Balanced $0,1$-words and the Galois group of $(x+1{)}^n-\lambda x^p$

We study the number of $0,1$-words where the fraction of 0 is “almost” fixed for any initial subword. It turns out that this study use and reveal the structure of the Galois group (the monodromy group) of the polynomials $(x+1{)}^n-\lambda x^p$. ($p$ is not necessary a prime here.)