Abstract
The Motzkin numbers Mn=∑k=0n(n2k)(2kk)/(k+1) (n=0,1,2,…) and the central trinomial coefficients Tn (n=0,1,2,…) 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:2n∑k=1n(2k+1)Mk2∈Z,n2(n2−1)6|∑k=0n−1k(k+1)(8k+9)TkTk+1, and also∑k=0n−1(k+1)(k+2)(2k+3)Mk23n−1−k=n(n+1)(n+2)MnMn−1.
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