Determining the asymptotic behavior of a sequence 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.
 A: One type of Catalan's $q$-analogue is due to Carlitz (see the paper for this and more)
$$C_{n+1}(q)=\sum_{k=0}^nC_k(q)\,C_{n-k}(q)\,q^{(k+1)(n-k)}, \qquad C_0:=1.$$
Blieberger and Kirschenhofer studied these Catalans, in equation (2), and the more related sequence $r_n$, in equation (5). They have found the asymptotics, on page 9, 
$$r_n\sim 2^{\frac{n^2+3n}2}\beta(1/2).$$
Now, your sequence is $a_n=2^nr_n$ and hence
$$a_n\sim 2^{\frac{n^2+5n}2}\beta(1/2);$$
where $\beta(1/2)\approx 0.7153374336\dots$. 
J. Furlinger and J. Hofbauer, $q$-Catalan Numbers, Jour. of Comb. Theory, Series A, 40(2):248–264, 1985.
J. Blieberger, P. Kirschenhofer,
Generalized Catalan Sequences Originating from the Analysis of Special Data Structures, Bulletin of the Institute of Combinatorics and its Applications, 71 (2014), 103-116.
A: For $b_k$:  Computing the first few terms and
searching into https://oeis.org returns A218222
and A088716.
A088716 is very close to your sequence. 
If $b_k$ is really A218222, from the comments:
$$ b_n \sim C 2^{n-1} (n-1)! (n-1)^2$$
