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

Abstract

Let $f_{n}= \sum_{i=0}^n {n \choose i}{ 2n-2i\choose n-i}$, $g_{n}= \sum_{i=1}^n {n\choose i}{2n-2i \choose n-i}$. Let $\{a_k\}_{k=1}$ be the set of all positive integers n, in increasing order, for which ${2n \choose n}$ 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 $a_k$, $b_k$, ${2n \choose n}$ mod 10, $ f_{n}$ mod 10, and $ g_{n}$ mod 10.