Explicit expression of the counting generating function for Gessel's walk
Résumé
We consider the so-called Gessel's walk, that is the planar random walk that is confined to the first quadrant and that can move in unit steps to the West, North-East, East and South-West. For this walk we make explicit the generating function of the number of paths starting at $(0,0)$ and ending at $(i,j)$ in time $k$.
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