$x^{2n} + a_{2n-1} (-1)^{2n-1} x^{2n-1} + \ldots + a_2 (-1)^2 x^2 + a_1 (-1)^1 x + a_ 0$

Consider the polynomial above such that $a_0 + a_1 + a_2 + \ldots + a_{2n-1} = 1600$ and it has non-negative integer roots $r_1, r_2, \ldots , r_{2n}$ .

For distinct prime $p_1,p_2, \ldots, p_{2n}$ , compute the number of factors of the integer which has the form

$p_1 ^{r_1} \times p_2 ^{r_2} \times \cdots \times p_{2n} ^{r_{2n}}.$

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