co.combinatorics - Is there a combinatorial reason that the (-1)st Catalan number is -1/2?

The $n$th Catalan number can be written in terms of factorials as

$$C_n = {(2n)! over (n+1)! n!}.$$
We can rewrite this in terms of gamma functions to define the Catalan numbers for complex $z$:
$$C(z) = {Gamma(2z+1) over Gamma(z+2) Gamma(z+1)}.$$
This function is analytic except where $2n+1, n+2$, or $n+1$ is a nonpositive integer -- that is, at $n = -1/2, -1, -3/2, -2, ldots$.

At $z = -2, -3, -4, ldots$, the numerator of the expression for $C(z)$ has a pole of order 1, but the denominator has a pole of order $2$, so $lim_{z to n} C(z) = 0$.

At $z = -1/2, -3/2, -5/2, ldots$, the denominator is just some real number and the numerator has a pole of order 1, so $C(z)$ has a pole of order $1$.

But at $z = -1$:
- $Gamma(2z+1)$ has a pole of order $1$ with residue $1/2$;
- $Gamma(z+2) = 1$;
- $Gamma(z+1)$ has a pole of order $1$ with residue $1$.
Therefore $lim_{z to -1} C(z) = 1/2$, so we might say that the $-1$st Catalan number is $-1/2$.

Is there an interpretation of this fact in terms of any of the countless combinatorial objects counted by the Catalan numbers?

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