Geodesic
A recursive relation for weighted Motzkin sequences
Journal of integer sequences, Tome 8 (2005) no. 1
We consider those lattice paths that use the steps $Up, Level$, and $Down$ with assigned weights $w, u$, and $v$. In probability theory, the total weight is 1. In combinatorics, we regard weight as the number of colors and normalize by setting $w=1$. The lattice paths generate Motzkin sequences. Here we give a combinatorial proof of a three-term recursion for a weighted Motzkin sequence and we find the radius of convergence.
Classification : 05A15, 05A19
Keywords: Motzkin paths, combinatorial identity 
@article{JIS_2005__8_1_a5,
     author = {Woan,  Wen-jin},
     title = {A recursive relation for weighted {Motzkin} sequences},
     journal = {Journal of integer sequences},
     year = {2005},
     volume = {8},
     number = {1},
     zbl = {1065.05012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2005__8_1_a5/}
}
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Woan,  Wen-jin. A recursive relation for weighted Motzkin sequences. Journal of integer sequences, Tome 8 (2005) no. 1. http://geodesic.mathdoc.fr/item/JIS_2005__8_1_a5/