Below are the sequences

           enumerating the number of ways of walking from the origin

                with positive unit steps, in the square lattice

                to the point (i,i), all the while staying in the

                             the region y>=b/a*x

              followed by the conjectured asymptotics (if found)

                                for 2<=b<=a<=, 6

a=, 2, b=, 1, seq.:[1, 3, 9, 31, 108, 391, 1431, 5319, 19926, 75252, 285750,

    1090491, 4177774, 16060401, 61916977, 239307063, 926929746, 3597296770,

    13984508500, 54448030092]

a=, 3, b=, 1, seq.:[1, 3, 10, 34, 121, 441, 1628, 6077, 22879, 86691, 330210,

    1263274, 4850434, 18681112, 72140344, 279226002, 1082960198, 4207716972,

    16374681426, 63814540806]

                                               n
                                        0.258 4
                               asymp.:, --------
                                           1/2
                                          n

a=, 3, b=, 2, seq.:[1, 2, 7, 23, 76, 278, 1014, 3693, 13843, 52130, 196501,

    748331, 2861495, 10961281, 42179044, 162768111, 629196267, 2438220602,

    9466942018, 36809980968]

a=, 4, b=, 1, seq.:[1, 3, 10, 35, 125, 456, 1688, 6315, 23810, 90331, 344433,

    1318824, 5067470, 19529520, 75458659, 292213035, 1133822400, 4407046010,

    17156364660, 66881920635]

                                               n
                                        0.272 4
                               asymp.:, --------
                                           1/2
                                          n

a=, 4, b=, 3, seq.:[1, 2, 5, 19, 66, 227, 785, 2966, 11112, 41405, 153990,

    587867, 2249896, 8605216, 32886853, 126814899, 490440726, 1898280587,

    7348515656, 28547980826]

a=, 5, b=, 1, seq.:[1, 3, 10, 35, 126, 461, 1709, 6399, 24145, 91663, 349707,

    1339636, 5149444, 19852080, 76727290, 297201616, 1153438866, 4484193837,

    17459839956, 68076029025]

                                               n
                                        0.277 4
                               asymp.:, --------
                                           1/2
                                          n

a=, 5, b=, 2, seq.:[1, 3, 9, 31, 111, 403, 1488, 5544, 20842, 78913, 300401,

    1148685, 4408481, 16972110, 65517824, 253515562, 982982117, 3818358316,

    14856289227, 57885980821]

a=, 5, b=, 3, seq.:[1, 2, 7, 23, 83, 299, 1077, 4002, 14960, 56554, 214860,

    818314, 3132617, 12032438, 46373879, 179198281, 693808774, 2691578130,

    10459819747, 40713660323]

a=, 5, b=, 4, seq.:[1, 2, 5, 14, 56, 202, 715, 2529, 8986, 34652, 132129,

    500042, 1885587, 7100744, 27416740, 106030460, 409479098, 1579081106,

    6083256428, 23636541784]

a=, 6, b=, 1, seq.:[1, 3, 10, 35, 126, 462, 1715, 6427, 24265, 92158, 351715,

    1347710, 5181729, 19980689, 77238255, 299227931, 1161464243, 4515951027,

    17585431976, 68572524650]

                                               n
                                        0.280 4
                               asymp.:, --------
                                           1/2
                                          n

a=, 6, b=, 5, seq.:[1, 2, 5, 14, 42, 174, 645, 2333, 8398, 30275, 109550,

    428455, 1654283, 6331712, 24121619, 91686628, 348195879, 1354938071,

    5278540414, 20526654977]

                            ------------------------

                          This took, 37.038, seconds.