# nt.number theory – A question on a real sequence

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 $$a_n equiv 0$$?