Let , M(n), be the Gouyou-Beuchamps numbers. The number of, n, -step 3D ballot-lattice paths starting at, [0, 0, 0, 0], equals (and is provable, but we only proved it semi-rigorously) M(n) The first, 30, terms are 1, 2, 4, 10, 25, 70, 196, 588, 1764, 5544, 17424, 56628, 184041, 613470, 2044900, 6952660, 23639044, 81662152, 282105616, 987369656, 3455793796, 12228193432, 43268992144, 154532114800, 551900410000, 1986841476000, 7152629313600, 25928281261800, 93990019574025, 342787130211150 The number of, n - 1, -step 3D ballot-lattice paths starting at, [1, 0, 0, 0], equals (and is provable, but we only proved it semi-rigorously) M(n) The first, 29, terms are 1, 2, 4, 10, 25, 70, 196, 588, 1764, 5544, 17424, 56628, 184041, 613470, 2044900, 6952660, 23639044, 81662152, 282105616, 987369656, 3455793796, 12228193432, 43268992144, 154532114800, 551900410000, 1986841476000, 7152629313600, 25928281261800, 93990019574025, 342787130211150 The number of, n - 2, -step 3D ballot-lattice paths starting at, [1, 1, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / 8 n\ |- 3/10 - ----| M(n) + 4/5 M(n - 1) + |-2 + ---| M(n - 2) \ 10 / \ 5 / The first, 28, terms are 1, 2, 5, 12, 33, 90, 266, 784, 2436, 7560, 24354, 78408, 259545, 858858, 2903758, 9815520, 33752004, 116046216, 404598844, 1410528080, 4974824036, 17544799272, 62484724680, 222526245312, 799151793680, 2869882132000, 10381246712100, 37551303896400, 136695424005225 The number of, n - 3, -step 3D ballot-lattice paths starting at, [1, 1, 1, 0], equals (and is provable, but we only proved it semi-rigorously) M(n - 2) The first, 27, terms are 1, 2, 4, 10, 25, 70, 196, 588, 1764, 5544, 17424, 56628, 184041, 613470, 2044900, 6952660, 23639044, 81662152, 282105616, 987369656, 3455793796, 12228193432, 43268992144, 154532114800, 551900410000, 1986841476000, 7152629313600, 25928281261800 The number of, n - 2, -step 3D ballot-lattice paths starting at, [2, 0, 0, 0], equals (and is provable, but we only proved it semi-rigorously) /13 n \ / 8 n\ |-- + ----| M(n) - 4/5 M(n - 1) + |2 - ---| M(n - 2) \10 10 / \ 5 / The first, 28, terms are 1, 2, 5, 13, 37, 106, 322, 980, 3108, 9864, 32274, 105633, 353925, 1186042, 4048902, 13823524, 47910148, 166059400, 582770812, 2045265716, 7253369396, 25724192872, 92047390120, 329374164688, 1187689682320, 4282747181600, 15547034549700, 56438715677625, 206091706205925 The number of, n - 3, -step 3D ballot-lattice paths starting at, [2, 1, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / 8 n\ |- 3/10 - ----| M(n) + 4/5 M(n - 1) + |-3 + ---| M(n - 2) \ 10 / \ 5 / The first, 27, terms are 1, 3, 8, 23, 65, 196, 588, 1848, 5796, 18810, 60984, 202917, 674817, 2290288, 7770620, 26799344, 92407172, 322936692, 1128422464, 3987454380, 14089005476, 50256531248, 179257253168, 644619678880, 2317981722000, 8394405236100, 30398674582800, 110767142743425 The number of, n - 4, -step 3D ballot-lattice paths starting at, [2, 1, 1, 0], equals (and is provable, but we only proved it semi-rigorously) M(n - 2) - M(n - 4) The first, 26, terms are 1, 3, 8, 21, 60, 171, 518, 1568, 4956, 15660, 51084, 166617, 556842, 1860859, 6339190, 21594144, 74709492, 258466572, 905707504, 3173688180, 11240823776, 39813198348, 142303921368, 508631417856, 1832309361200, 6600728903600, 23941439785800 The number of, n - 4, -step 3D ballot-lattice paths starting at, [2, 2, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / 2 n\ /47 2 n\ |- 3/70 - ----| M(n) + |- 1/7 - ---| M(n - 1) + |-- + ---| M(n - 2) \ 70 / \ 35 / \35 5 / / 32 n\ /332 96 n\ + |- 32/7 + ----| M(n - 3) + |--- - ----| M(n - 4) \ 35 / \35 35 / The first, 26, terms are 1, 2, 6, 17, 52, 156, 492, 1540, 4998, 16164, 53724, 178233, 604032, 2044900, 7041892, 24233924, 84569628, 294999640, 1041049672, 3672798116, 13085360496, 46610858672, 167432186992, 601350686736, 2175591416220, 7870099846600, 28651422814200 The number of, n - 5, -step 3D ballot-lattice paths starting at, [2, 2, 1, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / 