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On Motzkin numbers and central trinomial coefficients (1801.08905v3)
Published 26 Jan 2018 in math.CO and math.NT
Abstract: The Motzkin numbers $M_n=\sum_{k=0}n\binom n{2k}\binom{2k}k/(k+1)$ $(n=0,1,2,\ldots)$ and the central trinomial coefficients $T_n$ ($n=0,1,2,\ldots)$ given by the constant term of $(1+x+x{-1})n$, have many combinatorial interpretations. In this paper we establish the following surprising arithmetic properties of them with $n$ any positive integer: $$\frac2n\sum_{k=1}n(2k+1)M_k2\in\mathbb Z,$$ $$\frac{n2(n2-1)}6\,\bigg|\,\sum_{k=0}{n-1}k(k+1)(8k+9)T_kT_{k+1},$$ and also $$\sum_{k=0}{n-1}(k+1)(k+2)(2k+3)M_k23{n-1-k}=n(n+1)(n+2)M_nM_{n-1}.$$