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 (a*i,b*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, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715,

    8414640, 50067108, 300830572, 1822766520, 11124755664, 68328754959,

    422030545335, 2619631042665, 16332922290300, 102240109897695]

                                                 n
                                      0.24 (27/4)
                             asymp.:, ------------
                                           3/2
                                          n

a=, 3, b=, 1, seq.:[1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888,

    225568798, 1882933364, 15875338990, 134993766600, 1156393243320,

    9969937491420, 86445222719724, 753310723010608, 6594154339031800,

    57956002331347120]

                                            /256\n
                                       0.15 |---|
                                            \27 /
                              asymp.:, -----------
                                           3/2
                                          n

a=, 3, b=, 2, seq.:[2, 23, 377, 7229, 151491, 3361598, 77635093, 1846620581,

    44930294909, 1113015378438, 27976770344941, 711771461238122,

    18293652115906958, 474274581883631615, 12388371266483017545,

    325714829431573496525, 8613086428709348334675, 228925936056388155632081,

    6112355595348903438948155, 163869604996054172670563730]

                                            /3125\n
                                     0.0876 |----|
                                            \108 /
                            asymp.:, --------------
                                           3/2
                                          n

a=, 4, b=, 1, seq.:[1, 5, 35, 285, 2530, 23751, 231880, 2330445, 23950355,

    250543370, 2658968130, 28558343775, 309831575760, 3390416787880,

    37377257159280, 414741863546285, 4628362722856425, 51912988256282175,

    584909606696793885, 6617078646960613370]

                                           /3125\n
                                      0.11 |----|
                                           \256 /
                             asymp.:, ------------
                                           3/2
                                          n

a=, 4, b=, 3, seq.:[5, 227, 15090, 1182187, 101527596, 9247179818, 877362665128,

    85783306955099, 8582893111512001, 874542924575207352, 90437361732467946334,

    9467275300762187682554, 1001309098267187214993056,

    106836493655355495755649544, 11485688815900189437990930096,

    1242964338344397490958154292155, 135294163560516394214244049693547,

    14802386593672915358103823856915731, 1626958525109139734774927715918218672,

    179560107576192929210029836580628234040]

                                           /823543\n
                                    0.0486 |------|
                                           \ 6912 /
                           asymp.:, ----------------
                                           3/2
                                          n

a=, 5, b=, 1, seq.:[1, 6, 51, 506, 5481, 62832, 749398, 9203634, 115607310,

    1478314266, 19180049928, 251857119696, 3340843549855, 44700485049720,

    602574657427116, 8175951659117794, 111572030260242090, 1530312970340384580,

    21085148778264281865, 291705220704719165526]

                                            /46656\n
                                     0.0873 |-----|
                                            \3125 /
                            asymp.:, ---------------
                                           3/2
                                          n

a=, 5, b=, 2, seq.:[3, 76, 2803, 121637, 5782513, 291437249, 15297882929,

    827402061954, 45790180469312, 2580588279994441, 147592910517101281,

    8544927937132306600, 499811636639428519226, 29491983283370728013309,

    1753398440591481772556798, 104933899400256659634374549,

    6316334518803437568442071134, 382157894402929888870502301422,

    23227867059873378295000835711897, 1417626021933575573690480062573957]

                                           /823543\n
                                    0.0538 |------|
                                           \12500 /
                           asymp.:, ----------------
                                           3/2
                                          n

a=, 5, b=, 3, seq.:[7, 525, 58040, 7574994, 1084532963, 164734116407,

    26070940600055, 4252443527211637, 709846349042619913, 120679177855928146859,

    20822762876863605793639, 3637213213067542990001936,

    641912742432770594132245835, 114287840570892852593437353124,

    20502971288127330644273350110698, 3702570208952583655019682802997983,

    672532254932908563141660483607892405,

    122788872081951688284011943771535283927,

    22521647274404155444930377651363981888609,

    4147941699622771075000003992530419494184133]

                                          /16777216\n
                                   0.0400 |--------|
                                          \ 84375  /
                          asymp.:, ------------------
                                           3/2
                                          n

a=, 5, b=, 4, seq.:[14, 2529, 678338, 215205020, 74954297074, 27706901431307,

    10673513460281995, 4238336602544646985, 1722532440479713492905,

    713032912920984966665914, 299578160956737689390169226,

    127423760336300467698086830007, 54761998197408482500643162670932,

    23742924018375726263229745697113234, 10372639160995862858028814581769132028,

    4561630228928173937942703373111379977087,

    2017804841330203670735598373154752656772860,

    897177656970083321768073667118946270564224913,

    400752711842105711698489590738780696157143645906,

    179749957957898273181768638889177302565608486108515]

                                          /387420489\n
                                   0.0321 |---------|
                                          \ 800000  /
                          asymp.:, -------------------
                                           3/2
                                          n

a=, 6, b=, 1, seq.:[1, 7, 70, 819, 10472, 141778, 1997688, 28989675, 430321633,

    6503352856, 99726673130, 1547847846090, 24269405074740, 383846168712104,

    6116574500860880, 98106248306858715, 1582638261961640247,

    25661404527790252375, 417980115131315136400, 6836064539925118356600]

                                           /823543\n
                                    0.0718 |------|
                                           \46656 /
                           asymp.:, ----------------
                                           3/2
                                          n

a=, 6, b=, 5, seq.:[42, 30275, 32675894, 41801669777, 58757395707430,

    87692166325858602, 136424052628767703708, 218803633512952794993022,

    359207350359845275021268732, 600670500086196968953098493461,

    1019548167065036208950503773416034, 1752004991899521391756122592044282946,

    3042034836797842895970861639014826559726,

    5328794951781120970145587503377874801450823,

    9405934110291932630862957007572132670419945104,

    16713101444911927790135600872984714010286956998516,

    29870724321187795475481153878471983310142040026880262,

    53663512180991572550190646088070609863531258865434988521,

    96853404214631891450758158266402354962407823477474965710806,

    175528710156446338907262214568649539295226404764268637894542861]

                                         /285311670611\n
                                 0.02324 |------------|
                                         \ 145800000  /
                        asymp.:, -----------------------
                                           3/2
                                          n

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

                         This took, 108.310, seconds.