# Problem 438: Integer part of polynomial equation's solutions

![graphic](img438.gif)

For an n-tuple of integers t = (a1, ..., an), let (x1, ..., xn) be the
solutions of the polynomial equation xn + a1xn-1 + a2xn-2 + ... + an-1x
+ an = 0. Consider the following two conditions: x1, ..., xn are all
real. If x1, ..., xn are sorted, ⌊xi⌋ = i for 1 ≤ i ≤ n. (⌊·⌋: floor
function.) In the case of n = 4, there are 12 n-tuples of integers which
satisfy both conditions. We define S(t) as the sum of the absolute
values of the integers in t. For n = 4 we can verify that ∑S(t) = 2087
for all n-tuples t which satisfy both conditions. Find ∑S(t) for n = 7.