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Odd Catalan numbers modulo \(2^k\). (English) Zbl 1251.11010

Integers 12, No. 2, 161-165 (2012); 11, A55, 5 p. (2011).
Summary: This article proves a conjecture by S.-C. Liu and J. C.-C. Yeh [J. Integer Seq. 13, No. 5, Article ID 10.5.4, 26 p. (2010; Zbl 1230.05013)] about Catalan numbers, which states that odd Catalan numbers can take exactly \(k-1\) distinct values modulo \(2^k\), namely the values \(C_{2^1-1},\dots, C_{2^{k-1}-1}\).

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
11A07 Congruences; primitive roots; residue systems

Citations:

Zbl 1230.05013
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Online Encyclopedia of Integer Sequences:

Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).