8 n\ |- 1/10 - ----| M(n - 2) + 4/5 M(n - 3) + |- 31/5 + ---| M(n - 4) \ 10 / \ 5 / The first, 25, terms are 1, 3, 8, 23, 65, 196, 588, 1848, 5796, 18810, 60984, 202917, 674817, 2290288, 7770620, 26799344, 92407172, 322936692, 1128422464, 3987454380, 14089005476, 50256531248, 179257253168, 644619678880, 2317981722000, 8394405236100 The number of, n - 6, -step 3D ballot-lattice paths starting at, [2, 2, 2, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / 8 n\ |9/10 + ----| M(n - 4) - 4/5 M(n - 5) + |42/5 - ---| M(n - 6) \ 10 / \ 5 / The first, 24, terms are 1, 2, 5, 13, 37, 106, 322, 980, 3108, 9864, 32274, 105633, 353925, 1186042, 4048902, 13823524, 47910148, 166059400, 582770812, 2045265716, 7253369396, 25724192872, 92047390120, 329374164688, 1187689682320 The number of, n - 3, -step 3D ballot-lattice paths starting at, [3, 0, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / 16 n\ (8/5 + n/5) M(n) - 8/5 M(n - 1) + |5 - ----| M(n - 2) \ 5 / The first, 27, terms are 1, 2, 5, 14, 41, 126, 392, 1260, 4068, 13464, 44649, 151008, 511225, 1758614, 6052904, 21110804, 73652228, 259834120, 916843252, 3265915016, 11635187396, 41790858872, 150116911520, 543070003440, 1964765459600, 7152629313600, 26040041094825, 95324563462500 The number of, n - 4, -step 3D ballot-lattice paths starting at, [3, 1, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / 3 n\ /33 2 n\ / 187 6 n\ |- 9/35 - ---| M(n) + |-- + ---| M(n - 1) + |- --- + ---| M(n - 2) \ 35 / \35 35 / \ 35 5 / / 32 n\ / 297 96 n\ + |32/7 - ----| M(n - 3) + |- --- + ----| M(n - 4) \ 35 / \ 35 35 / The first, 26, terms are 1, 3, 9, 27, 84, 261, 838, 2688, 8856, 29160, 98109, 329967, 1129414, 3864861, 13418262, 46579104, 163657572, 574956252, 2040697204, 7242519180, 25930346976, 92833196148, 334883570520, 1207999617408, 4386504458680, 15927845832600, 58174280143425 The number of, n - 5, -step 3D ballot-lattice paths starting at, [3, 1, 1, 0], equals (and is provable, but we only proved it semi-rigorously) /11 n \ / 8 n\ |-- + ----| M(n - 2) - 4/5 M(n - 3) + |21/5 - ---| M(n - 4) \10 10 / \ 5 / The first, 25, terms are 1, 3, 9, 27, 81, 252, 784, 2520, 8100, 26730, 88209, 297297, 1002001, 3435432, 11778624, 40957488, 142420356, 501108660, 1763160100, 6265999740, 22268399076, 79819196688, 286105172544, 1033157567520, 3730846771600, 13560193073700 The number of, n - 5, -step 3D ballot-lattice paths starting at, [3, 2, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / 2 n\ /101 \ |- 3/70 - ----| M(n) + |- 1/7 - ---| M(n - 1) + |--- + n/2| M(n - 2) \ 70 / \ 35 / \70 / / 188 32 n\ /549 152 n\ + |- --- + ----| M(n - 3) + |--- - -----| M(n - 4) \ 35 35 / \35 35 / The first, 25, terms are 1, 3, 9, 29, 91, 296, 952, 3150, 10368, 34914, 117249, 401115, 1370083, 4751604, 16463304, 57770284, 202592468, 718112980, 2544375652, 9097906116, 32521853196, 117175655744, 422093433568, 1530971737340, 5552118124600, 20257017578100 The number of, n - 6, -step 3D ballot-lattice paths starting at, [3, 2, 1, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / 8 n\ |- 1/10 - ----| M(n - 2) + 4/5 M(n - 3) + |- 36/5 + ---| M(n - 4) \ 10 / \ 5 / The first, 24, terms are 1, 4, 13, 40, 126, 392, 1260, 4032, 13266, 43560, 146289, 490776, 1676818, 5725720, 19846684, 68768128, 241274540, 846316848, 3000084724, 10633211680, 38028337816, 135988261024, 490087564080, 1766081312000, 6407563760100 The number of, n - 7, -step 3D ballot-lattice paths starting at, [3, 2, 2, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / 8 n\ |9/10 + ----| M(n - 4) - 4/5 M(n - 5) + |37/5 - ---| M(n - 6) \ 10 / \ 5 / The first, 23, terms are 1, 3, 9, 27, 81, 252, 784, 2520, 8100, 26730, 88209, 297297, 1002001, 3435432, 11778624, 40957488, 142420356, 501108660, 1763160100, 6265999740, 22268399076, 79819196688, 286105172544, 1033157567520 The number of, n - 6, -step 3D ballot-lattice paths starting at, [3, 3, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / n \ /38 12 