On the Generating Functions enumerating walks confined to the Non-Negative H\ alf Line With step-sets that are subsets of , {-3, -2, -1, 0, 1, 2, 3} By Shalosh B. Ekhad It is well known, and not too hard to see, that the generating function enumerating walks confined to the non-negative half-line with ANY given step\ -set is always algebraic. This is true for both the total number and for the walk\ s that end-up at a specific location. In this article we will state the algebraic equations satisfied by generatin\ g functions that enumerate walks where the step-sets are subsets of {-3, -2, -1, 0, 1, 2, 3} that have at least one positive and at least one negative step. Theorem Number, 1, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-3, 1}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 1, 1, 1, 2, 3, 4, 5, 10, 16, 23, 31, 62, 102, 152, 213, 426, 712, 1084, 1556, 3112, 5255, 8116, 11843, 23686, 40288, 62866, 92842, 185684, 317548, 499376] and the first 30 terms of B(n) are [1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 22, 0, 0, 0, 140, 0, 0, 0, 969, 0, 0, 0, 7084, 0, 0, 0, 53820, 0, 0] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (4 t - 1) P + 3 t (-1 + 2 t) P + t (-3 + 4 t) P + t (-1 + 2 t) P = 0 and in Maple input format 1+(4*t-1)*P+3*t*(-1+2*t)*P^2+t^2*(-3+4*t)*P^3+t^3*(-1+2*t)*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 4 4 t Q - Q + 1 = 0 and in Maple input format t^4*Q^4-Q+1 = 0 Theorem Number, 2, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-3, 0, 1}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 2, 4, 8, 17, 39, 94, 230, 563, 1382, 3425, 8590, 21743, 55327, 141241, 361737, 930039, 2400620, 6217576, 16147130, 42030295, 109638992, 286607316, 750734144, 1970084891, 5178486924, 13632623181, 35939751026, 94876520819, 250782875410, 663677631323] and the first 30 terms of B(n) are [1, 1, 1, 1, 2, 6, 16, 36, 75, 163, 391, 991, 2498, 6150, 15016, 37116, 93481, 238137, 607921, 1550401, 3959335, 10155615, 26182267, 67753907, 175713561, 456422121, 1187771521, 3097869841, 8097629671, 21207212047, 55628797891] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (-1 + 5 t) P + 3 t (-1 + 3 t) P + t (-3 + 7 t) P + t (-1 + 3 t) P = 0 and in Maple input format 1+(-1+5*t)*P+3*t*(-1+3*t)*P^2+t^2*(-3+7*t)*P^3+t^3*(-1+3*t)*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 4 4 1 + (t - 1) Q + t Q = 0 and in Maple input format 1+(t-1)*Q+t^4*Q^4 = 0 Theorem Number, 3, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-2, 1}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 1, 1, 2, 3, 4, 8, 13, 19, 38, 64, 98, 196, 337, 531, 1062, 1851, 2974, 5948, 10468, 17060, 34120, 60488, 99658, 199316, 355369, 590563, 1181126, 2115577, 3540464, 7080928] and the first 30 terms of B(n) are [1, 0, 0, 1, 0, 0, 3, 0, 0, 12, 0, 0, 55, 0, 0, 273, 0, 0, 1428, 0, 0, 7752, 0, 0, 43263, 0, 0, 246675, 0, 0, 1430715] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 1 + (-1 + 3 t) P + t (-2 + 3 t) P + t (-1 + 2 t) P = 0 and in Maple input format 1+(-1+3*t)*P+t*(-2+3*t)*P^2+t^2*(-1+2*t)*P^3 = 0 and Q=Q(t) satisfies the following algebraic equation 3 3 1 - Q + t Q = 0 and in Maple input format 1-Q+t^3*Q^3 = 0 Theorem Number, 4, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-2, 0, 1}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 2, 4, 9, 22, 55, 139, 357, 930, 2445, 6473, 17248, 46221, 124449, 336439, 912842, 2484802, 6783289, 18565801, 50933984, 140033419, 385749984, 1064536922, 2942628418, 8146598481, 22585803575, 62700416569, 174278053995, 484973417867, 1351029974326, 3767520334032] and the first 30 terms of B(n) are [1, 1, 1, 2, 5, 11, 24, 57, 141, 349, 871, 2212, 5688, 14730, 38403, 