On Delannoy numbers and Schroder numbers

The nth Delannoy number and the nth Schroder number given by D-n = Sigma(n)(k=0) (n k) (n + k k) and S-n = Sigma(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 Sigma(p-1)(k=1) D-k/k(2) equivalent to 2(-1/p) Ep-3 (mod p) and Sigma(p-1)(k=1) S-k/m(k) equivalent to 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 in is any integer not divisible by p. We also conjecture that Sigma(p-1)(k=1) D-k(2)/k(2) equivalent to -2q(p)(2)(2) (mod p) where q(p)(2) denotes the Fermat quotient (2(p-1) - 1)/p. (C) 2011 Elsevier Inc. All rights reserved.