Conjecture of the general form of a power series

Question

Relcently I met a power series(Source Link-Eq(4.1)) of the type $$ f(x)=1-x+\frac{1}{2}x^2+\frac{1}{4}x^3-\frac{1}{8}x^4-\frac{35}{128}x^5-\frac{157}{1024}x^6+\cdots $$ where $x$ is supposed to be a small value. I just wonder that if anyone is power enough to GUESS the general behavior of the function? Maybe one can conjecture that it has the form $$ f(x)=1+\sum_{n=1}^{\infty}\frac{a_n}{4^{n-1}}x^n. $$ where $a_n's$ are some integers.