Assuming I have a series $a^m(k)$ defined as follows (m being a superscript - not an exponent):
$$ a^m(k) = a_k^{m} = \sum_{i=1}^{k} \, a_{i-1}^{m-1}\\ ...\\ a_k^{2} = \sum_{i=1}^{k} a_{i-1}^{1} \\ a_k^{1} = 1 \\ \text{and}\\ a_0^{1} = 1, ~a_0^{m} = 0 ~~ \forall\, m>1 $$ what is the asymptotic growth of $a^m(k)$?
My intuition says that $a^m(k) \in \mathcal{O}\left(k^{(m-1)}\right)$, as
- $a^1(k)$ is constant ($k^0$)
- $a^2(k)$ is linear ($k^1$)
- $a^3(k)$ is quadratic ($\frac{k(k-1)}{2}$)
But I don't know how to verify/falsify the general case.
Also, does someone have a good book/reference where to look up those kind of questions? I imagine this is some well known case and I just don't know how to properly search for it.
Thanks in advance and sorry if any of the terms or natations are off, I don't have a lot of formal math education and am not a native speaker.
