English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

The numbers $R_n$ and $W_n$ are defined as \begin{align*}R_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{1}{2k-1},\ \text{and}\W_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{3}{2k-3}. \end{align*} Weprove that, for any positive integer $n$ and odd prime $p$, there hold\begin{align*} \sum_{k=0}^{n-1}(2k+1)R_k^2 &\equiv 0 \pmod{n}, \\\sum_{k=0}^{p-1}(2k+1)R_k^2 &\equiv 4p(-1)^{\frac{p-1}{2}} -p^2 \pmod{p^3}, \\9\sum_{k=0}^{n-1}(2k+1)W_k^2 &\equiv 0 \pmod{n}, \\ \sum_{k=0}^{p-1}(2k+1)W_k^2&\equiv 12p(-1)^{\frac{p-1}{2}}-17p^2 \pmod{p^3}, \quad\text{if $p>3$.}\end{align*} The first two congruences were originally conjectured by Z.-W.Sun. Our proof is based on the multi-variable Zeilberger algorithm and thefollowing observation: $$ {2n\choose n}{n\choose k}{m\choose k}{k\choosem-n}\equiv 0\pmod{{2k\choose k}{2m-2k\choose m-k}}, $$ where $0\leqslantk\leqslant n\leqslant m \leqslant 2n$.

Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum