The Narayana numbers defined by $N_{n,k}=\frac{1}{k+1}\binom{n}{k} \binom{n+1}{k},$ for $n\geq0,$ $0\leq k \leq n,$ and $N_{n.k}=0$ else, can be constructed recursively via the recurrence $$\binom{n+2}{2} N_{n,k}=\binom{2n+2-k}{2}N_{n-1,k-1}+(k+1)(2n+1-k)N_{n-1,k}+\binom{k+2}{2} N_{n-1,k+1}$$ with $N_{0,k}=[k=0]$.
Is there a combinatorial interpretation of this recurrence? For example by interpreting $N_{n,k}$ as the number of semi-standard Young tableaux with shape $2^k$ and entries in $ \{ 1, 2,\dots, n \}$?
