Note that $(1+x)^n$ itself does not have Gaussian coefficients 
when you go that far from the central term: by Stirling, 
for fixed $\rho \in (0,1)$ the $x^{\rho n}$ coefficient
$\bigl( {n \atop \rho n} \bigr)$
is roughly proportional to $\exp H(\rho)$ where 
$H(\rho) = -\rho \log \rho - (1-\rho) \log(1-\rho)$.
Your graph looks like it could be something like this multiplied by $\cos cx$
for some $c \approx 1/6$.  The stationary-phase technique for finding the
asymptotic behavior of power-series coefficients often gives rise to
expressions such as this, though $H(r)$ might be a sum of more complicated
terms than just $-\rho \log \rho$ and $-(1-\rho) \log(1-\rho)$.