Let , M(n), be the Motzkin numbers The number of, n, -step 3D ballot-lattice paths starting at, [0, 0, 0], equals (and (rigorously!) proved) M(n) The first, 30, terms are list, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829, 1697385471211 The number of, n - 1, -step 3D ballot-lattice paths starting at, [1, 0, 0], equals (and (rigorously!) proved) M(n) The first, 29, terms are 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829, 1697385471211 The number of, n - 2, -step 3D ballot-lattice paths starting at, [1, 1, 0], equals (and (rigorously!) proved) M(n - 1) The first, 28, terms are 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829 The number of, n - 2, -step 3D ballot-lattice paths starting at, [2, 0, 0], equals (and (rigorously!) proved) M(n) - M(n - 1) The first, 28, terms are 1, 2, 5, 12, 30, 76, 196, 512, 1353, 3610, 9713, 26324, 71799, 196938, 542895, 1503312, 4179603, 11662902, 32652735, 91695540, 258215664, 728997192, 2062967382, 5850674704, 16626415975, 47337954326, 135015505407, 385719506620, 1103642686382 The number of, n - 3, -step 3D ballot-lattice paths starting at, [2, 1, 0], equals (and (rigorously!) proved) M(n - 1) - M(n - 3) The first, 27, terms are 1, 3, 7, 17, 42, 106, 272, 708, 1865, 4963, 13323, 36037, 98123, 268737, 739833, 2046207, 5682915, 15842505, 44315637, 124348275, 349911204, 987212856, 2791964574, 7913642086, 22477090679, 63964370301, 182353459733, 520735012027 The number of, n - 4, -step 3D ballot-lattice paths starting at, [2, 2, 0], equals (and (rigorously!) proved) M(n - 2) - M(n - 3) The first, 26, terms are 1, 2, 5, 12, 30, 76, 196, 512, 1353, 3610, 9713, 26324, 71799, 196938, 542895, 1503312, 4179603, 11662902, 32652735, 91695540, 258215664, 728997192, 2062967382, 5850674704, 16626415975, 47337954326, 135015505407 The number of, n - 3, -step 3D ballot-lattice paths starting at, [3, 0, 0], equals (and (rigorously!) proved) M(n) - 2 M(n - 1) + M(n - 3) The first, 27, terms are 1, 2, 5, 13, 34, 90, 240, 645, 1745, 4750, 13001, 35762, 98815, 274158, 763479, 2133396, 5979987, 16810230, 47379903, 133867389, 379085988, 1075754526, 3058710130, 8712773889, 24860863647, 71051135106, 203366046887, 582907674355 The number of, n - 4, -step 3D ballot-lattice paths starting at, [3, 1, 0], equals (and (rigorously!) proved) M(n - 1) - M(n - 2) - M(n - 3) The first, 26, terms are 1, 3, 8, 21, 55, 145, 385, 1030, 2775, 7525, 20526, 56288, 155103, 429261, 1192740, 3326136, 9306123, 26116353, 73496256, 207363645, 586449633, 1662204159, 4720914289, 13433688178, 38294551825, 109345686931, 312711733818 The number of, n - 5, -step 3D ballot-lattice paths starting at, [3, 2, 0], equals (and (rigorously!) proved) M(n - 2) - M(n - 3) - M(n - 4) The first, 25, terms are 1, 3, 8, 21, 55, 145, 385, 1030, 2775, 7525, 20526, 56288, 155103, 429261, 1192740, 3326136, 9306123, 26116353, 73496256, 207363645, 586449633, 1662204159, 4720914289, 13433688178, 38294551825, 109345686931 The number of, n - 6, -step 3D ballot-lattice paths starting at, [3, 3, 0], equals (and (rigorously!) proved) M(n - 3) - 2 M(n - 4) + M(n - 6) The first, 24, terms are 1, 2, 5, 13, 34, 90, 240, 645, 1745, 4750, 13001, 35762, 98815, 274158, 763479, 2133396, 5979987, 16810230, 47379903, 133867389, 379085988, 1075754526, 3058710130, 8712773889, 24860863647 The number of, n - 4, -step 3D ballot-lattice paths starting at, [4, 0, 0], equals (and (rigorously!) proved) M(n) - 3 M(n - 1) + M(n - 2) + 2 M(n - 3) The first, 26, terms are 1, 2, 5, 13, 35, 95, 260, 715, 1975, 5476, 15236, 42527, 119055, 334218, 940656, 2653851, 7504107, 21263550, 60371133, 171722343, 489304893, 1396505971, 3991859600, 11427175469, 32756583281, 94020359956, 270195940537 The number of, n - 5, -step 3D ballot-lattice paths starting at, [4, 1, 0], equals (and (rigorously!) proved) M(n - 1) - 2 M(n - 2) - M(n - 3) + 2 M(n - 4) The first, 25, terms are 1, 3, 8, 22, 60, 164, 449, 1233, 3397, 9391, 26049, 72491, 202359, 566541, 1590501, 4476675, 12630627, 35717001, 101214654, 287390448, 817538862, 2329712938, 6649806507, 19010188943, 54424719131, 156028092561 The number of, n - 6, -step 3D ballot-lattice paths starting at, [4, 2, 0], equals (and (rigorously!) proved) M(n - 2) - 2 M(n - 3) The first, 24, terms are 1, 3, 9, 25, 69, 189, 518, 1422, 3915, 10813, 29964, 83304, 232323, 649845, 1822824, 5126520, 14453451, 40843521, 115668105, 328233969, 933206967, 2657946907, 7583013474, 21668135850, 62007732605 The number of, n - 7, -step 3D ballot-lattice paths starting at, [4, 3, 0], equals (and (rigorously!) proved) M(n - 3) - 2 M(n - 4) - M(n - 5) + 2 M(n - 6) The first, 23, terms are 1, 3, 8, 22, 60, 164, 449, 1233, 3397, 9391, 26049, 72491, 202359, 566541, 1590501, 4476675, 12630627, 35717001, 101214654, 287390448, 817538862, 2329712938, 6649806507, 19010188943 The number of, n - 8, -step 3D ballot-lattice paths starting at, [4, 4, 0], equals (and (rigorously!) proved) M(n - 4) - 3 M(n - 5) + M(n - 6) + 2 M(n - 7) The first, 22, terms are 1, 2, 5, 13, 35, 95, 260, 715, 1975, 5476, 15236, 42527, 119055, 334218, 940656, 2653851, 7504107, 21263550, 60371133, 171722343, 489304893, 1396505971, 3991859600 The number of, n - 5, -step 3D ballot-lattice paths starting at, [5, 0, 0], equals (and (rigorously!) proved) M(n) - 4 M(n - 1) + 3 M(n - 2) + 3 M(n - 3) - 2 M(n - 4) The first, 25, terms are 1, 2, 5, 13, 35, 96, 266, 742, 2079, 5845, 16478, 46564, 131859, 374115, 1063350, 3027432, 8632923, 24654132, 70507689, 201914445, 578967109, 1662146662, 4777368962, 13746394338, 39595640825, 114167847976 The number of, n - 6, -step 3D ballot-lattice paths starting at, [5, 1, 0], equals (and (rigorously!) proved) M(n - 1) - 3 M(n - 2) + 4 M(n - 4) - M(n - 6) The first, 24, terms are 1, 3, 8, 22, 61, 170, 475, 1330, 3731, 10486, 29526, 83293, 235403, 666498, 1890372, 5370711, 15283563, 43560903, 124342440, 355437504, 1017419983, 2916105074, 8368465339, 24043809392, 69159496309 The number of, n - 7, -step 3D ballot-lattice paths starting at, [5, 2, 0], equals (and (rigorously!) proved) M(n - 2) - 3 M(n - 3) + M(n - 4) + 2 M(n - 5) - M(n - 6) The first, 23, terms are 1, 3, 9, 26, 74, 209, 588, 1652, 4641, 13048, 36729, 103544, 292383, 827022, 2343279, 6650640, 18906771, 53834751, 153523059, 438452874, 1253958412, 3591096377, 10297415054, 29563855484 The number of, n - 8, -step 3D ballot-lattice paths starting at, [5, 3, 0], equals (and (rigorously!) proved) M(n - 3) - 3 M(n - 4) + M(n - 5) + 2 M(n - 6) - M(n - 7) The first, 22, terms are 1, 3, 9, 26, 74, 209, 588, 1652, 4641, 13048, 36729, 103544, 292383, 827022, 2343279, 6650640, 18906771, 53834751, 153523059, 438452874, 1253958412, 3591096377, 10297415054 The number of, n - 9, -step 3D ballot-lattice paths starting at, [5, 4, 0], equals (and (rigorously!) proved) M(n - 4) - 3 M(n - 5) + 4 M(n - 7) - M(n - 9) The first, 21, terms are 1, 3, 8, 22, 61, 170, 475, 1330, 3731, 10486, 29526, 83293, 235403, 666498, 1890372, 5370711, 15283563, 43560903, 124342440, 355437504, 1017419983, 2916105074 The number of, n - 10, -step 3D ballot-lattice paths starting at, [5, 5, 0], equals (and (rigorously!) proved) M(n - 5) - 4 M(n - 6) + 3 M(n - 7) + 3 M(n - 8) - 2 M(n - 9) The first, 20, terms are 1, 2, 5, 13, 35, 96, 266, 742, 2079, 5845, 16478, 46564, 131859, 374115, 1063350, 3027432, 8632923, 24654132, 70507689, 201914445, 578967109 The number of, n - 6, -step 3D ballot-lattice paths starting at, [6, 0, 0], equals (and (rigorously!) proved) M(n) - 5 M(n - 1) + 6 M(n - 2) + 3 M(n - 3) - 6 M(n - 4) + M(n - 6) The first, 24, terms are 1, 2, 5, 13, 35, 96, 267, 749, 2114, 5992, 17038, 48566, 138712, 396852, 1137060, 3262212, 9370569, 26946786, 77572005, 223529605, 644726679, 1861263888, 5377928999, 15551831433, 45008351667 The number of, n - 7, -step 3D ballot-lattice paths starting at, [6, 1, 0], equals (and (rigorously!) proved) M(n - 1) - 4 M(n - 2) + 2 M(n - 3) + 6 M(n - 4) - 3 M(n - 5) - 2 M(n - 6) The first, 23, terms are 1, 3, 8, 22, 61, 171, 482, 1364, 3870, 11002, 31328, 89332, 255060, 729132, 2086776, 5979072, 17150025, 49244139, 141543312, 407244766, 1172841769, 3380862991, 9754534738, 28168465356 The number of, n - 8, -step 3D ballot-lattice paths starting at, [6, 2, 0], equals (and (rigorously!) proved) M(n - 2) - 4 M(n - 3) + 3 M(n - 4) + 3 M(n - 5) - 2 M(n - 6) - M(n - 7) The first, 22, terms are 1, 3, 9, 26, 75, 215, 615, 1756, 5010, 14290, 40766, 116348, 332280, 949716, 2716860, 7779456, 22297353, 63971307, 183715161, 528115090, 1519599103, 4376605739, 12616633923 The number of, n - 9, -step 3D ballot-lattice paths starting at, [6, 3, 0], equals (and (rigorously!) proved) M(n - 3) - 4 M(n - 4) + 3 M(n - 5) + 4 M(n - 6) - 5 M(n - 7) + M(n - 9) The first, 21, terms are 1, 3, 9, 27, 79, 229, 659, 1889, 5402, 15430, 44054, 125786, 359296, 1026936, 2937444, 8409540, 24097737, 69118635, 198442329, 570286939, 1640469427, 4723363073 The number of, n - 10, -step 3D