For $k,n\in \mathbb{N_+}$ and $k\leq n$, define $s(n,k)$ recursively by: $$ \begin{align} & s(1,1)=1\\ & \\ & s(n,1)=s(n-1,n-1)\\ & \\ & s(n,2)=\frac{n-2}{n-1}s(n-1,1)\\ & \\ & s(n,k)=\frac{n-k}{n-1}s(n-1,k-1)+\frac{k-2}{n-1}s(n-1,k-2) \end{align} $$
When I plot $s(n,k)$ against $k$ for fixed $n$'s, I always get a curve that looks sinusoidal:
Is there an explanation for why it should be so? I understand that a 2nd order linear recurrence relation with constant coefficients can have periodic solutions, but this function is apparently not such an example, since you have $s(n-1,k-1)$ and $s(n-1,k-2)$ on the right hand side instead of $s(n,\cdot)$, not to mention the changing coefficients.