Geodesic
A natural extension of Catalan numbers
Journal of integer sequences, Tome 11 (2008) no. 3
A Dyck path is a lattice path in the plane integer lattice Z $\times $ Z consisting of steps (1,1) and (1,-1), each connecting diagonal lattice points, which never passes below the $x$-axis. The number of all Dyck paths that start at (0,0) and finish at ($2n,0$) is also known as the $n$th Catalan number. In this paper we find a closed formula, depending on a non-negative integer $t$ and on two lattice points $p_{1}$ and $p_{2}$, for the number of Dyck paths starting at $p_{1}$, ending at $p_{2}$, and touching the $x$-axis exactly $t$ times. Moreover, we provide explicit expressions for the corresponding generating function and bivariate generating function.
Classification : 05A15, 11B83, 11Y55
Keywords: Catalan numbers, Dyck paths 
@article{JIS_2008__11_3_a7,
     author = {Solomon,  Noam and Solomon,  Shay},
     title = {A natural extension of {Catalan} numbers},
     journal = {Journal of integer sequences},
     year = {2008},
     volume = {11},
     number = {3},
     zbl = {1148.05009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2008__11_3_a7/}
}
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Solomon,  Noam; Solomon,  Shay. A natural extension of Catalan numbers. Journal of integer sequences, Tome 11 (2008) no. 3. http://geodesic.mathdoc.fr/item/JIS_2008__11_3_a7/