On Delannoy numbers and Schröder numbers

Abstract

The nth Delannoy number and the nth Schröder number given by D n = ∑ k = 0 n ( n k ) ( n + k k ) and S n = ∑ k = 0 n ( n k ) ( n + k k ) 1 k + 1 respectively arise naturally from enumerative combinatorics. Let p be an odd prime. We mainly show that ∑ k = 1 p − 1 D k k 2 ≡ 2 ( − 1 p ) E p − 3 ( mod p ) and ∑ k = 1 p − 1 S k m k ≡ m 2 − 6 m + 1 2 m ( 1 − ( m 2 − 6 m + 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 ∑ k = 1 p − 1 D k 2 k 2 ≡ − 2 q p ( 2 ) 2 ( mod p ) where q p ( 2 ) denotes the Fermat quotient ( 2 p − 1 − 1 ) / p .