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.



Determining the asymptotic behavior of a sequence
One type of Catalan's $q$-analogue is due to Carlitz ([see the paper for this and more](http://homepage.univie.ac.at/josef.hofbauer/85jct_catalan.pdf))
$$C\_{n+1}(q)=\sum\_{k=0}^nC\_k(q)\,C\_{n-k}(q)\,q^{(k+1)(n-k)}, \qquad C\_0:=1.$$
[Blieberger and Kirschenhofer](http://institute.unileoben.ac.at/mathstat/personal/(2014)%20Generalized%20Catalan%20Sequences%20%5BEtc%5D.pdf) 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.