ballot-lattice paths starting at, [6, 4, 0], equals (and (rigorously!) proved) M(n - 4) - 4 M(n - 5) + 3 M(n - 6) + 3 M(n - 7) - 2 M(n - 8) - M(n - 9) The first, 20, terms are 1, 3, 9, 26, 75, 215, 615, 1756, 5010, 14290, 40766, 116348, 332280, 949716, 2716860, 7779456, 22297353, 63971307, 183715161, 528115090, 1519599103 The number of, n - 11, -step 3D ballot-lattice paths starting at, [6, 5, 0], equals (and (rigorously!) proved) M(n - 5) - 4 M(n - 6) + 2 M(n - 7) + 6 M(n - 8) - 3 M(n - 9) - 2 M(n - 10) The first, 19, terms are 1, 3, 8, 22, 61, 171, 482, 1364, 3870, 11002, 31328, 89332, 255060, 729132, 2086776, 5979072, 17150025, 49244139, 141543312, 407244766 The number of, n - 12, -step 3D ballot-lattice paths starting at, [6, 6, 0], equals (and (rigorously!) proved) M(n - 6) - 5 M(n - 7) + 6 M(n - 8) + 3 M(n - 9) - 6 M(n - 10) + M(n - 12) The first, 18, terms are 1, 2, 5, 13, 35, 96, 267, 749, 2114, 5992, 17038, 48566, 138712, 396852, 1137060, 3262212, 9370569, 26946786, 77572005 The number of, n - 7, -step 3D ballot-lattice paths starting at, [7, 0, 0], equals (and (rigorously!) proved) M(n) - 6 M(n - 1) + 10 M(n - 2) + M(n - 3) - 12 M(n - 4) + 3 M(n - 5) + 3 M(n - 6) The first, 23, terms are 1, 2, 5, 13, 35, 96, 267, 750, 2122, 6036, 17238, 49380, 141792, 407928, 1175436, 3391497, 9796761, 28327866, 81986293, 237481913, 688422119, 1997066008, 5797296695, 16839886311 The number of, n - 8, -step 3D ballot-lattice paths starting at, [7, 1, 0], equals (and (rigorously!) proved) M(n - 1) - 5 M(n - 2) + 5 M(n - 3) + 7 M(n - 4) - 9 M(n - 5) - 3 M(n - 6) + 3 M(n - 7) The first, 22, terms are 1, 3, 8, 22, 61, 171, 483, 1372, 3913, 11193, 32088, 92148, 264993, 762945, 2198862, 6343137, 18313863, 52917873, 153021916, 442812234, 1282296779, 3715782337, 10774462471 The number of, n - 9, -step 3D ballot-lattice paths starting at, [7, 2, 0], equals (and (rigorously!) proved) M(n - 2) - 5 M(n - 3) + 6 M(n - 4) + 3 M(n - 5) - 6 M(n - 6) The first, 21, terms are 1, 3, 9, 26, 75, 216, 622, 1791, 5157, 14850, 42768, 123201, 355017, 1023426, 2951640, 8517102, 24590007, 71035623, 205330321, 593874660, 1718716329, 4977165776 The number of, n - 10, -step 3D ballot-lattice paths starting at, [7, 3, 0], equals (and (rigorously!) proved) M(n - 3) - 5 M(n - 4) + 6 M(n - 5) + 4 M(n - 6) - 9 M(n - 7) + 3 M(n - 9) The first, 20, terms are 1, 3, 9, 27, 80, 235, 686, 1994, 5779, 16716, 48291, 139404, 402273, 1160706, 3349401, 9667641, 27914511, 80636271, 233048719, 673901463, 1949805558 The number of, n - 11, -step 3D ballot-lattice paths starting at, [7, 4, 0], equals (and (rigorously!) proved) M(n - 4) - 5 M(n - 5) + 6 M(n - 6) + 4 M(n - 7) - 9 M(n - 8) + 3 M(n - 10) The first, 19, terms are 1, 3, 9, 27, 80, 235, 686, 1994, 5779, 16716, 48291, 139404, 402273, 1160706, 3349401, 9667641, 27914511, 80636271, 233048719, 673901463 The number of, n - 12, -step 3D ballot-lattice paths starting at, [7, 5, 0], equals (and (rigorously!) proved) M(n - 5) - 