n\ |- 1/140 - ---| M(n) + |- 1/70 - ---| M(n - 1) + |-- + ----| M(n - 3) \ 420/ \ 210/ \35 35 / / 73 19 n\ /2428 64 n\ + |- -- + ----| M(n - 4) + |---- - ----| M(n - 5) \ 10 70 / \105 15 / / 2902 568 n\ + |- ---- + -----| M(n - 6) \ 105 105 / The first, 24, terms are 1, 2, 6, 18, 59, 190, 635, 2100, 7120, 24000, 82467, 282414, 982267, 3409406, 11986832, 42087760, 149382740, 529737000, 1895932480, 6781385000, 24449292492, 88109596344, 319732704482, 1159873901656, 4233270933080 The number of, n - 7, -step 3D ballot-lattice paths starting at, [3, 3, 1, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / 2 n\ /31 \ |- 1/70 - ----| M(n - 2) + |- 1/35 - ---| M(n - 3) + |-- + n/2| M(n - 4) \ 70 / \ 35 / \70 / / 32 n\ /853 152 n\ + |- 36/5 + ----| M(n - 5) + |--- - -----| M(n - 6) \ 35 / \35 35 / The first, 23, terms are 1, 3, 9, 29, 91, 296, 952, 3150, 10368, 34914, 117249, 401115, 1370083, 4751604, 16463304, 57770284, 202592468, 718112980, 2544375652, 9097906116, 32521853196, 117175655744, 422093433568, 1530971737340 The number of, n - 8, -step 3D ballot-lattice paths starting at, [3, 3, 2, 0], equals (and is provable, but we only proved it semi-rigorously) / 3 n\ / 2 n\ / 6 n\ |3/35 - ---| M(n - 4) + |5/7 + ---| M(n - 5) + |- 71/7 + ---| M(n - 6) \ 35 / \ 35 / \ 5 / /288 32 n\ / 681 96 n\ + |--- - ----| M(n - 7) + |- --- + ----| M(n - 8) \35 35 / \ 35 35 / The first, 22, terms are 1, 3, 9, 27, 84, 261, 838, 2688, 8856, 29160, 98109, 329967, 1129414, 3864861, 13418262, 46579104, 163657572, 574956252, 2040697204, 7242519180, 25930346976, 92833196148, 334883570520 The number of, n - 9, -step 3D ballot-lattice paths starting at, [3, 3, 3, 0], equals (and is provable, but we only proved it semi-rigorously) / 16 n\ (2/5 + n/5) M(n - 6) - 8/5 M(n - 7) + |121/5 - ----| M(n - 8) \ 5 / The first, 21, terms are 1, 2, 5, 14, 41, 126, 392, 1260, 4068, 13464, 44649, 151008, 511225, 1758614, 6052904, 21110804, 73652228, 259834120, 916843252, 3265915016, 11635187396, 41790858872 The number of, n - 4, -step 3D ballot-lattice paths starting at, [4, 0, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / 2 n\ / 89 2 n\ /362 22 n\ |13/7 + ---| M(n) + |- -- - ---| M(n - 1) + |--- - ----| M(n - 2) \ 7 / \ 35 35 / \35 5 / / 32 n\ /297 96 n\ + |- 32/7 + ----| M(n - 3) + |--- - ----| M(n - 4) \ 35 / \35 35 / The first, 26, terms are 1, 2, 5, 14, 42, 131, 422, 1380, 4608, 15489, 52899, 181258, 629200, 2188043, 7692542, 27073124, 96176548, 341887000, 1225217812, 4392668216, 15860511896, 57283715372, 208186432920, 756765842192, 2766124854920, 10112195262225, 37150283319075 The number of, n - 5, -step 3D ballot-lattice paths starting at, [4, 1, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ /38 4 n\ / 276 3 n\ |- 3/14 - ----| M(n) + |-- + ---| M(n - 1) + |- --- + ---| M(n - 2) \ 14 / \35 35 / \ 35 5 / /376 64 n\ / 993 304 n\ + |--- - ----| M(n - 3) + |- --- + -----| M(n - 4) \35 35 / \ 35 35 / The first, 25, terms are 1, 3, 9, 28, 89, 290, 952, 3186, 10692, 36465, 124509, 431002, 1492777, 5231226, 18337176, 64929800, 229943428, 821475564, 2934983428, 10566441120, 38042943876, 137888718088, 499801011296, 1822375153820, 6644880936400, 24357069491625 The number of, n - 6, -step 3D ballot-lattice paths starting at, [4, 1, 1, 0], equals (and is provable, but we only proved it semi-rigorously) / 33 n\ (6/5 + n/5) M(n - 2) - 8/5 M(n - 3) + |21/2 - ----| M(n - 4) + 4/5 M(n - 5) \ 10 / / 8 n\ + |- 42/5 + ---| M(n - 6) \ 5 / The first, 24, terms are 1, 3, 9, 28, 89, 286, 938, 3088, 10356, 34785, 118734, 405592, 1404689, 4866862, 17061902, 59828704, 211923972, 750783852, 2683144204, 9589921680, 34537489476, 124392718648, 451022613320, 1635391294912, 5964939631280 The number of, n - 6, -step 3D ballot-lattice paths starting at, [4, 2, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / 11 n\ /54 3 n\ |- 1/28 - ----| M(n) + |- 9/70 - ----| M(n - 1) + |-- + ---| M(n - 2) \ 84 / \ 210 / \35 5 / / 254 4 n\ /2113 87 n\ + |- --- + ---| M(n - 3) + |---- - ----| M(n - 4) \ 35 7 / \ 70 14 / / 2428 64 n\ /2902 568 n\ + |- ---- + ----| M(n - 5) + |---- - -----| M(n - 6) \ 105 15 / \105 105 / The first, 24, terms are 1, 3, 10, 33, 111, 370, 1255, 4236, 14528, 49689, 172359, 596893, 2092519, 7328178, 25936768, 91736580, 327455700, 1168321804, 4201888912, 15107256516, 54698025436, 197995576200, 721151468778, 2626162910944, 9616182884920 The number of, n - 7, -step 3D ballot-lattice paths starting at, [4, 2, 1, 0], equals (and is provable, but we only proved it semi-rigorously) / 3 n\ /29 2 n\ / 121 11 n\ |- 3/35 - ---| M(n - 2) + |-- + ---| M(n - 3) + |- --- + ----| M(n - 4) \ 35 / \35 35 / \ 14 10 / / 32 n\ / 783 152 n\ + |36/5 - ----| M(n - 5) + |- --- + -----| M(n - 6) \ 35 / \ 35 35 / The first, 23, terms are 1, 4, 14, 47, 155, 516, 1708, 5748, 19296, 65835, 224334, 775489, 2678819, 9369360, 32755580, 115747424, 408896852, 1457926392, 5197253464, 18676977580, 67109003276, 242836180400, 878625452720, 3198814776360 The number of, n - 8, -step 3D ballot-lattice paths starting at, [4, 2, 2, 0], equals (and is provable, but we only proved it semi-rigorously) /57 13 n\ / 53 2 n\ /579 14 n\ |-- + ----| M(n - 4) + |- -- - ---| M(n - 5) + |--- - ----| M(n - 6) \70 70 / \ 35 35 / \35 5 / / 288 32 n\ /716 96 n\ + |- --- + ----| M(n - 7) + |--- - ----| M(n - 8) \ 35 35 / \35 35 / The first, 22, terms are 1, 3, 10, 33, 108, 352, 1164, 3844, 12918, 43389, 148104, 505417, 1749176, 6052904, 21200036, 74247108, 262741596, 929737276, 3319595032, 11852191716, 42648025936, 153458778048, 555970075632 The number of, n - 7, -step 3D ballot-lattice paths starting at, [4, 3, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / n \ / n \ |- 1/140 - ---| M(n) + |- 1/70 - ---| M(n - 1) + |1/70 + ----| M(n - 2) \ 420/ \ 210/ \ 70 / /39 2 n\ / 271 8 n\ /3184 544 n\ + |-- + ---| M(n - 3) + |- --- - ---| M(n - 4) + |---- - -----| M(n - 5) \35 5 / \ 35 35 / \105 105 / / 5461 1024 n\ + |- ---- + ------| M(n - 6) \ 105 105 / The first, 23, terms are 1, 3, 9, 30, 99, 339, 1148, 3970, 13632, 47553, 165165, 581152, 2039323, 7235228, 25624456, 91612456, 327144532, 1177819500, 4237009348, 15351386376, 55587743148, 202557048738, 737780468088, 2702299195740 The number of, n - 8, -step 3D ballot-lattice paths starting at, [4, 3, 1, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / 2 n\ /12 3 n\ |- 1/70 - ----| M(n - 2) + |- 1/35 - ---| M(n - 3) + |-- + ---| M(n - 4) \ 70 / \ 35 / \35 5 / / 32 n\ /1217 208 n\ + |-8 + ----| M(n - 5) + |---- - -----| M(n - 6) - M(n - 8) \ 35 / \ 35 35 / The first, 22, terms are 1, 4, 14, 47, 160, 535, 1820, 6140, 21060, 71925, 249282, 861883, 3018158, 10553543, 37310130, 131779440, 469885780, 1674419760, 6016159240, 21606535900, 78159948272, 282649378748, 1028655979828 The number of, n - 9, -step 3D ballot-lattice paths starting at, [4, 3, 2, 0], equals (and is provable, but we only proved it semi-rigorously) / 3 n\ / 2 n\ / 759 11 n\ |3/35 - ---| M(n - 4) + |5/7 + ---| M(n - 5) + |- --- + ----| M(n - 6) \ 35 / \ 35 / \ 70 10 / /316 32 n\ / 1087 152 n\ + |--- - ----| M(n - 7) + |- ---- + -----| M(n - 8) \35 35 / \ 35 35 / The first, 21, terms are 1, 4, 14, 47, 155, 516, 1708, 5748, 19296, 65835, 224334, 775489, 2678819, 9369360, 32755580, 115747424, 408896852, 1457926392, 5197253464, 18676977580, 67109003276, 242836180400 The number of, n - 10, -step 3D ballot-lattice paths starting at, [4, 3, 3, 0], equals (and is provable, but we only proved it semi-rigorously) /237 33 n\ (2/5 + n/5) M(n - 6) - 8/5 M(n - 7) + |--- - ----| M(n - 8) + 4/5 M(n - 9) \10 10 / / 8 n\ + |- 74/5 + ---| M(n - 10) \ 5 / The first, 20, terms are 1, 3, 9, 28, 89, 286, 938, 3088, 10356, 34785, 118734, 405592, 1404689, 4866862, 17061902, 59828704, 211923972, 750783852, 2683144204, 9589921680, 34537489476 The number of, n - 8, -step 3D ballot-lattice paths starting at, [4, 4, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / n \ / 17 n \ |- 1/770 - ----| M(n) + |- 