Estimating the behavior for large $n$ I want to find how these coefficients increase/decrease as $n$ increases:
$$ C_n = \frac{1}{n!} \left[(n+\alpha)^{n-\alpha-\frac{1}{2}}\right]$$ 
with $\alpha=\frac{1}{br-1}$ and $0\leq b,r \leq 1$. 
I used the Stirling's Approximation factorial  $n!\sim \sqrt{2\pi n} n^n e^{-n}$ and got:
$$ C_n = \frac{1}{\sqrt{2\pi n} n^n e^{-n}} \left[(n+\alpha)^{n-\alpha-\frac{1}{2}}\right]$$ 
I can't proceed any further. I would greatly appreciate any comment!

Take logarithms.  $\alpha$ can be any negative number; if you are interested in $n\gg-\alpha$, then $\ln(n+\alpha)\approx\ln n+(\alpha/n)$, so 
$$ \ln C_n\approx (n-\alpha-\frac12)(\ln n+\alpha/n)-(n+\frac12)\ln n+n-\ln(2\pi)/2\\
\approx(-\alpha-1)\ln n+n+\alpha-\ln(2\pi)/2\\
C_n\approx\frac{1}{\sqrt{2\pi}}n^{-\alpha-1}e^{n+\alpha}$$
