The Filbert matrix (Q2735227)

The Hankel matrix \(A=(a_{ij})\) defined by \(a_{ij}=(i+j-1)^{-1}\) for \(i,j=1,2,\dots\). is known as the Hilbert matrix. \textit{Man-Duen Choi} [Am. Math. Mon. 90, 301-312 (1983; Zbl 0546.47007)] explored many properties of the Hilbert matrix, including the inverse, which appears to have integer entries. Choi asked what sort of coincidence it is if the inverse of a matrix of reciprocals of integers has integer entries. NEWLINENEWLINENEWLINEThe present author shows that the inverses of the Hankel matrices based on the reciprocals of the Fibonacci numbers, the binomial coefficients \({i+j\choose 2}\) and the binomial coefficients \({i+j+2\choose 3}\) all have integer entries. The Hankel matrix based on the reciprocals of the Fibonacci numbers is termed the Filbert matrix, because its inverse bears a striking resemblance to the inverse of the Hilbert matrix.