Let $\mu \in \mathbb{N}_0^n$ be a multi index and set $$P(X_1, \dots, X_n) = X_1^{\mu_1}(X_1 + X_2)^{\mu_2} \cdots (X_1 + \cdots +X_n)^{\mu_n} = \prod_{j=1}^n (X_1 + \cdots + X_j)^{\mu_j}.$$ Since $P$ is a homogeneous polynomial of degree $|\mu| = \mu_1 + \dots + \mu_n$, it is clear that $$ P(X_1, \dots, X_n) = \sum_{|\alpha| = |\mu|} C(\alpha, \mu) \,X_1^{\alpha_1} \cdots X_n^{\alpha_n}$$ for certain coefficients $C(\alpha, \mu)$.

While we expect that there should be a closed expression for $C(\alpha, \mu)$ in terms of binomial coefficients, we were unable to obtain such a formula. Does someone know the answer or a reference to the literature?