Moderate# Series Solution with Nonzero Center

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Lisa is curious about the behavior of the solutions for the equation $y''+(2x+1)y'+2y=0$ near $x=2$, which is an ordinary point.

She thinks there exists a solution of the form $y=\sum_{n=0}^{\infty}a_n(x-2)^n$.

The recurrence relation Lisa will find could be: