Congruences concerning generalized central trinomial coefficients

Abstract

For any n ∈ N = {0, 1, 2, . . .} and b, c ∈ Z, the generalized central trinomial coefficient T n (b, c) denotes the coefficient of x n in the expansion of (x 2 + bx + c) n . Let p be an odd prime. In this paper, we determine the summation p−1 k=0 T k (b, c) 2 /m k modulo p 2 for integers m with certain restrictions. As applications, we confirm some conjectural congruences of Sun [Sci. China Math. 57 (2014), 1375-1400.