Let ${a_n}_{nge1}$ be a real sequence that decays faster than any algebraic speed, that is, $lim_{nto infty} n^pa_n = 0$ for every positive integer $p$. Assume that $$sum_{nge 1}(n+1)^kn^ka_n = 0$$ for every integer $k ge 0$.

**Question:** Can we conclude that $a_n equiv 0$?