I'm trying to come up with the formula describing the number of paths on hexagonal lattice of length $2n$ that start at the origin $O$ and go back to $O$ but doing so for the first time at step $2n$ (i.e. first return to the origin).

Suppose I already have a formula for the number of such paths but without the condition of returning for the first time at step $2n$. Is there a way to go through this formula to the one I'm looking for? For example, through generating functions, etc?

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