The number of minimal lattice paths restricted by two parallel lines

AbstractWe deal with non-decreasing paths on the non-negative quadrant of the integral square lattice, called by minimal lattice paths, from (0,0) to a point (n, m) restricted by two parallel lines with an incline k (⩾0). We express the generating functions of the number of these distinct minimal lattice paths in terms of the polynomials ϕk(n, x) = ∑l = 0[nk]n − kll(−x)l, n⩾)Formulas obtained thus include the generating function of the so-called higher Catalan number Ck(n) or Ballot numbers as the special case.The number of minimal lattice paths for k=1 is given as an explicit form by expanding the corresponding generating function