5 M(n - 6) + 6 M(n - 7) + 3 M(n - 8) - 6 M(n - 9) The first, 18, terms are 1, 3, 9, 26, 75, 216, 622, 1791, 5157, 14850, 42768, 123201, 355017, 1023426, 2951640, 8517102, 24590007, 71035623, 205330321 The number of, n - 13, -step 3D ballot-lattice paths starting at, [7, 6, 0], equals (and (rigorously!) proved) M(n - 6) - 5 M(n - 7) + 5 M(n - 8) + 7 M(n - 9) - 9 M(n - 10) - 3 M(n - 11) + 3 M(n - 12) The first, 17, terms are 1, 3, 8, 22, 61, 171, 483, 1372, 3913, 11193, 32088, 92148, 264993, 762945, 2198862, 6343137, 18313863, 52917873 The number of, n - 14, -step 3D ballot-lattice paths starting at, [7, 7, 0], equals (and (rigorously!) proved) M(n - 7) - 6 M(n - 8) + 10 M(n - 9) + M(n - 10) - 12 M(n - 11) + 3 M(n - 12) + 3 M(n - 13) The first, 16, terms are 1, 2, 5, 13, 35, 96, 267, 750, 2122, 6036, 17238, 49380, 141792, 407928, 1175436, 3391497, 9796761 The number of, n - 8, -step 3D ballot-lattice paths starting at, [8, 0, 0], equals (and (rigorously!) proved) M(n) - 7 M(n - 1) + 15 M(n - 2) - 4 M(n - 3) - 19 M(n - 4) + 12 M(n - 5) + 6 M(n - 6) - 3 M(n - 7) The first, 22, terms are 1, 2, 5, 13, 35, 96, 267, 750, 2123, 6045, 17292, 49644, 142935, 412491, 1192635, 3453624, 10014003, 29068420, 84459997, 245609885, 714769229, 2081514358, 6065423840 The number of, n - 9, -step 3D ballot-lattice paths starting at, [8, 1, 0], equals (and (rigorously!) proved) M(n - 1) - 6 M(n - 2) + 9 M(n - 3) + 6 M(n - 4) - 18 M(n - 5) + 9 M(n - 7) - M(n - 9) The first, 21, terms are 1, 3, 8, 22, 61, 171, 483, 1373, 3922, 11246, 32342, 93226, 269216, 778584, 2254437, 6534549, 18957297, 55039507, 159909908, 464892514, 1352339329, 3936032807 The number of, n - 10, -step 3D ballot-lattice paths starting at, [8, 2, 0], equals (and (rigorously!) proved) M(n - 2) - 6 M(n - 3) + 10 M(n - 4) + M(n - 5) - 12 M(n - 6) + 3 M(n - 7) + 3 M(n - 8) - M(n - 9) The first, 20, terms are 1, 3, 9, 26, 75, 216, 623, 1799, 5201, 15050, 43582, 126281, 366093, 1061802, 3080925, 8943294, 25971087, 75449911, 219282629, 637570100, 1854518449 The number of, n - 11, -step 3D ballot-lattice paths starting at, [8, 3, 0], equals (and (rigorously!) proved) M(n - 3) - 6 M(n - 4) + 10 M(n - 5) + 2 M(n - 6) - 16 M(n - 7) + 6 M(n - 8) + 5 M(n - 9) - 2 M(n - 10) The first, 19, terms are 1, 3, 9, 27, 80, 236, 693, 2029, 5927, 17285, 50347, 146521, 426153, 1238979, 3601380, 10467414, 30424407, 88441141, 257137583, 747789005 The number of, n - 12, -step 3D ballot-lattice paths starting at, [8, 4, 0], equals (and (rigorously!) proved) M(n - 4) - 6 M(n - 5) + 10 M(n - 6) + 2 M(n - 7) - 15 M(n - 8) + 2 M(n - 9) + 8 M(n - 10) - M(n - 12) The first, 18, terms are 1, 3, 9, 27, 81, 241, 713, 2099, 6157, 18011, 52582, 153286, 446393, 1299039, 3778557, 10987869, 31948527, 92894461, 270128813 The number of, n - 13, -step 3D ballot-lattice paths starting at, [8, 5, 0], equals (and (rigorously!) proved) M(n - 5) - 6 M(n - 6) + 10 M(n - 7) + 2 M(n - 8) - 16 M(n - 9) + 6 M(n - 10) + 5 M(n - 11) - 