Given $\alpha\geq0$ we consider the sequence $$ C_k=k^\alpha\sum_{j=0}^{k-1}C_jC_{k-1-j} $$ with $C_0=1$. I'm interested in upper bounds (in terms of $\alpha$) for such a sequence. I know that when $\alpha=0$ the previous sequence reduces to the Catalan numbers where the bound is

$$ O(4^k/k^{3/2}), $$ but I wonder whether something is known for positive $\alpha$.