100829, 266333, 706997, 1885165, 5047522, 13565203, 36578497, 98934826, 268342933, 729709432, 1989021256, 5433518806, 14873285506, 40790118487, 112064912455, 308390452661] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 1 + (4 t - 1) P + t (-2 + 5 t) P + t (-1 + 3 t) P = 0 and in Maple input format 1+(4*t-1)*P+t*(-2+5*t)*P^2+t^2*(-1+3*t)*P^3 = 0 and Q=Q(t) satisfies the following algebraic equation 3 3 1 + (t - 1) Q + t Q = 0 and in Maple input format 1+(t-1)*Q+t^3*Q^3 = 0 Theorem Number, 5, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-1, 1}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462, 924, 1716, 3432, 6435, 12870, 24310, 48620, 92378, 184756, 352716, 705432, 1352078, 2704156, 5200300, 10400600, 20058300, 40116600, 77558760, 155117520] and the first 30 terms of B(n) are [1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, 132, 0, 429, 0, 1430, 0, 4862, 0, 16796, 0, 58786, 0, 208012, 0, 742900, 0, 2674440, 0, 9694845] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 1 + (-1 + 2 t) P + t (-1 + 2 t) P = 0 and in Maple input format 1+(-1+2*t)*P+t*(-1+2*t)*P^2 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 1 - Q + t Q = 0 and in Maple input format 1-Q+t^2*Q^2 = 0 Theorem Number, 6, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-1, 0, 1}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17303, 49721, 143365, 414584, 1201917, 3492117, 10165779, 29643870, 86574831, 253188111, 741365049, 2173243128, 6377181825, 18730782252, 55062586341, 161995031226, 476941691177, 1405155255055, 4142457992363, 12219350698880, 36064309311811] and the first 30 terms of B(n) are [1, 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] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 1 + (-1 + 3 t) P + t (-1 + 3 t) P = 0 and in Maple input format 1+(-1+3*t)*P+t*(-1+3*t)*P^2 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 1 + (t - 1) Q + t Q = 0 and in Maple input format 1+(t-1)*Q+t^2*Q^2 = 0 Theorem Number, 7, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-3, -2, 1}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 1, 1, 2, 4, 6, 11, 24, 44, 80, 171, 348, 659, 1361, 2887, 5749, 11696, 24952, 51668, 105636, 223884, 473976, 983484, 2073906, 4428707, 9331731, 19691733, 42120404, 89754642, 190354836, 407176953] and the first 30 terms of B(n) are [1, 0, 0, 1, 1, 0, 3, 7, 4, 12, 45, 55, 77, 286, 546, 728, 1960, 4760, 7548, 15504, 39729, 75582, 140448, 336490, 723327, 1366200, 2992990, 6758895, 13522275, 28094040, 63183315] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (4 t - 1) P + 3 t (-1 + 2 t) P + t (5 t - 3) P + t (-1 + 3 t) P = 0 and in Maple input format 1+(4*t-1)*P+3*t*(-1+2*t)*P^2+t^2*(5*t-3)*P^3+t^3*(-1+3*t)*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 3 3 4 4 t Q + t Q + 1 - Q = 0 and in Maple input format t^3*Q^3+t^4*Q^4+1-Q = 0 Theorem Number, 8, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-3, -2, 0, 1}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 2, 4, 9, 23, 62, 169, 466, 1309, 3738, 10789, 31380, 91924, 271112, 804296, 2397881, 7179965, 21583826, 65116174, 197083235, 598242061, 1820795134, 5555315095, 16987756366, 52055637556, 159822266632, 491570261041, 1514467365105, 4673180549083, 14441103681714, 44687322158441] and the first 30 terms of B(n) are [1, 1, 1, 2, 6, 16, 39, 99, 271, 763, 2146, 6062, 17359, 50337, 147057, 431874, 1275273, 3786649, 11298031, 33846202, 101762937, 306997821, 929038518, 2819426688, 8578433304, 26163061776, 79970186791, 244938841096, 751646959402, 2310683396056, 7115199919151] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (5 t - 1) P + 3 t (-1 + 3 t) P + t (-3 + 8 t) P + t (4 t - 1) P = 0 and in Maple input