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{\bf Guoce Xin and Jing-Feng Xu}
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{\bf A Short Approach to Catalan Numbers Modulo $2^r$}
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We notice that two combinatorial interpretations of the well-known
Catalan numbers $C_n=(2n)!/n!(n+1)!$ naturally give rise to a
recursion for $C_n$. This recursion is ideal for the study of the
congruences of $C_n$ modulo $2^r$, which attracted a lot of interest
recently. We present short proofs of some known results, and improve
Liu and Yeh's recent classification of $C_n$ modulo $2^r$. The
equivalence $C_{n}\equiv_{2^r} C_{\bar n}$ is further reduced to
$C_{n}\equiv_{2^r} C_{\tilde{n}}$ for simpler $\tilde{n}$. Moreover,
by using connections between weighted Dyck paths and Motzkin paths, we
find new classes of combinatorial sequences whose $2$-adic order is
equal to that of $C_n$, which is one less than the sum of the digits
of the binary expansion of $n+1$.
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