My goal is to compute 

\begin{equation}
I = \det \left(\mathbf{I} + \mathbf{A}\right)
\end{equation}
where $\mathbf{A}$ is a $n \times n$ checkerboard matrix filled with Catalan numbers:
$$
\left\{
 \begin{array}{crc}
 \mathbf{A}\_{ij} = C\_{p-1} \alpha^{2(p-1)} &\mbox{ if }& i+j=2p \mbox{, and with }C\_p= \frac{1}{p+1}\binom {2p} {p}\\
 \mathbf{A}\_{ij} = 0 &\mbox{ if }& i+j \mbox{ is odd.}
 \end{array}
\right.
$$
with $\alpha >0$ a parameter.

Numerically, it seems that $I$ has a limit when $n\to +\infty$ if $\alpha<1/2$ and diverges to $\infty$ otherwise.

Any idea ?