I've encountered the following sequences

$$ a_k=2^{k+1}\sum_{j=0}^{k-1}a_{k-1-j}a_j,\;a_0=1 $$

$$ b_k=(k+1)\sum_{j=0}^{k-1}b_{k-1-j}b_j,\;b_0=1. $$

I would like to have an estimate of the growth of these sequences as $k$ grows.

After looking here and there, I found the Catalan's numbers defined by $$ C_k=\sum_{j=0}^{k-1}C_{k-1-j}C_j,\;C_0=1. $$ They have an asymptotic growth of $$ C_k=2^{k+1}k^{-3/2}. $$

I tried (unsuccesfully) to manipulate my original series to recover some form of the Catalan's numbers.

Any idea is very welcome.