This is a continuation of some questions asked by Johann Cigler: Number of bounded Dyck paths with "negative length" and Number of bounded Dyck paths with negative length as Hankel determinants.

Let $\mathcal{D}(k,n)$ denote the following planar directed graph: It has $k+1$ vertices in the leftmost column and $n+1$ vertices in the bottom row. It always has an odd number of columns, and even number of rows. Also, all edges are directed from left to right.

For $0\leq i \leq k+1$, let $C(k,i;n)$ denote the number of $i$-tuples of nonintersecting lattice paths in $\mathcal{D}(k,n)$ which connect the bottom $i$ vertices of the leftmost column to the bottom $i$ vertices of the rightmost column.

Note that these tuples of nonintersecting lattice paths could also be called *$i$-fans of $(2k+1-2(i-1))$-bounded Dyck paths of semilength $n$*.

There is of course a Lindström-Gessel-Viennot determinantal expression for $C(k,i;n)$.

**Conjecture/Proposition**: As a function of $n$, $C(k,i;n)$ satisfies a linear recurrence with constant coefficients.

The reason this should be true is via a "transfer matrix"-style argument. We can make $\mathcal{D}(k,n+1)$ from $\mathcal{D}(k,n)$ by adding two columns on the right; and if we consider $i$-tuples of nonintersecting lattice paths in $\mathcal{D}(k,n)$ that start at the bottom $i$ vertices of the leftmost column, there are finitely many patterns of sinks at which they could terminate at; and in turn there are a fixed number of ways to continue these patterns for the two additional columns.

If that is indeed so, then we can define $C(k,i;-n)$ at negative values via the recurrence.

**Question**: Do we have the "reciprocity" result that $C(k,i;-n)=C(k,k+1-i;n+1)$?

The resolution of the previous questions implies that this is true for $i=1$ (and it is trivially true for $i=0$).