format 1+(5*t-1)*P+3*t*(-1+3*t)*P^2+t^2*(-3+8*t)*P^3+t^3*(4*t-1)*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 3 3 4 4 1 + (t - 1) Q + t Q + t Q = 0 and in Maple input format 1+(t-1)*Q+t^3*Q^3+t^4*Q^4 = 0 Theorem Number, 9, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-3, -1, 1}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 1, 2, 3, 7, 12, 29, 53, 131, 249, 624, 1218, 3082, 6133, 15630, 31563, 80883, 165237, 425297, 877037, 2265475, 4708230, 12198185, 25517232, 66277826, 139422362, 362917931, 767131683, 2000611629, 4246747092, 11093412621] and the first 30 terms of B(n) are [1, 0, 1, 0, 3, 0, 11, 0, 46, 0, 207, 0, 979, 0, 4797, 0, 24138, 0, 123998, 0, 647615, 0, 3428493, 0, 18356714, 0, 99229015, 0, 540807165, 0, 2968468275] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (4 t - 1) P + t (-3 + 7 t) P + 3 t (-1 + 2 t) P + t (-1 + 3 t) P = 0 and in Maple input format 1+(4*t-1)*P+t*(-3+7*t)*P^2+3*t^2*(-1+2*t)*P^3+t^3*(-1+3*t)*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 4 4 t Q + t Q - Q + 1 = 0 and in Maple input format t^2*Q^2+t^4*Q^4-Q+1 = 0 Theorem Number, 10, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-3, -1, 0, 1}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 2, 5, 13, 36, 103, 303, 908, 2762, 8500, 26414, 82751, 261053, 828488, 2643124, 8471342, 27262602, 88059386, 285377001, 927602475, 3023345730, 9878600702, 32351633803, 106172562048, 349120048709, 1150065186718, 3794902329355, 12541808255557, 41510400418046, 137578618578488, 456568531055761] and the first 30 terms of B(n) are [1, 1, 2, 4, 10, 26, 72, 204, 593, 1753, 5263, 15995, 49127, 152231, 475359, 1494287, 4724903, 15017767, 47954400, 153764322, 494892393, 1598241869, 5177492708, 16820048006, 54785449703, 178873019471, 585312014446, 1919212203652, 6305054573239, 20750506952675, 68405627426506] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (5 t - 1) P + t (-3 + 10 t) P + 3 t (-1 + 3 t) P + t (4 t - 1) P = 0 and in Maple input format 1+(5*t-1)*P+t*(-3+10*t)*P^2+3*t^2*(-1+3*t)*P^3+t^3*(4*t-1)*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 4 4 1 + (t - 1) Q + t Q + t Q = 0 and in Maple input format 1+(t-1)*Q+t^2*Q^2+t^4*Q^4 = 0 Theorem Number, 11, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-2, -1, 1}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 1, 2, 4, 8, 18, 39, 89, 204, 472, 1110, 2616, 6231, 14909, 35861, 86705, 210364, 512480, 1252350, 3069638, 7544818, 18589202, 45907708, 113608590, 281698359, 699748003, 1741102844, 4338995332, 10828981851, 27063384783, 67722954114] and the first 30 terms of B(n) are [1, 0, 1, 1, 2, 5, 8, 21, 42, 96, 222, 495, 1177, 2717, 6435, 15288, 36374, 87516, 210494, 509694, 1237736, 3014882, 7370860, 18059899, 44379535, 109298070, 269766655, 667224480, 1653266565, 4103910930, 10203669285] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 1 + (-1 + 3 t) P + 2 t (-1 + 2 t) P + t (-1 + 3 t) P = 0 and in Maple input format 1+(-1+3*t)*P+2*t*(-1+2*t)*P^2+t^2*(-1+3*t)*P^3 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 3 3 1 - Q + t Q + t Q = 0 and in Maple input format 1-Q+t^2*Q^2+t^3*Q^3 = 0 Theorem Number, 12, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-2, -1, 0, 1}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 2, 5, 14, 41, 124, 384, 1210, 3865, 12482, 40677, 133572, 441468, 1467296, 4900760, 16439370, 55357305, 187050302, 633998079, 2154950454, 7343407521, 25082709012, 85858848820, 294480653064, 1011871145116, 3482837144984, 12006861566684, 41454180382688, 143320499084136, 496148919813504, 1719671983751856] and the first 30 terms of B(n) are [1, 1, 2, 5, 13, 36, 104, 309, 939, 2905, 9118, 28964, 92940, 300808, 980864, 3219205, 