Suppose there is an integer sequence $\{a_0,a_1...a_n...\}$ and a family of polynomials is defined as follows:

$p_0 = a_0$

$p_1 = a_0x+a_1$

$p_2 = a_0x^2+a_1x+a_2$

$p_n = a_0x^n+a_1x^{n-1}+...+a_n$

...

Where the coefficients are the elements of the sequence. The characteristics of $p_n$ are interesting because $p_n(0)=a_n$ and $p_n(1)=\sum_{n}a_n$

For instance, applied to the Möebius function:

$p_0 = 1$

$p_1 = x-1$

$p_2 = x^2-x-1$

$p_3 = x^3-x^2-x$

$p_4 = x^4-x^3-x^2-1$

...

$p_n(0)=\mu(n)$ ($n^{th}$ element of the Möbius sequence) and $p_n(1)=\sum_{n}\mu(n)$ is the partial summation up to $n$ of the Möbius function, in other words, it is the Merten's function.

This is the graph of the first $100$ polynomials, $p_0$ to $p_{99}$

And this is a zoom of the segment [0,1] so it is possible to see the ramifications of each polynomial from the position of the last $\mu(n)$ to the position of the last $\sum_n\mu(n)$. The shape of the paths is quite curious because it is a representation of the surjective-only diagram of $\mu(n) \to \sum_n \mu(n)$ for each $n$ in $[0,100]$.

I would like to ask the following questions:

Is it possible to know something about the properties (e.g. the convergence of the accumulated sum) of a sequence as the $lim_{n \to \infty}p_n(1)=lim_{n \to \infty}\sum_{n}a_n$ by the calculation of the shape of the generic polynomial generated with the elements of the sequence as coefficients? e.g. finding the the shape of the "limit" polynomial $p_n$ when $n \to \infty$?

I tried to find some papers about this kind of approach, to understand if it leads to something or it is just visually interesting. Are there any papers regarding the generation of polynomials by using the elements of sequences as coefficients? Thank you!