  Koroljuk gave a summation formula for counting the number of lattice paths
from $(0,0)$ to $(m,n)$ with $(1,0), (0,1)$-steps in the plane that stay
strictly above the line $y=k(x-d)$, where $k$ and $d$ are positive integers. In
this paper we obtain an explicit formula for the number of lattice paths from
$(a,b)$ to $(m,n)$ above the diagonal $y=kx-r$, where $r$ is a rational number.
Our result slightly generalizes Koroljuk's formula, while the former can be
essentially derived from the latter. However, our proof uses a recurrence with
respect to the starting points, and hereby presents a new approach to
Koroljuk's formula.
