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Polynomial Having Values of a Sequence

Source: Turkey TST 2016 P4

April 10, 2016
algebrapolynomial

Problem Statement

A sequence of real numbers a0,a1,…a_0, a_1, \dots satisfies the condition∑n=0man⋅(−1)n⋅(mn)=0\sum\limits_{n=0}^{m}a_n\cdot(-1)^n\cdot\dbinom{m}{n}=0for all large enough positive integers mm. Prove that there exists a polynomial PP such that an=P(n)a_n=P(n) for all n≥0n\ge0.
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