Geodesic
A combinatorial interpretation for a super-Catalan recurrence
Journal of integer sequences, Tome 8 (2005) no. 1
Nicholas Pippenger and Kristin Schleich have recently given a combinatorial interpretation for the second-order super-Catalan numbers $ (u_{n})_{n\ge 0}=(3,2,3,6,14,36,...)$: they count "aligned cubic trees" on $ n$ interior vertices. Here we give a combinatorial interpretation of the recurrence $ u_{n} = \sum_{k=0}^{n/2-1}\binom{n-2}{2k}2^{n-2-2k}u_{k}\,:$ it counts these trees by number of deep interior vertices where "deep interior" means "neither a leaf nor adjacent to a leaf".
Classification : 05A19, 05A15
Keywords: super-Catalan, aligned cubic tree 
@article{JIS_2005__8_1_a6,
     author = {Callan,  David},
     title = {A combinatorial interpretation for a {super-Catalan} recurrence},
     journal = {Journal of integer sequences},
     year = {2005},
     volume = {8},
     number = {1},
     zbl = {1064.05011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2005__8_1_a6/}
}
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Callan,  David. A combinatorial interpretation for a super-Catalan recurrence. Journal of integer sequences, Tome 8 (2005) no. 1. http://geodesic.mathdoc.fr/item/JIS_2005__8_1_a6/