Congruences involving $\left(\genfrac{}{}{0ex}{}{4k}{2k}\right)$ and $\left(\genfrac{}{}{0ex}{}{3k}{k}\right)$

Abstract

Let $p$ be a prime greater than 3. In the paper we mainly determine ${\sum }_{k=0}^{[p/4]}\left(\genfrac{}{}{0ex}{}{4k}{2k}\right)(-1{)}^k$, ${\sum }_{k=0}^{[p/3]}\left(\genfrac{}{}{0ex}{}{3k}{k}\right),{\sum }_{k=0}^{[p/3]}\left(\genfrac{}{}{0ex}{}{3k}{k}\right)(-1{)}^k$ and ${\sum }_{k=0}^{[p/3]}\left(\genfrac{}{}{0ex}{}{3k}{k}\right)(-3{)}^k$ modulo $p$, where $[x]$ is the greatest integer not exceeding $x$.

24 pages

References in corpus (1)

• Various congruences involving binomial coefficients and higher-order Catalan numbers

Cited by in corpus (1)

• Cubic congruences and sums involving $\left(\genfrac{}{}{0ex}{}{3k}{k}\right)$