2/1155 - ----| M(n - 1) + |---- + ---| M(n - 2) \ 2310/ \ 2310/ \2310 154/ / 31 38 n\ /222 13 n\ + |---- - ----| M(n - 3) + |--- + ----| M(n - 4) \1155 1155/ \385 35 / / 10414 4 n\ /17246 884 n\ + |- ----- - ---| M(n - 5) + |----- - -----| M(n - 6) \ 1155 33 / \ 385 165 / / 96704 4672 n\ /21001 8768 n\ + |- ----- + ------| M(n - 7) + |----- - ------| M(n - 8) \ 1155 385 / \ 385 1155 / The first, 22, terms are 1, 2, 6, 18, 60, 199, 688, 2360, 8276, 28845, 102060, 359590, 1281852, 4556409, 16353194, 58576596, 211552692, 762971560, 2771373592, 10056480968, 36720886788, 133985978856, 491596750572 The number of, n - 9, -step 3D ballot-lattice paths starting at, [4, 4, 1, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / n \ / n \ |- 1/420 - ---| M(n - 2) + |- 1/210 - ---| M(n - 3) + |- 1/70 + ----| M(n - 4) \ 420/ \ 210/ \ 70 / /11 2 n\ / 8 n\ /1424 544 n\ + |-- + ---| M(n - 5) + |- 51/7 - ---| M(n - 6) + |---- - -----| M(n - 7) \35 5 / \ 35 / \ 35 105 / / 2503 1024 n\ + |- ---- + ------| M(n - 8) \ 35 105 / The first, 21, terms are 1, 3, 9, 30, 99, 339, 1148, 3970, 13632, 47553, 165165, 581152, 2039323, 7235228, 25624456, 91612456, 327144532, 1177819500, 4237009348, 15351386376, 55587743148, 202557048738 The number of, n - 10, -step 3D ballot-lattice paths starting at, [4, 4, 2, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ /17 11 n\ / 3 n\ |1/84 - ----| M(n - 4) + |--- - ----| M(n - 5) + |- 6/7 + ---| M(n - 6) \ 84 / \210 210 / \ 5 / / 334 4 n\ /3853 87 n\ + |- --- + ---| M(n - 7) + |---- - ----| M(n - 8) \ 35 7 / \ 70 14 / / 844 64 n\ /5174 568 n\ + |- --- + ----| M(n - 9) + |---- - -----| M(n - 10) \ 21 15 / \105 105 / The first, 20, terms are 1, 3, 10, 33, 111, 370, 1255, 4236, 14528, 49689, 172359, 596893, 2092519, 7328178, 25936768, 91736580, 327455700, 1168321804, 4201888912, 15107256516, 54698025436 The number of, n - 11, -step 3D ballot-lattice paths starting at, [4, 4, 3, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / 4 n\ / 402 3 n\ |3/14 - ----| M(n - 6) + |2/5 + ---| M(n - 7) + |- --- + ---| M(n - 8) \ 14 / \ 35 / \ 35 5 / / 64 n\ / 2817 304 n\ + |152/7 - ----| M(n - 9) + |- ---- + -----| M(n - 10) \ 35 / \ 35 35 / The first, 19, terms are 1, 3, 9, 28, 89, 290, 952, 3186, 10692, 36465, 124509, 431002, 1492777, 5231226, 18337176, 64929800, 229943428, 821475564, 2934983428, 10566441120 The number of, n - 12, -step 3D ballot-lattice paths starting at, [4, 4, 4, 0], equals (and is provable, but we only proved it semi-rigorously) / 2 n\ / 73 2 n\ /1594 22 n\ |- 3/7 + ---| M(n - 8) + |- -- - ---| M(n - 9) + |---- - ----| M(n - 10) \ 7 / \ 35 35 / \ 35 5 / / 416 32 n\ / 96 n\ + |- --- + ----| M(n - 11) + |213/7 - ----| M(n - 12) \ 35 35 / \ 35 / The first, 18, terms are 1, 2, 5, 14, 42, 131, 422, 1380, 4608, 15489, 52899, 181258, 629200, 2188043, 7692542, 27073124, 96176548, 341887000, 1225217812 The number of, n - 5, -step 3D ballot-lattice paths starting at, [5, 0, 0, 0], equals (and is provable, but we only proved it semi-rigorously) /29 5 n\ / 127 6 n\ /638 \ |-- + ---| M(n) + |- --- - ---| M(n - 1) + |--- - 5 n| M(n - 2) \14 14 / \ 35 35 / \35 / / 536 96 n\ / 80 n\ + |- --- + ----| M(n - 3) + |258/7 - ----| M(n - 4) \ 35 35 / \ 7 / The first, 25, terms are 1, 2, 5, 14, 42, 132, 428, 1422, 4797, 16434, 56749, 198198, 695266, 2461316, 8735948, 31246748, 111943572, 403742248, 1457684788, 5294070776, 19240771496, 70297714832, 256964830896, 943749701100, 3467314325825, 12793213827450 The number of, n - 6, -step 3D ballot-lattice paths starting at, [5, 1, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / 5 n\ /17 \ / 372 \ |- 5/28 - ---| M(n) + |-- + n/6| M(n - 1) + |- --- - n/5| M(n - 2) \ 84 / \14 / \ 35 / / 12 n\ / 2417 91 n\ + |98/5 - ----| M(n - 3) + |- ---- + ----| M(n - 4) \ 5 / \ 35 5 / /2344 64 n\ / 404 80 n\ + |---- - ----| M(n - 5) + |- --- + ----| M(n - 6) \105 15 / \ 21 21 / The first, 24, terms are 1, 3, 9, 28, 90, 296, 993, 3368, 11581, 40035, 139909, 490292, 