On a Nesbitt-Carlitz determinant (Q2227813)

Consider the \(n\times n\) matrix \(A=(a_{ij})\), \(a_{ij}=\binom{n+1}{2i-j}\). \textit{A. M. Nesbitt} [Educ. Times 57, 449 (1904)] announced without proof, and \textit{J. D. Niblett} [Am. Math. Mon. 59, 171--174 (1952; Zbl 0046.01003)] proved, that \(\det{A}=2^\frac{n(n+1)}{2}\). Applying the dual Jacobi-Trudi determinant presentation of certain Schur function (see [\textit{I. G. Macdonald}, Symmetric functions and Hall polynomials. 2nd ed. Oxford: Clarendon Press (1995; Zbl 0824.05059)]), the present author proves that the eigenvalues of \(A\) are \(2,2^2,\dots,2^n\). He also gives a cubic generalization of this result.