10626023, 35252867, 117485454, 393133485, 1320357501, 4449298136, 15038769672, 50973266380, 173214422068, 589998043276, 2014026871496, 6889055189032, 23608722350440, 81049178840528, 278700885572096] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 1 + (4 t - 1) P + 2 t (-1 + 3 t) P + t (4 t - 1) P = 0 and in Maple input format 1+(4*t-1)*P+2*t*(-1+3*t)*P^2+t^2*(4*t-1)*P^3 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 3 3 1 + (t - 1) Q + t Q + t Q = 0 and in Maple input format 1+(t-1)*Q+t^2*Q^2+t^3*Q^3 = 0 Theorem Number, 13, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-3, -2, -1, 1}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 1, 2, 4, 9, 20, 48, 114, 281, 694, 1747, 4417, 11299, 29037, 75198, 195583, 511477, 1342638, 3539040, 9358692, 24829370, 66056232, 176207941, 471153500, 1262628841, 3390568265, 9122279345, 24586773895, 66377698870, 179479380925, 486003769120] and the first 30 terms of B(n) are [1, 0, 1, 1, 3, 5, 14, 28, 74, 168, 432, 1045, 2684, 6721, 17355, 44408, 115502, 299812, 785570, 2060094, 5434475, 14362841, 38114760, 101360402, 270373303, 722696570, 1936398635, 5198249550, 13982513625, 37674988080, 101685303765] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (4 t - 1) P + t (-3 + 7 t) P + t (-3 + 7 t) P + t (4 t - 1) P = 0 and in Maple input format 1+(4*t-1)*P+t*(-3+7*t)*P^2+t^2*(-3+7*t)*P^3+t^3*(4*t-1)*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 3 3 4 4 -Q + t Q + t Q + t Q + 1 = 0 and in Maple input format -Q+t^2*Q^2+t^3*Q^3+t^4*Q^4+1 = 0 Theorem Number, 14, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-3, -2, -1, 0, 1}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 2, 5, 14, 42, 131, 420, 1375, 4576, 15431, 52603, 180957, 627340, 2189430, 7685785, 27118855, 96123508, 342099955, 1221979374, 4379357895, 15742077045, 56742085710, 205041235750, 742647580815, 2695585363122, 9803561513316, 35720226039252, 130373533268780, 476607031823524, 1744949420105892, 6397589935451431] and the first 30 terms of B(n) are [1, 1, 2, 5, 14, 41, 125, 393, 1265, 4147, 13798, 46476, 158170, 543050, 1878670, 6542330, 22915999, 80682987, 285378270, 1013564805, 3613262795, 12924536005, 46373266470, 166856922125, 601928551824, 2176616383346, 7888184659826, 28645799759632, 104224861693855, 379885129946864, 1386926469714491] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (-1 + 5 t) P + t (-3 + 10 t) P + t (-3 + 10 t) P + t (-1 + 5 t) P = 0 and in Maple input format 1+(-1+5*t)*P+t*(-3+10*t)*P^2+t^2*(-3+10*t)*P^3+t^3*(-1+5*t)*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 3 3 4 4 1 + (t - 1) Q + t Q + t Q + t Q = 0 and in Maple input format 1+(t-1)*Q+t^2*Q^2+t^3*Q^3+t^4*Q^4 = 0 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, 2} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, 0, 2} Theorem Number, 15, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-1, 2}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 1, 2, 4, 7, 14, 28, 53, 106, 212, 412, 824, 1648, 3241, 6482, 12964, 25655, 51310, 102620, 203812, 407624, 815248, 1622744, 3245488, 6490976, 12938689, 25877378, 51754756, 103262837, 206525674, 413051348] and the first 30 terms of B(n) are [1, 0, 0, 1, 0, 0, 3, 0, 0, 12, 0, 0, 55, 0, 0, 273, 0, 0, 1428, 0, 0, 7752, 0, 0, 43263, 0, 0, 246675, 0, 0, 1430715] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 1 + (-1 + 3 t) P + 3 t (-1 + 2 t) P + t (-1 + 2 t) P = 0 and in Maple input format 1+(-1+3*t)*P+3*t*(-1+2*t)*P^2+t*(-1+2*t)^2*P^3 = 0 and Q=Q(t) satisfies the following algebraic equation 3 3 1 - Q + t Q = 0 and in Maple input format 1-Q+t^3*Q^3 = 0 Theorem Number, 16, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-1, 0, 