An $\mathbf{R}$-sequence is a sequence $(c_1,\ldots,c_n)$ of integers $c_i\geq -1$ such that $c_1+\cdots +c_n=0$ and for all $1\leq i \leq n$, $c_1+\ldots+c_i\geq 0$. For each $n\in \mathbb{N}$, determine the number of $\mathbf{R}$-sequences of length $n$.
An approach I have been considering is to compute the number of $\mathbf{R}$-sequences by constructing a surjective function $f:\mathcal{M}\rightarrow\mathbf{R}$, where $\mathcal{M}$ is the set of Motzkin paths (2D lattice paths that start at $(0,0)$ with NE,E,SE steps, ending at $(n,0)$ and never going below the $x$-axis) and determining the size of the pre-image $|f^{-1}(R)|$ for each $R\in \mathbf{R}$. Indeed, given a Motzkin path $\mathcal{P}$, for each strictly rising NE-subsequence $r_1,\ldots,r_\ell$ of $\mathcal{P}$ there are $comp(\ell)=2^{\ell-1}$ ways to construct that corresponding segment of the corresponding $\mathbf{R}$-sequence. So if $\mathcal{P}$ has $k$ rising NE-subsequences each with length $\ell_k$, there are $$\frac{2^{\sum_i^k \ell_i}}{2^k},$$ possible $\mathbf{R}$-sequences. So we have reduced the problem to determining the number and length of each of such rising subsequences in the set of Motzkin paths. For completeness, the generating function for class of Motzkin paths is given by $$M(x)=\sum_{n=0}^\infty \sum_{k=0}^{\lfloor n/2\rfloor}\binom{n}{2k}\frac{1}{k+1}\binom{2k}{n}x^n,$$ where $$m_{n,k}=\sum_{k=0}^{\lfloor n/2\rfloor}\binom{n}{2k}\frac{1}{k+1}\binom{2k}{n},$$ is the number of Motzkin paths of length $n$ with $k$ NE-upsteps.
Looking for any insights into making this work, or another valid approach to counting these objects.
