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Catalan Numbers Modulo 2^k

By Shu-chung Liu and Jean C.-C. Yeh


In this paper, we develop a systematic tool to calculate the congruences of some combinatorial numbers involving n!. Using this tool, we re-prove Kummer’s and Lucas’ theorems in a unique concept, and classify the congruences of the Catalan numbers cn (mod 64). To achieve the second goal, cn (mod 8) and cn (mod 16) are also classified. Through the approach of these three congruence problems, we develop several general properties. For instance, a general formula with powers of 2 and 5 can evaluate cn (mod 2 k) for any k. An equivalence cn ≡ 2 k c¯n is derived, where ¯n is the number obtained by partially truncating some runs of 1 and runs of 0 in the binary string [n]2. By this equivalence relation, we show that not every number in [0, 2 k − 1] turns out to be a residue of cn (mod 2 k) for k ≥ 2

Year: 2010
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