Note: Theorem 1 stated in the paper is incorrect (as it does not satisfy the initial condition of the difference equations). However, a significant amount of recent numerical and other evidence suggests it is correct in a more restricted form. Thus there is the following conjecture:

Conjecture: Theorem 1 is correct so long as every terminal vertex of the paths is reachable from all of the initial vertices.

Since corollary 1 satisfies the "reachability" condition it is conjectured to be correct as it is stated.

Definition [Reachable]. A vertex A is reachable from vertex B if there exists a binomial lattice path from A to B.

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