  Ozeki and Prodinger showed that the odd power sum of the first several
consecutive Fibonacci numbers of even order is equal to a polynomial evaluated
at certain Fibonacci number of odd order. We prove that this polynomial and its
derivative both vanish at $1$, and will be an integer polynomial after
multiplying it by a product of the first consecutive Lucas numbers of odd
order. This presents an affirmative answer to a conjecture of Melham.
