Congruences involving generalized central trinomial coefficients

Abstract

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 $D_n = \sum\nolimits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} \left( {\begin{array}{*{20}c} {n + k} \\ k \\ \end{array} } \right)$ in combinatorics. We investigate congruences involving generalized central trinomial coefficients systematically. Here are some typical results: For each n = 1, 2, 3, …, we have

$$\sum\limits_{k = 0}^{n - 1} {\left( {2k + 1} \right)T_k \left( {b,c} \right)^2 \left( {b^2 - 4c} \right)^{n - 1 - k} \equiv 0 \left( {\bmod n^2 } \right)}$$

and in particular $\left. {n^2 } \right|\sum\nolimits_{k = 0}^{n - 1} {\left( {2k + 1} \right)D_k^2 }$ is an odd prime then

$$\sum\limits_{k = 0}^{p - 1} {T_k^2 \equiv \left( {\frac{{ - 1}} {p}} \right)\left( {\bmod p} \right) and } \sum\limits_{k = 0}^{p - 1} {D_k^2 \equiv \left( {\frac{2} {p}} \right)\left( {\bmod p} \right), }$$

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.