The last digit of \(\binom {2n}n\) and \(\sum \binom ni\binom{2n-2i}{n-1}\) (Q1378555)

Summary: 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 \(a_k\), \(b_k\), \( {2n\choose n}\bmod 10\), \( f_n\bmod 10\), and \( g_{n}\bmod 10\).