2}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 2, 5, 14, 40, 115, 334, 978, 2877, 8490, 25121, 74492, 221264, 658104, 1959582, 5840343, 17420200, 51994267, 155275804, 463942247, 1386779219, 4146772454, 12403738865, 37112281769, 111068502374, 332475797690, 995438371814, 2980881596636, 8927771504402, 26742524394719, 80115508271702] and the first 30 terms of B(n) are [1, 1, 1, 2, 5, 11, 24, 57, 141, 349, 871, 2212, 5688, 14730, 38403, 100829, 266333, 706997, 1885165, 5047522, 13565203, 36578497, 98934826, 268342933, 729709432, 1989021256, 5433518806, 14873285506, 40790118487, 112064912455, 308390452661] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 1 + (-1 + 4 t) P + 3 t (-1 + 3 t) P + t (-1 + 3 t) P = 0 and in Maple input format 1+(-1+4*t)*P+3*t*(-1+3*t)*P^2+t*(-1+3*t)^2*P^3 = 0 and Q=Q(t) satisfies the following algebraic equation 3 3 1 + (-1 + t) Q + t Q = 0 and in Maple input format 1+(-1+t)*Q+t^3*Q^3 = 0 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, 2} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, 0, 2} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -1, 2} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -1, 0, 2} Theorem Number, 17, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-2, -1, 2}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 1, 3, 6, 16, 40, 103, 278, 726, 1994, 5351, 14719, 40368, 111304, 309102, 857134, 2395283, 6687149, 18756944, 52671482, 148215700, 418005810, 1180050981, 3338554839, 9453743939, 26813409447, 76130652939, 216402498694, 615812103260, 1753960391836, 5000735264430] and the first 30 terms of B(n) are [1, 0, 1, 1, 2, 7, 8, 38, 58, 199, 452, 1149, 3277, 7650, 22696, 55726, 157502, 416967, 1128026, 3122336, 8365304, 23402737, 63505268, 176860650, 487957967, 1353427722, 3774616133, 10483218667, 29371164344, 81965145468, 230030965231 ] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 1 + (-1 + 6 t) P + t (-7 + 17 t) P + t (37 t + 2 - 18 t) P 2 4 2 2 5 + t (-1 + 3 t) (-7 + 17 t) P + t (-1 + 6 t) (-1 + 3 t) P 3 3 6 + t (-1 + 3 t) P = 0 and in Maple input format 1+(-1+6*t)*P+t*(-7+17*t)*P^2+t*(37*t^2+2-18*t)*P^3+t^2*(-1+3*t)*(-7+17*t)*P^4+t ^2*(-1+6*t)*(-1+3*t)^2*P^5+t^3*(-1+3*t)^3*P^6 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 2 3 4 4 4 5 6 6 1 - Q - t Q + t (2 + t) Q - t Q - t Q + t Q = 0 and in Maple input format 1-Q-t^2*Q^2+t^2*(2+t)*Q^3-t^4*Q^4-t^4*Q^5+t^6*Q^6 = 0 Theorem Number, 18, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-2, -1, 0, 2}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 2, 6, 19, 63, 216, 755, 2680, 9623, 34866, 127257, 467293, 1724691, 6393308, 23789052, 88809758, 332512768, 1248190418, 4696360933, 17707253819, 66890649798, 253122673582, 959366755811, 3641419906864, 13840140394431, 52668462492448, 200660896823348, 765318940708460, 2921856889904855, 11165659939059786, 42706471956704738] and the first 30 terms of B(n) are [1, 1, 2, 5, 13, 38, 116, 368, 1203, 4016, 13642, 46987, 163696, 575816, 2042175, 7294299, 26215927, 94736708, 344015468, 1254647606, 4593682529, 16878510120, 62215957762, 230007985382, 852612196852, 3168359595108, 11800740083576, 44045606325107, 164721039237571, 617148978777583, 2316181581852586] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 1 + (7 t - 1) P + t (-7 + 24 t) P + t (2 - 22 t + 57 t ) P 2 4 2 2 5 + t (4 t - 1) (-7 + 24 t) P + t (7 t - 1) (4 t - 1) P 3 3 6 + t (4 t - 1) P = 0 and in Maple input format 1+(7*t-1)*P+t*(-7+24*t)*P^2+t*(2-22*t+57*t^2)*P^3+t^2*(4*t-1)*(-7+24*t)*P^4+t^2 *(7*t-1)*(4*t-1)^2*P^5+t^3*(4*t-1)^3*P^6 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 2 3 4 4 4 5 6 6 1 + (t - 1) Q - t Q - t (-2 + t) Q - t Q + t (t - 1) Q + t Q = 0 and in Maple input