Identities and Inequalities for Sums Involving Binomial Coefficients

摘要

We present identities and inequalities for sums involving the partial sums of binomial coe?cients. One of our results states that T n = n X k=0 ? n k ◆ k k X j=0 ? n j ◆ = n2 2n?2 + n 2 ? 2n n ◆ (n = 0,1,2,...), p T n?1 T n+1 < T n < T n?1 + T n+1 2 (n = 1,2,3,...) and T m + T n ? T m+n (m,n = 0,1,2,...) with equality if and only if m = 0 or n = 0.