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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 $\alpha$.

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

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