Greatly extended series have been derived for moments of the pair-connectedness for bond and site percolation on the directed square and triangular lattices. The length of the various series has been at least doubled to more than 110 (100) terms for the square-lattice bond (site) problem and more than 55 terms for the bond and site problems on the triangular lattice. Analysis of the series leads to very accurate estimates for the critical parameters and generally seems to rule out simple rational values for the critical exponents. The values of the critical exponents for the average cluster size, parallel and perpendicular connectedness lengths are estimated by γ = 2.277 69(4), ν∥ = 1.733 825(25) and ν⊥ = 1.096 844(14), respectively. An improved estimate for the percolation probability exponent is obtained from the scaling relation β = (ν∥ + ν⊥ - γ)/2 = 0.276 49(4). In all cases the leading correction to scaling term is analytic.