  Let p be an odd prime and let a be a positive integer. In this paper we
investigate the sum $\sum_{k=0}^{p^a-1}\binom{hp^a-1}{k}\binom{2k}{k}/m^k$ mod
p^2, where h,m are p-adic integers with m\not=0 (mod p). For example, we show
that if h\not=0 (mod p) and p^a>3 then $$
sum_{k=0}^{p^a-1}\binom{hp^a-1}{k}\binom{2k}{k}(-h/2)^k
=(\frac{1-2h}{p^a})(1+h((4h-2)^{p-1}/h^{p-1}-1)) (mod p^2),$$ where (-) denotes
the Jacobi symbol. Here is another remarkable congruence: If p>3 then
$$\sum_{k=0}^{p^a-1}\binom{p^a-1}{k}\binom{2k}{k}(-1)^k =3^{p-1}(\frac{p^a}3)
(mod p^2).$$