2 M(n - 12) The first, 17, terms are 1, 3, 9, 27, 80, 236, 693, 2029, 5927, 17285, 50347, 146521, 426153, 1238979, 3601380, 10467414, 30424407, 88441141 The number of, n - 14, -step 3D ballot-lattice paths starting at, [8, 6, 0], equals (and (rigorously!) proved) M(n - 6) - 6 M(n - 7) + 10 M(n - 8) + M(n - 9) - 12 M(n - 10) + 3 M(n - 11) + 3 M(n - 12) - M(n - 13) The first, 16, terms are 1, 3, 9, 26, 75, 216, 623, 1799, 5201, 15050, 43582, 126281, 366093, 1061802, 3080925, 8943294, 25971087 The number of, n - 15, -step 3D ballot-lattice paths starting at, [8, 7, 0], equals (and (rigorously!) proved) M(n - 7) - 6 M(n - 8) + 9 M(n - 9) + 6 M(n - 10) - 18 M(n - 11) + 9 M(n - 13) - M(n - 15) The first, 15, terms are 1, 3, 8, 22, 61, 171, 483, 1373, 3922, 11246, 32342, 93226, 269216, 778584, 2254437, 6534549 The number of, n - 16, -step 3D ballot-lattice paths starting at, [8, 8, 0], equals (and (rigorously!) proved) M(n - 8) - 7 M(n - 9) + 15 M(n - 10) - 4 M(n - 11) - 19 M(n - 12) + 12 M(n - 13) + 6 M(n - 14) - 3 M(n - 15) The first, 14, terms are 1, 2, 5, 13, 35, 96, 267, 750, 2123, 6045, 17292, 49644, 142935, 412491, 1192635 The number of, n - 9, -step 3D ballot-lattice paths starting at, [9, 0, 0], equals (and (rigorously!) proved) M(n) - 8 M(n - 1) + 21 M(n - 2) - 13 M(n - 3) - 25 M(n - 4) + 30 M(n - 5) + 6 M(n - 6) - 12 M(n - 7) + M(n - 9) The first, 21, terms are 1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17302, 49709, 143275, 414051, 1199187, 3479454, 10111123, 29420490, 85699977, 249876715, 729175029, 2129391033 The number of, n - 10, -step 3D ballot-lattice paths starting at, [9, 1, 0], equals (and (rigorously!) proved) M(n - 1) - 7 M(n - 2) + 14 M(n - 3) + 2 M(n - 4) - 29 M(n - 5) + 11 M(n - 6) + 18 M(n - 7) - 6 M(n - 8) - 3 M(n - 9) The first, 20, terms are 1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11256, 32406, 93555, 270699, 784707, 2278188, 6622506, 19271659, 56132131, 163623592, 477287316, 1393092239 The number of, n - 11, -step 3D ballot-lattice paths starting at, [9, 2, 0], equals (and (rigorously!) proved) M(n - 2) - 7 M(n - 3) + 15 M(n - 4) - 4 M(n - 5) - 19 M(n - 6) + 12 M(n - 7) + 6 M(n - 8) - 3 M(n - 9) - M(n - 10) The first, 19, terms are 1, 3, 9, 26, 75, 216, 623, 1800, 5210, 15104, 43846, 127424, 370656, 1079001, 3143052, 9160536, 26711641, 77923615, 227410601, 663917210 The number of, n - 12, -step 3D ballot-lattice paths starting at, [9, 3, 0], equals (and (rigorously!) proved) M(n - 3) - 7 M(n - 4) + 15 M(n - 5) - 3 M(n - 6) - 24 M(n - 7) + 18 M(n - 8) + 9 M(n - 9) - 10 M(n - 10) + M(n - 12) The first, 18, terms are 1, 3, 9, 27, 80, 236, 694, 2037, 5971, 17486, 51171, 149666, 437569, 1278915, 3737217, 10919436, 31902607, 93207499, 272329871 The number of, n - 13, -step 3D ballot-lattice paths starting at, [9, 4, 0], equals (and (rigorously!) proved) M(n - 4) - 7 M(n - 5) + 15 M(n - 6) - 3 M(n - 7) - 23 M(n - 8) + 14 M(n - 9) + 11 M(n - 10) - 5 M(n - 11) - 2 M(n - 12) The