Are Recurrence Relations Related?

Algebra Level 4

Let x1,x2,x3,x4x_1, x_2, x_3, x_4 be the roots of the polynomial x4+2x3+3x2+4x+5x^4+2x^3+3x^2+4x+5. Define an=k=14kxkn\displaystyle a_n=\sum_{k=1}^{4}kx_k^{n}. Compute

a2018+3a2016+5a2014a2017+2a2015  .\dfrac{a_{2018}+3a_{2016}+5a_{2014}}{a_{2017}+2a_{2015}} \; .


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