Let $n$ be a positive integer and let $g(n)$ be a given strictly increasing integer function such that $0<g(n)<n$ for all $n$. Also the sequence $ |g(n) - n|$ is unbounded as $n$ grows.
Let $f(1)=1$ and further $$ f(n)=\sum_{k=1}^{g(n)}f(k) $$
Now I know that if $g(n)$ is close to $\dfrac{n}{2}$ then $f(n) = O(\exp(\ln(n)^{2-o(1)}))$
See : http://oeis.org/A000123 and http://oeis.org/A018819
The question is how to find the asymptotic to $f(n)$ (like above) when given $g(n)$ ?
For instance I assume if $g(n)$ is close to $\dfrac{n}{a}$ for $0.4<a<0.6$ then also $f(n) = O(\exp(\ln(n)^{2-o(1)}))$ but that could be completely wrong.
What happens for $g(n)$ being close to $ A n^B $ or $n - A n^B$ ?
Or $g(n)$ close to $\ln(n)$ or $n - \ln(n)$ ? Can we still express such slow/fast growth ?
How about $g(n)$ being the prime counting function ?
Of course I know some trivial brute boundaries such as $f(n) < 2^n$ or $f(n)<fib(n)$ , but Im looking for better boundaries.
I hope this question is not too general.