first, 17, terms are 1, 3, 9, 27, 81, 242, 720, 2134, 6305, 18581, 54648, 160468, 470613, 1378872, 4037088, 11813472, 34555543, 101051401 The number of, n - 14, -step 3D ballot-lattice paths starting at, [9, 5, 0], equals (and (rigorously!) proved) M(n - 5) - 7 M(n - 6) + 15 M(n - 7) - 3 M(n - 8) - 23 M(n - 9) + 14 M(n - 10) + 11 M(n - 11) - 5 M(n - 12) - 2 M(n - 13) The first, 16, terms are 1, 3, 9, 27, 81, 242, 720, 2134, 6305, 18581, 54648, 160468, 470613, 1378872, 4037088, 11813472, 34555543 The number of, n - 15, -step 3D ballot-lattice paths starting at, [9, 6, 0], equals (and (rigorously!) proved) M(n - 6) - 7 M(n - 7) + 15 M(n - 8) - 3 M(n - 9) - 24 M(n - 10) + 18 M(n - 11) + 9 M(n - 12) - 10 M(n - 13) + M(n - 15) The first, 15, terms are 1, 3, 9, 27, 80, 236, 694, 2037, 5971, 17486, 51171, 149666, 437569, 1278915, 3737217, 10919436 The number of, n - 16, -step 3D ballot-lattice paths starting at, [9, 7, 0], equals (and (rigorously!) proved) M(n - 7) - 7 M(n - 8) + 15 M(n - 9) - 4 M(n - 10) - 19 M(n - 11) + 12 M(n - 12) + 6 M(n - 13) - 3 M(n - 14) - M(n - 15) The first, 14, terms are 1, 3, 9, 26, 75, 216, 623, 1800, 5210, 15104, 43846, 127424, 370656, 1079001, 3143052 The number of, n - 17, -step 3D ballot-lattice paths starting at, [9, 8, 0], equals (and (rigorously!) proved) M(n - 8) - 7 M(n - 9) + 14 M(n - 10) + 2 M(n - 11) - 29 M(n - 12) + 11 M(n - 13) + 18 M(n - 14) - 6 M(n - 15) - 3 M(n - 16) The first, 13, terms are 1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11256, 32406, 93555, 270699, 784707 The number of, n - 18, -step 3D ballot-lattice paths starting at, [9, 9, 0], equals (and (rigorously!) proved) M(n - 9) - 8 M(n - 10) + 21 M(n - 11) - 13 M(n - 12) - 25 M(n - 13) + 30 M(n - 14) + 6 M(n - 15) - 12 M(n - 16) + M(n - 18) The first, 12, terms are 1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17302, 49709, 143275 The number of, n - 10, -step 3D ballot-lattice paths starting at, [10, 0, 0], equals (and (rigorously!) proved) M(n) - 9 M(n - 1) + 28 M(n - 2) - 27 M(n - 3) - 27 M(n - 4) + 59 M(n - 5) - 5 M(n - 6) - 30 M(n - 7) + 6 M(n - 8) + 4 M(n - 9) The first, 20, terms are 1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17303, 49720, 143352, 414480, 1201266, 3488617, 10148831, 29567846, 86253123, 251887713, 736298794 The number of, n - 11, -step 3D ballot-lattice paths starting at, [10, 1, 0], equals (and (rigorously!) proved) M(n - 1) - 8 M(n - 2) + 20 M(n - 3) - 6 M(n - 4) - 40 M(n - 5) + 34 M(n - 6) + 25 M(n - 7) - 24 M(n - 8) - 6 M(n - 9) + 4 M(n - 10) The first, 19, terms are 1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11257, 32417, 93631, 271116, 786696, 2286819, 6657499, 19406487, 56631557, 165416718, 483565144 The number of, n - 12, -step 3D ballot-lattice paths starting at, [10, 2, 0], equals (and (rigorously!) proved) M(n - 2) - 8 M(n - 3) + 21 M(n - 4) - 13 M(n - 5) - 25 M(n - 6) + 30 M(n - 7) + 6 M(n - 8) - 12 M(n - 9) The first, 18, terms are 1, 3, 9, 26, 75, 