The main result of the paper is that if a sequence of complex numbers \((a_{n})_{n\geqslant 0}\) satisfies the following conditions \[ \sum_{\substack{ 0\leqslant k\leqslant n\\k\text{ even}}} \binom{n}{k}a_{k}=O(n^{r}) \quad \text{and} \quad \sum_{\substack{ 0\leqslant k\leqslant n\\k\text{ odd}}} \binom{n}{k}a_{k}=O(n^{r}) \quad \text{as} \quad n\rightarrow\infty \] for some integer \(r\geqslant 0\), then \(a_{n}=0\) for all \(n>r\). As an application, there is deduced a localized form of a theorem of \textit{G. R. Allan} about nilpotent elements in Banach algebras [Stud. Math. 121, No. 2, 185-192 (1996; Zbl 0862.46029)], and this in turn leads to some invariant-subspace theorems. As a further application, there is proved a variant of Carleman's theorem on the unique determination of probability measures by their moments. The paper concludes with some generalization of the main result.