It frequently happens that, in some famillies of polynomials with positive coefficients, the coefficients of large polynomials look like a bell curve and tend to the distribution function of the Gaussian law. One obvious example is given by the familly $(1+x)^n$.
I know some examples of famillies of polynomials where the coefficients are not positive, but where plotting the coefficients give a nice oscillating curve, as the one displayed below.
My question is the following:
Is there any known such oscillating function, with some kind of universal property ?
In other words, what is the function one can see in this picture ?
As far as I can tell, this does not seem to be given by a product of $\exp(-x^2)$ by trigonometric functions, because the local maxima do not fit very well on a parabola.
Here is a graph of the log of the absolute value of the coefficients, compared with a parabola.