catalan numbers – Upper bounds for a sequence of integers

Given $alphageq0$ we consider the sequence
$$
C_k=k^alphasum_{j=0}^{k-1}C_jC_{k-1-j}
$$

with $C_0=0$. 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 $alpha$.