216, 623, 1800, 5211, 15114, 43911, 127764, 372216, 1085553, 3168882, 9257656, 27063711, 79163595, 231677431 The number of, n - 13, -step 3D ballot-lattice paths starting at, [10, 3, 0], equals (and (rigorously!) proved) M(n - 3) - 8 M(n - 4) + 21 M(n - 5) - 12 M(n - 6) - 31 M(n - 7) + 40 M(n - 8) + 7 M(n - 9) - 24 M(n - 10) + 2 M(n - 11) + 4 M(n - 12) The first, 17, terms are 1, 3, 9, 27, 80, 236, 694, 2038, 5980, 17540, 51436, 150820, 442209, 1296543, 3801423, 11145841, 32680869, 95828559 The number of, n - 14, -step 3D ballot-lattice paths starting at, [10, 4, 0], equals (and (rigorously!) proved) M(n - 4) - 8 M(n - 5) + 21 M(n - 6) - 12 M(n - 7) - 30 M(n - 8) + 35 M(n - 9) + 13 M(n - 10) - 22 M(n - 11) - 2 M(n - 12) + 4 M(n - 13) The first, 16, terms are 1, 3, 9, 27, 81, 242, 721, 2142, 6349, 18782, 55473, 163624, 482106, 1419237, 4175004, 12274657, 36071451 The number of, n - 15, -step 3D ballot-lattice paths starting at, [10, 5, 0], equals (and (rigorously!) proved) M(n - 5) - 8 M(n - 6) + 21 M(n - 7) - 12 M(n - 8) - 30 M(n - 9) + 36 M(n - 10) + 8 M(n - 11) - 16 M(n - 12) The first, 15, terms are 1, 3, 9, 27, 81, 243, 727, 2169, 6453, 19151, 56715, 167661, 494910, 1459134, 4297698, 12648238 The number of, n - 16, -step 3D ballot-lattice paths starting at, [10, 6, 0], equals (and (rigorously!) proved) M(n - 6) - 8 M(n - 7) + 21 M(n - 8) - 12 M(n - 9) - 30 M(n - 10) + 35 M(n - 11) + 13 M(n - 12) - 22 M(n - 13) - 2 M(n - 14) + 4 M(n - 15) The first, 14, terms are 1, 3, 9, 27, 81, 242, 721, 2142, 6349, 18782, 55473, 163624, 482106, 1419237, 4175004 The number of, n - 17, -step 3D ballot-lattice paths starting at, [10, 7, 0], equals (and (rigorously!) proved) M(n - 7) - 8 M(n - 8) + 21 M(n - 9) - 12 M(n - 10) - 31 M(n - 11) + 40 M(n - 12) + 7 M(n - 13) - 24 M(n - 14) + 2 M(n - 15) + 4 M(n - 16) The first, 13, terms are 1, 3, 9, 27, 80, 236, 694, 2038, 5980, 17540, 51436, 150820, 442209, 1296543 The number of, n - 18, -step 3D ballot-lattice paths starting at, [10, 8, 0], equals (and (rigorously!) proved) M(n - 8) - 8 M(n - 9) + 21 M(n - 10) - 13 M(n - 11) - 25 M(n - 12) + 30 M(n - 13) + 6 M(n - 14) - 12 M(n - 15) The first, 12, terms are 1, 3, 9, 26, 75, 216, 623, 1800, 5211, 15114, 43911, 127764, 372216 The number of, n - 19, -step 3D ballot-lattice paths starting at, [10, 9, 0], equals (and (rigorously!) proved) M(n - 9) - 8 M(n - 10) + 20 M(n - 11) - 6 M(n - 12) - 40 M(n - 13) + 34 M(n - 14) + 25 M(n - 15) - 24 M(n - 16) - 6 M(n - 17) + 4 M(n - 18) The first, 11, terms are 1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11257, 32417, 93631 The number of, n - 20, -step 3D ballot-lattice paths starting at, [10, 10, 0], equals (and (rigorously!) proved) M(n - 10) - 9 M(n - 11) + 28 M(n - 12) - 27 M(n - 13) - 27 M(n - 14) + 59 M(n - 15) - 5 M(n - 16) - 30 M(n - 17) + 6 M(n - 18) + 4 M(n - 19) The first, 10, terms are 1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17303 this took, 111.904, second of CPU time