Consider \(f(x) = \sum\limits_{n=x}^{x^2} n\) for all positive integers \(x\).

It can be represented by a polynomial of degree four with complex roots \(a_1, a_2, a_3, a_4\).

Evaluate the sum of the coefficients of \(g(x)\) such that its roots are \(a_1^2, a_2^2, a_3^2, a_4^2\).

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