One of the many interpretations of Dyck Paths is the number of lattice paths from $(0,0)$ to $(n,n)$, staying at or below the diagonal $y=x$, using only 2 kinds of line segments (1 unit right, or 1 unit up).
Dyck Paths from Wolfram Mathworld: http://mathworld.wolfram.com/DyckPath.html
The number of Dyck Paths is given by the Catalan numbers.
My question is, if we generalize this so that the 2 allowed types of line segments are ($1$ unit to the right) and ($K$ units upward), now how many "K-Dyck paths" are possible?
Can we further generalize this to allow $K+1$ kinds of step types: ($1$ unit right) vs. (any of $1, 2, \ldots, K$ units upward)?