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

Walter Shur

paper

The Last Digit of $(n 2 n)$ and $\sum (i n) (n - i 2 n - 2 i)$

arXiv:1302.0188

Abstract

Let $f_{n} = \sum_{i = 0}^{n} (i n) (n - i 2 n - 2 i)$ , $g_{n} = \sum_{i = 1}^{n} (i n) (n - i 2 n - 2 i)$ . Let ${a_{k}}_{k = 1}$ be the set of all positive integers n, in increasing order, for which $(n 2 n)$ 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 $a_{k}$ , $b_{k}$ , $(n 2 n) m o d 10$ , $f_{n} m o d 10$ , and $g_{n} m o d 10$ .

The Last Digit of (n2n​) and ∑(in​)(n−i2n−2i​)

Abstract

The Last Digit of $(n 2 n)$ and $\sum (i n) (n - i 2 n - 2 i)$