p-adic congruences motivated by series

Let p>5p>5 be a prime. Motivated by the known formulae∑k=1∞(−1)kk3(2kk)=−25ζ(3)and∑k=0∞(2kk)2(2k+1)16k=4Gπ (where G=∑k=0∞(−1)k/(2k+1)2 is the Catalan constant), we show that∑k=1(p−1)/2(−1)kk3(2kk)≡−2Bp−3(modp),∑k=(p+1)/2p−1(2kk)2(2k+1)16k≡−74p2Bp−3(modp3) and∑k=0(p−3)/2(2kk)2(2k+1)16k≡−2qp(2)−pqp(2)2+512p2Bp−3(modp3), where B0,B1,B2,…B0,B1,B2,… are Bernoulli numbers and qp(2)qp(2) is the Fermat quotient (2p−1−1)/p(2p−1−1)/p.