2486 on Delannoy Numbers and Schröder Numbers

The nth Delannoy number and the nth Schröder number given by D n = n k=0 n k n + k k and S n = n k=0 n k n + k k 1 k + 1 respectively arise naturally from enumerative combinatorics. Let p be an odd prime. We mainly show that p−1 k=1 D k k 2 ≡ 2 −1 p E p−3 (mod p) and p−1 k=1 S k m k ≡ m 2 − 6m + 1 2m 1 − m 2 − 6m + 1 p (mod p), where (−) is the Legendre symbol, E 0 , E 1 , E 2 ,. .. are Euler numbers, and m is any integer not divisible by p. We also conjecture that p−1 k=1 D 2 k k 2 ≡ −2q p (2) 2 (mod p) where q p (2) denotes the Fermat quotient (2 p−1 − 1)/p.