Self-Convolutions of Generalized Narayana Numbers

For the Fibonacci numbers $F_n$, we have the self-convolution formula $5{\sum }_{i=0}^n{F}_i{F}_{n-i}=(2n)F_{n+1}-(n+1)F_n$. We find the corresponding self-convolution formula for the Narayana numbers $R_n$ which satisfy $R_n=R_{n-1}+R_{n-3}$, and then generalize it to the $k$-step Narayana numbers ${\mathcal{R}}_n$ with order-$k$ recurrence formula ${\mathcal{R}}_n={\mathcal{R}}_{n-1}+{\mathcal{R}}_{n-k}$.