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

TL;DR

This paper investigates the last digit properties of central binomial coefficients and related sums, providing explicit formulas for their behavior modulo 10 and identifying specific integers with non-divisibility properties by 5.

Contribution

It introduces simple formulas for the last digit of binomial coefficients and sums, and characterizes integers where these are not divisible by 5, advancing understanding of their modular properties.

Findings

Formulas for the last digit of inom{2n}{n} and related sums.

Characterization of integers where inom{2n}{n} and sums are not divisible by 5.

Explicit sequences of such integers based on divisibility properties.

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$.