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How many ways, exactly, can a Chess King, always moving forward (i.e. withsteps [1,0],[0,1],[1,1]) walk to [1,2]? Thanks to the amazingApagodu-Zeilberger extension of the Almkvist-Zeilberger algorithm, adapted inthis article for combinatorial applications, this 104492-digit number, can becomputed in less than 33 seconds. But not just this particular number. Manyother numbers that come up in enumerative combinatorics, can be computed justas efficiently

Linear-Time and Constant-Space Algorithms to compute Multi-Sequences that arise in Enumerative Combinatorics (and Elsewhere)