A supercongruence involving Delannoy numbers and Schröder numbers

The Delannoy numbers and Schröder numbers are given byDn=∑k=0n(nk)(n+kk)andSn=∑k=0n(nk)(n+kk)1k+1, respectively. Let p>3p>3 be a prime. We mainly prove that∑k=1p−1DkSk≡2p3Bp−3−2pHp−1⁎(modp4), where BnBn is the n  -th Bernoulli number and these Hn⁎ are the alternating harmonic numbers given by Hn⁎=∑k=1n(−1)kk. This supercongruence was originally conjectured by Z.-W. Sun in 2011.