesting a convergency radius slightly larger than $1/3$ for $A(x)$.

Is there an easy argument ensuring that $A(x)$ has strictly positive convergency radius? Are there computable upper/lower bounds for the convergency radius of $A(x)$?