  A Filbert matrix is a matrix whose (i,j) entry is 1/F_(i+j-1), where F_n is
the nth Fibonacci number. The inverse of the n by n Filbert matrix resembles
the inverse of the n by n Hilbert matrix, and we prove that it shares the
property of having integer entries. We prove that the matrix formed by
replacing the Fibonacci numbers with the Fibonacci polynomials has entries
which are integer polynomials. We also prove that certain Hankel matrices of
reciprocals of binomial coefficients have integer entries, and we conjecture
that the corresponding matrices based on Fibonomial coefficients have integer
entries. Our method is to give explicit formulae for the inverses.
