A binomial identity related to Calkin's (Q1297421)

In a recent paper \textit{N. J. Calkin} [Discrete Math. 131, 335-337 (1994; Zbl 0799.05005)] had given a closed expression for the sum of the cubes of partial sums of binomial coefficients. In this note the author considers the sum of the same terms, but with alternative signs, i.e. he shows for an odd number \(n\) \[ \sum_{s=0}^{n}(-1)^s \left(\sum_{k=0}^s {n\choose k}\right)^3 = -n2^{3n-1}-3(-1)^{(n-1)/2}2^{n-1}{n-1 \choose (n-1)/2}. \] (Remark. Obviously, the formula in Theorem 2 contains a misprint, instead of \(s\) please read \(k\)).