We consider some combinatorial questions about the number of certain lattice walks in the plane restricted to the first quadrant. We will show how a long-standing open conjecture of Gessel about these walks was recently proven with computer algebra, in a joint effort with Doron Zeilberger and Christoph Koutschan. We will also report on large-scale computations done together with Alin Bostan, which indicate that a much stronger assertion than Gessel's original conjecture might be true.
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