1734018, 6142136, 21931130, 78378144, 282095892, 1015877772, 3681408004, 13345765680, 48653203176, 177412716448, 650201071722, 2383326730544, 8775946975425 The number of, n - 7, -step 3D ballot-lattice paths starting at, [5, 1, 1, 0], equals (and is provable, but we only proved it semi-rigorously) / 2 n\ / 2 n\ /642 23 n\ |9/7 + ---| M(n - 2) + |- 17/7 - ---| M(n - 3) + |--- - ----| M(n - 4) \ 7 / \ 35 / \35 5 / / 32 n\ / 134 16 n\ + |- 24/5 + ----| M(n - 5) + |- --- + ----| M(n - 6) \ 35 / \ 35 35 / The first, 23, terms are 1, 3, 9, 28, 90, 296, 988, 3348, 11421, 39435, 136609, 478192, 1676818, 5933928, 21020220, 75065744, 268234772, 965383692, 3475824964, 12594596880, 45648527976, 166395574048, 606648930672, 2223054851480 The number of, n - 7, -step 3D ballot-lattice paths starting at, [5, 2, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / n \ /113 47 n\ |- 1/35 - ---| M(n) + |- 4/35 - ----| M(n - 1) + |--- + ----| M(n - 2) \ 105/ \ 21 / \70 70 / / 4 n\ / 248 n\ + |- 46/5 + ---| M(n - 3) + |326/7 - -----| M(n - 4) \ 35 / \ 35 / / 6368 1088 n\ /10712 2048 n\ + |- ---- + ------| M(n - 5) + |----- - ------| M(n - 6) \ 105 105 / \ 105 105 / The first, 23, terms are 1, 3, 10, 34, 116, 400, 1380, 4810, 16761, 58971, 207394, 735878, 2610036, 9332180, 33356544, 120095820, 432280420, 1566143020, 5672993704, 20669661480, 75298829776, 275758239640, 1009756990136, 3715068912820 The number of, n - 8, -step 3D ballot-lattice paths starting at, [5, 2, 1, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / 4 n\ / 346 2 n\ |- 1/14 - ----| M(n - 2) + |6/7 + ---| M(n - 3) + |- --- + ---| M(n - 4) \ 14 / \ 35 / \ 35 5 / / 64 n\ / 2224 416 n\ + |16 - ----| M(n - 5) + |- ---- + -----| M(n - 6) \ 35 / \ 35 35 / The first, 22, terms are 1, 4, 14, 48, 164, 560, 1926, 6624, 23001, 79860, 279994, 981552, 3472612, 12284272, 43818996, 156291200, 561641444, 2018140176, 7300526104, 26407756480, 96097859216, 349684099776, 1279305150380 The number of, n - 9, -step 3D ballot-lattice paths starting at, [5, 2, 2, 0], equals (and is provable, but we only proved it semi-rigorously) /51 19 n\ / 78 4 n\ /934 \ |-- + ----| M(n - 4) + |- -- - ---| M(n - 5) + |--- - 4 n| M(n - 6) \70 70 / \ 35 35 / \35 / / 576 64 n\ /1432 192 n\ + |- --- + ----| M(n - 7) + |---- - -----| M(n - 8) \ 35 35 / \ 35 35 / The first, 21, terms are 1, 3, 10, 34, 116, 396, 1352, 4650, 15993, 55539, 192874, 676390, 2372084, 8395244, 29712904, 106036684, 378420068, 1360559980, 4891778152, 17705048616, 64081375696, 233314698544 The number of, n - 8, -step 3D ballot-lattice paths starting at, [5, 3, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / 3 n \ / 29 n \ / 17 n\ |- 9/1540 - ----| M(n) + |- ---- - ---| M(n - 1) + |7/330 + ----| M(n - 2) \ 1540/ \ 2310 231/ \ 770 / /1289 566 n\ / 667 6 n\ + |---- + -----| M(n - 3) + |- --- - ---| M(n - 4) \1155 1155 / \ 77 5 / /18226 460 n\ / 4342 24316 n\ + |----- - -----| M(n - 5) + |- ---- + -------| M(n - 6) \ 385 77 / \ 33 1155 / /96704 4672 n\ / 20616 8768 n\ + |----- - ------| M(n - 7) + |- ----- + ------| M(n - 8) \1155 385 / \ 385 1155 / The first, 22, terms are 1, 3, 10, 34, 119, 414, 1462, 5132, 18217, 64395, 229810, 817850, 2935218, 10514504, 37949132, 136788496, 496381028, 1799618028, 6563853544, 23924726280, 87676213678, 321145110484, 1182046465340 The number of, n - 9, -step 3D ballot-lattice paths starting at, [5, 3, 1, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / 11 n\ / 24 n\ |- 1/84 - ----| M(n - 2) + |- 1/42 - ----| M(n - 3) + |9/35 + ----| M(n - 4) \ 84 / \ 210 / \ 35 / / 319 18 n\ /1871 256 n\ + |- --- + ----| M(n - 5) + |---- - -----| M(n - 6) \ 35 35 / \ 35 35 / / 1424 544 n\ /2398 1024 n\ + |- ---- + -----| M(n - 7) + |---- - ------| M(n - 8) \ 35 105 / \ 35 105 / The first, 21, terms are 1, 4, 15, 54, 190, 669, 2332, 8192, 28629, 100980, 355135, 1260402, 4465318, 15953938, 56936100, 204755616, 735785908, 2662300368, 9627897436, 35033678200, 127430779128, 466087094946 The number of, n - 10, -step 3D ballot-lattice paths starting at, [5, 3, 2, 0], equals (and is provable, but we only proved it semi-rigorously) /31 31 n\ /19 23 n\ / 374 2 n\ |--- - ----| M(n - 4) + |-- + ----| M(n - 5) + |- --- + ---| M(n - 6) \420 420 / \30 210 / \ 35 5 / /678 52 n\ / 429 n\ + |--- - ----| M(n - 7) + |- 481/5 + -----| M(n - 8) \35 35 / \ 35 / /4136 64 n\ / 724 80 n\ + |---- - ----| M(n - 9) + |- --- + ----| M(n - 10) \105 15 / \ 21 21 / The first, 20, terms are 1, 4, 15, 54, 190, 660, 2295, 7940, 27685, 96300, 338107, 1185558, 4195334, 14834820, 52902070, 188563440, 677272180, 2431830960, 8791859740, 31778613400, 115572327672 The number of, n - 11, -step 3D ballot-lattice paths starting at, [5, 3, 3, 0], equals (and is provable, but we only proved it semi-rigorously) /13 19 n\ / 4 n\ /1214 \ |-- + ----| M(n - 6) + |-2 - ---| M(n - 7) + |---- - 4 n| M(n - 8) \70 70 / \ 35 / \ 35 / / 704 64 n\ /1816 192 n\ + |- --- + ----| M(n - 9) + |---- - -----| M(n - 10) \ 35 35 / \ 35 35 / The first, 19, terms are 1, 3, 10, 34, 116, 396, 1352, 4650, 15993, 55539, 192874, 676390, 2372084, 8395244, 29712904, 106036684, 378420068, 1360559980, 4891778152, 17705048616 The number of, n - 9, -step 3D ballot-lattice paths starting at, [5, 4, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / n \ / 41 n\ |- 1/770 - ----| M(n) + |- 2/1155 - ----| M(n - 1) + |3/308 + ----| M(n - 2) \ 2310/ \ 2310/ \ 4620/ / 73 13 n\ /13 5 n\ / 10777 86 n\ + |---- - ----| M(n - 3) + |-- + ---| M(n - 4) + |- ----- - ----| M(n - 5) \2310 462 / \22 14 / \ 1155 165 / /20051 5924 n\ / 20528 4000 n\ + |----- - ------| M(n - 6) + |- ----- + ------| M(n - 7) \ 385 1155 / \ 165 231 / /48534 20032 n\ + |----- - -------| M(n - 8) \ 385 1155 / The first, 21, terms are 1, 3, 9, 30, 100, 349, 1212, 4306, 15213, 54507, 194425, 700700, 2517086, 9117966, 32952140, 119940236, 435827028, 1593554092, 5819471620, 21369500412, 78398235708, 289039701834 The number of, n - 10, -step 3D ballot-lattice paths starting at, [5, 4, 1, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / n \ / n \ |- 1/420 - ---| M(n - 2) + |- 1/210 - ---| M(n - 3) + |- 1/35 + ----| M(n - 4) \ 420/ \ 210/ \ 35 / / 16 n\ / 29 n\ + |8/35 + ----| M(n - 5) + |- 45/7 - ----| M(n - 6) \ 35 / \ 35 / /1768 128 n\ / 4136 1648 n\ + |---- - -----| M(n - 7) + |- ---- + ------| M(n - 8) \ 35 21 / \ 35 105 / The first, 20, terms are 1, 4, 14, 48, 169, 588, 2080, 7296, 25905, 91476, 326326, 1160016, 4160442, 14886872, 53688856, 193320192, 700981060, 2538950544, 9253564984, 33699101632, 123409730810 The number of, n - 11, -step 3D ballot-lattice paths starting at, [5, 4, 2, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ /17 11 n\ / 39 24 n\ |1/84 - ----| M(n - 4) + |--- - ----| M(n - 5) + |- -- + ----| M(n - 6) \ 84 / \210 210 / \ 35 35 / / 18 n\ /2383 256 n\ + |- 71/7 + ----| M(n - 7) + |---- - -----| M(n - 8) \ 35 / \ 35 35 / / 1072 544 n\ /9242 1024 n\ + |- ---- + -----| M(n - 9) + |---- - ------| M(n - 10) \ 21 105 / \105 105 / The first, 19, terms are 1, 4, 15, 54, 190, 669, 2332, 8192, 28629, 100980, 355135, 1260402, 4465318, 15953938, 56936100, 204755616, 735785908, 2662300368, 9627897436, 35033678200 The number of, n - 12, -step 3D ballot-lattice paths starting at, [5, 4, 3, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ /4 n \ /2 n 402\ |- ---- + 3/14| M(n - 6) + |--- + 2/5| M(n - 7) + |--- - ---| M(n - 8) \ 14 / \35 / \ 5 35 / / 64 n 816\ /416 n 3888\ + |- ---- + ---| M(n - 9) + |----- - ----| M(n - 10) \ 35 35 / \ 35 35 / The first, 18, terms are 1, 4, 14, 48, 164, 560, 1926, 6624, 23001, 79860, 279994, 981552, 3472612, 12284272, 43818996, 156291200, 561641444, 2018140176, 7300526104 The number of, n - 13, -step 3D ballot-lattice paths starting at, [5, 4, 4, 0], equals (and is provable, but we only proved it semi-rigorously) /2 n \ / 2 n 73\ / 23 n 1608\ |--- - 3/7| M(n - 8) + |- --- - --| M(n - 9) + |- ---- + ----| M(n - 10) \ 7 / \ 35 35/ \ 5 35 / /32 n \ /16 n \ + |---- - 