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Problem 479

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Roots on the Rise

Let a_k, b_k, and c_k represent the three solutions (real or complex numbers) to the expression 1/x = (k/x)²(k+x²) - kx.

For instance, for k = 5, we see that {a₅, b₅, c₅} is approximately {5.727244, -0.363622+2.057397i, -0.363622-2.057397i}.

Let S(n) = Σ (a_k+b_k)^p(b_k+c_k)^p(c_k+a_k)^p for all integers p, k such that 1 ≤ p, k ≤ n.

Interestingly, S(n) is always an integer. For example, S(4) = 51160.

Find S(10⁶) modulo 1 000 000 007.

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逐渐增长的根

记a_k，b_k，c_k分别是方程1/x = (k/x)²(k+x²) - kx在复数域内的三个根。

例如，对于k = 5，{a₅, b₅, c₅}的近似值为{5.727244, -0.363622+2.057397i, -0.363622-2.057397i}。

对于任意整数p，k满足1 ≤ p, k ≤ n，记S(n) = Σ (a_k+b_k)^p(b_k+c_k)^p(c_k+a_k)^p。

有趣的是，S(n)永远是个整数。例如，S(4) = 51160。

求S(10⁶) modulo 1 000 000 007。

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