Congruences involving generalized central trinomial coefficients

TL;DR

This paper systematically investigates congruences involving generalized central trinomial coefficients, revealing new divisibility properties and Legendre symbol relations, and proposing several conjectures related to prime representations.

Contribution

It introduces new congruence relations for generalized central trinomial coefficients, including divisibility by squares of integers and primes, and explores connections with quadratic forms and Legendre symbols.

Findings

Sum of weighted squares of coefficients is divisible by n^2.

Sum of squares of Delannoy numbers modulo p relates to Legendre symbol.

New conjectures on prime representations by quadratic forms.

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 $T_n=T_n(1,1)\ (n=0,1,2,\ldots)$ are the usual central trinomial coefficients, and $T_n(3,2)$ coincides with the Delannoy number $D_n=\sum_{k=0}^n\binom nk\binom{n+k}k$ in combinatorics. We investigate congruences involving generalized central trinomial coefficients systematically. Here are some typical results: For each $n=1,2,3,\ldots$ we have $$\sum_{k=0}^{n-1}(2k+1)T_k(b,c)^2(b^2-4c)^{n-1-k}\equiv0\pmod{n^2}$$ and in particular $n^2\mid\sum_{k=0}^{n-1}(2k+1)D_k^2$; if $p$ is an odd prime then $$\sum_{k=0}^{p-1}T_k^2\equiv\left(\frac{-1}p\right)\ \pmod{p}\ \ \ {\rm and}\ \ \ \sum_{k=0}^{p-1}D_k^2\equiv\left(\frac 2p\right)\ \pmod{p},$$ 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.