Congruences involving generalized central trinomial coefficients

For integers b and c the generalized central trinomial coefficient T (n) (b,c) denotes the coefficient of x(n) in the expansion of (x(2) + bx + c)(n) . Those T-n = T-n (1,1) (n = 0,1,2, ... ) are the usual central trinomial coefficients, and T-n (3,2) coincides with the Delannoy number D-u = Sigma(n)(k=0) ((n)(k)) ((n+k)(k)) in combinatorics. We investigate congruences involving generalized central trinomial coefficients systematically. Here are some typical results: For each n = 1, 2, 3, ... , we have Sigma(n-1)(k=0) (2k+1)T-k(b,c)(2)(b(2)-4c)(n-1-k) equivalent to 0 (mod n(2)) and in particular n(2) vertical bar Sigma(n-1)(k=0)(2k+1)D-k(2); if p is an odd prime then Sigma(p-1)(k=0) T-k(2) equivalent to (-1/p) (mod p) and Sigma(p-1)(k=0) D-k(2) equivalent to (2/p) (mod p), where (-) denotes the Legendre symbol. We also raise several conjectures some of which involve parameters in the representations of primes by certain binary quadratic forms.