A Discrete Convolution on the Generalized Hosoya Triangle

The generalized Hosoya triangle is an arrangement of numbers where each entry is a product of two generalized Fibonacci numbers. We define a discrete convolution C based on the entries of the genealized Hosoya triangle. We use C and generating functions to prove that the sum of every k-th entry in the n-th row or diagonal of generalized Hosoya triangle, beginning on the left with the first entry, is a linear combination of rational functions on Fibonacci numbers and Lucas numbers. A simple formula is given for a particular case of this convolution. We also show that C summarizes several sequences in the OEIS. As an application, we use our convolution to enumerate many statistics in combinatorics.