# 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 $$alpha$$.