Arithmetic properties of Delannoy numbers and Schröder numbers

DefineDn(x)=∑k=0n(nk)2xk(x+1)n−kforn=0,1,2,… andsn(x)=∑k=1n1n(nk)(nk−1)xk−1(x+1)n−kforn=1,2,3,…. Then Dn(1) is the n-th central Delannoy number Dn, and sn(1) is the n-th little Schröder number sn. In this paper we obtain some surprising arithmetic properties of Dn(x) and sn(x). We show that1n∑k=0n−1Dk(x)sk+1(x)∈Z[x(x+1)]for alln=1,2,3,…. Moreover, for any odd prime p and p-adic integer x≢0,−1(modp), we establish the supercongruence∑k=0p−1Dk(x)sk+1(x)≡0(modp2). As an application we confirm Conjecture 5.5 in [S14a], in particular we prove that1n∑k=0n−1TkMk(−3)n−1−k∈Zfor alln=1,2,3,…, where Tk is the k-th central trinomial coefficient and Mk is the k-th Motzkin number.