Professor Christopher French cowrote an article, "Hankel Transforms of Linear Combinations of Catalan Numbers" with math majors Michael Dougherty, Benjamin Saderholm, and Wenyang Qian, all math majors in the class of 2012. The article appeared on April 20 in the Journal of Integer Sequences, Vol. 14 (2011), Article 11.5.1. In this paper, the authors consider the result of applying a certain operation called the Hankel Transform to certain sequences of numbers obtained in a natural way from the famous Catalan number sequence. Other authors had shown that when one applies the Hankel Transform to the sequence of sums of adjacent Catalan numbers, the resulting sequence consists of every other Fibonacci number. This paper generalizes that result to more general linear combinations of Catalan numbers. The research for this work was done during the summer of 2010, as part of a MAP that Dougherty, Saderholm, and Qian participated in.