Congruences for Franel numbers

The Franel numbers given by fn=∑k=0n(nk)3(n=0,1,2,…)(n=0,1,2,…) play important roles in both combinatorics and number theory. In this paper we initiate the systematic investigation of fundamental congruences for the Franel numbers. We mainly establish for any prime p>3p>3 the following congruences:∑k=0p−1(−1)kfk≡(p3)(modp2),∑k=0p−1(−1)kkfk≡−23(p3)(modp2),∑k=1p−1(−1)kkfk≡0(modp2),∑k=1p−1(−1)kk2fk≡0(modp).