Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum

Define the numbers RnRn and WnWn byRn=∑k=0n(n+k2k)(2kk)12k−1,andWn=∑k=0n(n+k2k)(2kk)32k−3. We prove that, for any positive integer n and odd prime p, there hold∑k=0n−1(2k+1)Rk2≡0(modn),∑k=0p−1(2k+1)Rk2≡4p(−1)p−12−p2(modp3),9∑k=0n−1(2k+1)Wk2≡0(modn),∑k=0p−1(2k+1)Wk2≡12p(−1)p−12−17p2(modp3),ifp>3. The first two congruences were originally conjectured by Z.-W. Sun.