Geodesic
A discrete convolution on the generalized Hosoya triangle
Journal of integer sequences, Tome 18 (2015) no. 1
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 generalized 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.
Keywords: hosoya triangle, generalized Fibonacci number, convolution, non-decreasing Dyck path, Fibonacci binary word 
@article{JIS_2015__18_1_a5,
     author = {Czabarka,  \'Eva and Fl\'orez,  Rigoberto and Junes,  Leandro},
     title = {A discrete convolution on the generalized {Hosoya} triangle},
     journal = {Journal of integer sequences},
     year = {2015},
     volume = {18},
     number = {1},
     zbl = {1310.11021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2015__18_1_a5/}
}
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Czabarka,  Éva; Flórez,  Rigoberto; Junes,  Leandro. A discrete convolution on the generalized Hosoya triangle. Journal of integer sequences, Tome 18 (2015) no. 1. http://geodesic.mathdoc.fr/item/JIS_2015__18_1_a5/