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2018 PUMaC Algebra A7

Source:

November 25, 2018
PuMACalgebra

Problem Statement

Let the sequence {an}n=−2∞\left \{ a_n \right \}_{n = -2}^\infty satisfy a−1=a−2=0,a0=1a_{-1} = a_{-2} = 0, a_0 = 1, and for all non-negative integers nn, n2=∑k=0nan−kak−1+∑k=0nan−kak−2n^2 = \sum_{k = 0}^n a_{n - k}a_{k - 1} + \sum_{k = 0}^n a_{n - k}a_{k - 2} Given a2018a_{2018} is rational, find the maximum integer mm such that 2m2^m divides the denominator of the reduced form of a2018a_{2018}.
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