On Delannoy numbers and Schröder numbers

The nth Delannoy number and the nth Schröder number given byDn=∑k=0n(nk)(n+kk)andSn=∑k=0n(nk)(n+kk)1k+1 respectively arise naturally from enumerative combinatorics. Let p be an odd prime. We mainly show that∑k=1p−1Dkk2≡2(−1p)Ep−3(modp) and∑k=1p−1Skmk≡m2−6m+12m(1−(m2−6m+1p))(modp), where (−)(−) is the Legendre symbol, E0,E1,E2,…E0,E1,E2,… are Euler numbers, and m is any integer not divisible by p. We also conjecture that∑k=1p−1Dk2k2≡−2qp(2)2(modp) where qp(2)qp(2) denotes the Fermat quotient (2p−1−1)/p(2p−1−1)/p.