DE-BUNDESWETTBEWERB 2014 Problem 6

For all positive integers $m$ and $k$ with $m\ge k$ , define $a_{m,k}=\binom{m}{k-1}-3^{m-k}$ . Determine all sequences of real numbers $\{x_1, x_2, x_3, \ldots\}$ , such that each positive integer $n$ satisfies the equation $$a_{n,1}x_1+ a_{n,2}x_2+ \cdots + a_{n,n}x_n = 0$$