The Ramanujan Journal

, Volume 48, Issue 2, pp 233–244

# Proof of some congruence conjectures of Guo and Liu

Article

## Abstract

Let n and r be positive integers. Define the numbers $$S_{n}^{(r)}$$ by $$S_{n}^{(r)}=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) (2k+1)^r.$$ In this paper we prove some conjectures of Guo and Liu which extend some conjectures of Sun (Two new kinds of numbers and related divisibility results, 2014), such as: There exist integers $$a_{2r-1}$$ and $$b_r$$, independent of n, such that
\begin{aligned} a_{2r-1}\sum _{k=0}^{n-1}S_k^{(2r-1)}\equiv 0 \, (\mathrm{mod}\, n^2) \quad \text{ and } \quad b_r\sum _{k=0}^{n-1}kS_k^{(r)}\equiv 0\, (\mathrm{mod}\, n^2). \end{aligned}
By the Zeilberger algorithm, we find that for all $$0\le j<n$$,
\begin{aligned} (2j+1)\left( {\begin{array}{c}2j\\ j\end{array}}\right) \sum _{k=j}^{n-1}(2k-j+1)\left( {\begin{array}{c}k\\ j\end{array}}\right) ^2\equiv 0\, (\mathrm{mod}\, n^2). \end{aligned}

## Keywords

Central binomial coefficients Congruences Bernoulli numbers Zeilberger algorithm

## Mathematics Subject Classification

11B65 11B68 05A10 11A07

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