Recurrence equation $g(k+n)=\sum_{j=1}^ng(k+n-j)$ and exponential function

Recurrence equation $g(k+n)=\sum_{j=1}^ng(k+n-j)$ and exponential function

pIf I have the recurrence equation: $$g(k+n)=\sum_{j=1}^ng(k+n-j)$$ with
$g(h)=1$ for $0\le h\lt (n-1)$, is it possible to find a value of $n$ such
that: $$\lim_{k\to\infty}\frac{\exp(k)}{g(k)}=0?$$ Thanks./p