Arithmetic properties of Delannoy numbers and Schroder numbers

Define & para;& para;D-n{x) = Sigma(k=0n) ((n)(k))(2 )x(k )(x + 1)(n-k )for n = 0,1, 2 ,...& para;& para;and & para;& para;s(n)(x) = Sigma k=1(n )1/n(k(n))1/n<((n)(k))((n)(k-1))x(k-1) >(x+1)(n-k) >for n = 1, 2, 3, ...& para;& para;Then D-n (1) is the n-th central Delannoy number D-n, and s(n) (1) is the n-th little Schroder number S-n. In this paper we obtain some surprising arithmetic properties of D-n(x) and S-n(x). We show that & para;& para;1/n(Sigma k=0)n-1( D)k((x) s)k+1((x) is an element of Z[x(x+1)] for all n=1, 2, 3, ...& para;& para;Moreover, for any odd prime p and p-adic integer x not equivalent to 0, -1 (mod p), we establish the supercongruence)& para;& para;Sigma k=0p-1( D)k((x) s)k+1(x)( equivalent to 0 (mod p)2().& para;& para;As an application we confirm Conjecture 5.5 in [S14a], in particular )(we prove that & para;& para;1/n)( Sigma)k=0(n-1T)k(M)k((-3))( is an element of Z for all n = 1, 2, 3, ...,& para;& para;where T)k( is the k-th central trinomial coefficient and M)k( is the k-th Motzkin number. (C) 2017 Elsevier Inc. All rights reserved.)