The Last Digit of $\binom{2n}{n}$ and $\sum\binom{n}{i}\binom{2n-2i}{n-i}$

Abstract

Let $f_{n}=\sum_{i=0}^n \binom{n}{i}\binom{2n-2i}{n-i}$, $g_{n}= \sum_{i=1}^n \binom{n}{i}\binom{2n-2i}{n-i}$. Let $\{a_k\}_{k=1}$ be the set of all positive integers n, in increasing order, for which $\binom{2n}{n}$ is not divisible by 5, and let $\{b_k\}_{k=1}$ be the set of all positive integers n, in increasing order, for which $g_n$ is not divisible by 5. This note finds simple formulas for $a_k$, $b_k$, $\binom{2n}{n}\ mod\ 10$, $ f_{n}\ mod\ 10$, and $ g_{n}\ mod\ 10$.