72/7| M(n - 11) + |---- - 46/7| M(n - 12) \ 35 / \ 35 / The first, 17, terms are 1, 3, 9, 28, 90, 296, 988, 3348, 11421, 39435, 136609, 478192, 1676818, 5933928, 21020220, 75065744, 268234772, 965383692 The number of, n - 10, -step 3D ballot-lattice paths starting at, [5, 5, 0, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / n \ / 71 3 n \ |- 1/4004 - -----| M(n) + |- 1/4290 - -----| M(n - 1) + |----- + ----| M(n - 2) \ 12012/ \ 30030/ \30030 1430/ / 20 53 n \ /2129 383 n\ + |---- - -----| M(n - 3) + |----- - -----| M(n - 4) \3003 15015/ \60060 8580 / / 391 411 n\ / 5638 2920 n\ + |- ---- + -----| M(n - 5) + |- ---- - ------| M(n - 6) \ 2002 770 / \ 715 3003 / /208574 29052 n\ / 203243 347491 n\ + |------ - -------| M(n - 7) + |- ------ + --------| M(n - 8) \ 3003 5005 / \ 1001 15015 / /3521032 81472 n\ / 1535572 169424 n\ + |------- - -------| M(n - 9) + |- ------- + --------| M(n - 10) \ 15015 3003 / \ 15015 15015 / The first, 20, terms are 1, 2, 6, 18, 60, 200, 699, 2436, 8701, 30954, 111769, 402094, 1461614, 5297292, 19353126, 70544240, 258843156, 948116520, 3492528586, 12848325008, 47501042796 The number of, n - 11, -step 3D ballot-lattice paths starting at, [5, 5, 1, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / n \ |- 1/2310 - ----| M(n - 2) + |- 1/1155 - ----| M(n - 3) \ 2310/ \ 2310/ / 37 41 n\ /29 13 n\ + |- ---- + ----| M(n - 4) + |--- - ----| M(n - 5) \ 4620 4620/ \330 462 / / 19 5 n\ / 3191 86 n\ + |- --- + ---| M(n - 6) + |- ---- - ----| M(n - 7) \ 154 14 / \ 385 165 / /72001 5924 n\ / 61232 4000 n\ + |----- - ------| M(n - 8) + |- ----- + ------| M(n - 9) \1155 1155 / \ 385 231 / /185666 20032 n\ + |------ - -------| M(n - 10) \ 1155 1155 / The first, 19, terms are 1, 3, 9, 30, 100, 349, 1212, 4306, 15213, 54507, 194425, 700700, 2517086, 9117966, 32952140, 119940236, 435827028, 1593554092, 5819471620, 21369500412 The number of, n - 12, -step 3D ballot-lattice paths starting at, [5, 5, 2, 0], equals (and is provable, but we only proved it semi-rigorously) / 3 n \ / n \ /17 n 31 \ |- ---- + 3/1540| M(n - 4) + |- --- + 1/210| M(n - 5) + |---- - ---| M(n - 6) \ 1540 / \ 231 / \770 462/ /566 n 65\ / 6 n 1487\ + |----- - --| M(n - 7) + |- --- - ----| M(n - 8) \1155 77/ \ 5 385 / / 460 n 3918\ /24316 n 83078\ + |- ----- + ----| M(n - 9) + |------- - -----| M(n - 10) \ 77 55 / \ 1155 385 / / 4672 n 1984\ /8768 n 19384\ + |- ------ + ----| M(n - 11) + |------ - -----| M(n - 12) \ 385 15 / \ 1155 231 / The first, 18, terms are 1, 3, 10, 34, 119, 414, 1462, 5132, 18217, 64395, 229810, 817850, 2935218, 10514504, 37949132, 136788496, 496381028, 1799618028, 6563853544 The number of, n - 13, -step 3D ballot-lattice paths starting at, [5, 5, 3, 0], equals (and is provable, but we only proved it semi-rigorously) / n \ / n \ / 169 47 n\ |- --- + 1/35| M(n - 6) + |6/35 - ----| M(n - 7) + |- --- + ----| M(n - 8) \ 105 / \ 21 / \ 70 70 / / 346 4 n\ /3118 248 n\ + |- --- + ---| M(n - 9) + |---- - -----| M(n - 10) \ 35 35 / \ 35 35 / / 12896 1088 n\ /4600 2048 n\ + |- ----- + ------| M(n - 11) + |---- - ------| M(n - 12) \ 105 105 / \ 21 105 / The first, 17, terms are 1, 3, 10, 34, 116, 400, 1380, 4810, 16761, 58971, 207394, 735878, 2610036, 9332180, 33356544, 120095820, 432280420, 1566143020 The number of, n - 14, -step 3D ballot-lattice paths starting at, [5, 5, 4, 0], equals (and is provable, but we only proved it semi-rigorously) / 5 n 25\ / 316\ |- --- + --| M(n - 8) + (n/6 - 5/42) M(n - 9) + |- n/5 - ---| M(n - 10) \ 84 84/ \ 35 / / 12 n\ /91 n 7513\ + |194/5 - ----| M(n - 11) + |---- - ----| M(n - 12) \ 5 / \ 5 35 / / 64 n 1976\ / 80 n\ + |- ---- + ----| M(n - 13) + |- 348/7 + ----| M(n - 14) \ 15 35 / \ 21 / The first, 16, terms are 1, 3, 9, 28, 90, 296, 993, 3368, 11581, 40035, 139909, 490292, 1734018, 6142136, 21931130, 78378144, 282095892 The number of, n - 15, -step 3D ballot-lattice paths starting at, [5, 5, 5, 0], equals (and is provable, but we only proved it semi-rigorously)