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Integer part of polynomial equation's solutions

Problem 438

Published on Sunday, 29th September 2013, 01:00 am; Solved by 101

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 ≤ in. (⌊·⌋: 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.