Geodesic
Dyck paths with no peaks at height $k$
Journal of integer sequences, Tome 4 (2001) no. 1.

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Summary: A Dyck path of length 2n is a path in two-space from (0,0) to (2n,0) which uses only steps (1,1) (north-east) and (1,-1) (south-east). Further, a Dyck path does not go below the x-axis. A peak on a Dyck path is a node that is immediately preceded by a north-east step and immediately followed by a south-east step. A peak is at height k if its y-coordinate is k. Let $G_k(x)$ be the generating function for the number of Dyck paths of length 2n with no peaks at height k with k >= 1. It is known that $G_1(x)$ is the generating function for the Fine numbers (sequence A000957). In this paper, we derive the recurrence
Keywords: Dyck paths, Catalan number, fine number, generating function 
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     title = {Dyck paths with no peaks at height $k$},
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Peart, Paul; Woan, Wen-Jin. Dyck paths with no peaks at height $k$. Journal of integer sequences, Tome 4 (2001) no. 1. http://geodesic.mathdoc.fr/item/JIS_2001__4_1_a5/