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I'm looking for the coefficients $a_0,\ldots,a_k$ of the polynomial $$f(x)=\prod_{i=1}^r(1+x+\cdots +x^{k_i-1})=\prod_{i=1}^r\frac{1-x^{k_i}}{1-x}$$ Since $f(1/x)=x^{-k}f(x)$ where $k = \deg(f)=\displaystyle\sum_{i=1}^r(k_i-1)$, I know that $a_j = a_{k-j}$ for all $0 \le j \le k$. Futhermore, by using the binomial series I can show $a_j=\displaystyle\binom{r+j-1}{j}$ for $j < \min_i (k_i-1)$.

But I'm struggled with the other coefficients. Has anyone an idea on how to find the correct formula ?

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One expression for the answer is $$a_k=\#\lbrace(d_1,\dots,d_r)|\forall i,~0\leq d_i\leq k_i-1~\mathrm{and}~d_1+\cdots +d_r=k\rbrace$$ but this pretty non explicit... –  Olivier Bégassat Jul 8 '12 at 21:47
With no constraints whatever on the $k_i$, I can't imagine there is any useful answer to this question. –  Gerry Myerson Jul 8 '12 at 23:22

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