On Motzkin numbers and central trinomial coefficients

The Motzkin numbers M-n = n-expressionry sumexpressiontion (n)(k=0)(n2k)(2kk)/(k+1) (n = 0,1,2, horizontexpressionl ellipsis ) and the central trinomial coefficients T-n (n = 0,1,2, horizontexpressionl ellipsis ) 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: 2/n n-expressionry sumexpressiontion (n)(k=1)(2k+1)M-k(2)is an element of Z,<br />n(2)(n(2)-1)6| n-expressionry sumexpressiontion (n-1)(k=0)k(k+1)(8k+9)TkTk+1,<br />and also<br /> n-expressionry sumexpressiontion (n-1)(k=0)(k+1)(k+2)(2k+3)M(k)(2)3(n-1-k )= n(n+1)(n+2)MnMn-1. (C) 2022 Elsevier Inc. All rights reserved.