If I understood your question correctly, the numbers you're looking for are called Ballot numbers. The number of paths from (0,0) to (m,n) (where m>n) which stay below the diagonal is \frac{m-n}{m+n}\binom{m+n}{m}
Moreover, if m>r⋅n, then the number of lattice paths from (0,0) to (m,n) which stay below the line x=r⋅y is \fra{m-rn}{m+n}\binom{m+n}{m}. (I haven't worked this out, but Ira Gessel says so in Introduction to Lattice Path Enumeration)