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Koroljuk gave a summation formula for counting the number of lattice pathsfrom $(0,0)$ to $(m,n)$ with $(1,0), (0,1)$-steps in the plane that staystrictly above the line $y=k(x-d)$, where $k$ and $d$ are positive integers. Inthis 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 beessentially derived from the latter. However, our proof uses a recurrence withrespect to the starting points, and hereby presents a new approach toKoroljuk's formula.

Koroljuk's formula for counting lattice paths revisited