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?



Explicit Expansion for Certain Product of homogeneous Polynomials
We can express the coefficient as a product of binomial coefficients by expanding by the binomial theorem, starting with the rightmost factor. 


For example, with $n=3$ we have
$$
\def\m#1{{\mu\_{#1}}}
\def\i#1{{i\_{#1}}}
\begin{aligned}
X\_1^\m1(X\_1+X\_2)^\m2(X\_1+&X\_2+X\_3)^\m3
 = X\_1^\m1(X\_1+X\_2)^\m2\sum\_{\i3}\binom{\m3}{\i3} (X\_1+X\_2)^{\m3-\i3}X\_3^\i3\\
 &= X\_1^\m1\sum\_{\i3}\binom{\m3}{\i3}(X\_1+X\_2)^{\m2+\m3-\i3}X\_3^\i3\\
 &=\sum\_{\i2,\i3}X\_1^{\m1+\m2+\m3-\i2-\i3}X\_2^{\i2}X\_3^{\i3}
 \binom{\m2+\m3-\i3}{\i2}\binom{\m3}{\i3}.
\end{aligned}
$$
The formula for general $n$ is similar.


