# Upper bounds for a sequence of integers

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 $$\alpha$$.

• maybe $C_0=1$? they seem to be all equal to 0 now Feb 19, 2021 at 8:43
• That must be true- edited accordingly. Feb 19, 2021 at 17:37
• For $\alpha=1$, this sequence seems to be OEIS A000699.
– Seva
Feb 19, 2021 at 21:02
• Oh, yes, sorry. That 0 was a typo. Nad yes, when $\alpha=0$ that sequence is well-known. Feb 27, 2021 at 16:54