format 1+(t-1)*Q-t^2*Q^2-t^2*(-2+t)*Q^3-t^4*Q^4+t^4*(t-1)*Q^5+t^6*Q^6 = 0 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, -1, 2} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, -1, 0, 2} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, 0, 3} Theorem Number, 19, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-1, 3}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 1, 2, 4, 8, 15, 30, 60, 120, 236, 472, 944, 1888, 3754, 7508, 15016, 30032, 59924, 119848, 239696, 479392, 957815, 1915630, 3831260, 7662520, 15317956, 30635912, 61271824, 122543648, 245033476, 490066952] and the first 30 terms of B(n) are [1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 22, 0, 0, 0, 140, 0, 0, 0, 969, 0, 0, 0, 7084, 0, 0, 0, 53820, 0, 0] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (4 t - 1) P + 6 t (-1 + 2 t) P + 4 t (-1 + 2 t) P + t (-1 + 2 t) P = 0 and in Maple input format 1+(4*t-1)*P+6*t*(-1+2*t)*P^2+4*t*(-1+2*t)^2*P^3+t*(-1+2*t)^3*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 4 4 1 - Q + t Q = 0 and in Maple input format 1-Q+t^4*Q^4 = 0 Theorem Number, 20, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-1, 0, 3}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 2, 5, 14, 41, 121, 357, 1055, 3129, 9312, 27773, 82928, 247793, 740881, 2216493, 6634463, 19866273, 59505338, 178277877, 534225710, 1601126729, 4799420852, 14388106941, 43138138556, 129346661761, 387864271722, 1163136393045, 3488221407614, 10461566353001, 31376601429332, 94108597075949] and the first 30 terms of B(n) are [1, 1, 1, 1, 2, 6, 16, 36, 75, 163, 391, 991, 2498, 6150, 15016, 37116, 93481, 238137, 607921, 1550401, 3959335, 10155615, 26182267, 67753907, 175713561, 456422121, 1187771521, 3097869841, 8097629671, 21207212047, 55628797891] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (-1 + 5 t) P + 6 t (-1 + 3 t) P + 4 t (-1 + 3 t) P + t (-1 + 3 t) P = 0 and in Maple input format 1+(-1+5*t)*P+6*t*(-1+3*t)*P^2+4*t*(-1+3*t)^2*P^3+t*(-1+3*t)^3*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 4 4 1 + (t - 1) Q + t Q = 0 and in Maple input format 1+(t-1)*Q+t^4*Q^4 = 0 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, 0, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -1, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -1, 0, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, -1, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, -1, 0, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, -1, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, -1, 0, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, 1, 2} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, 0, 1, 2} Theorem Number, 21, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-2, 1, 2}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 2, 5, 14, 38, 110, 313, 911, 2653, 7761, 22832, 67174, 198525, 586886, 1739137, 5158679, 15318781, 45540713, 135471010, 403357000, 1201586864, 3581782768, 10681943955, 31871082167, 95131856283, 284056425778, 848465193837, 2535055660918, 7576384832642, 22648674094492, 67720746798648] and the first 30 terms of B(n) are [1, 0, 1, 1, 2, 7, 8, 38, 58, 199, 452, 1149, 3277, 7650, 22696, 55726, 157502, 416967, 1128026, 3122336, 8365304, 23402737, 63505268, 176860650, 487957967, 1353427722, 3774616133, 10483218667, 29371164344, 81965145468, 230030965231 ] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 1 + (9 t - 1) P + t (-9 + 32 t) P + t (2 - 29 t + 70 t ) P 2 4 2 2 5 + t (-9 + 32 t) (-1 + 3 t) P + t (9 t - 1) (-1 + 3 t) P 3 3 6 + t (-1 + 3 t) P = 0 and in Maple input format 1+(9*t-1)*P+t*(-9+32*t)*P^2+t*(2-29*t+70*t^2)*P^3+t^2*(-9+32*t)*(-1+3*t)*P^4+t^ 2*(9*t-1)*(-1+3*t)^2*P^5+t^3*(-1+3*t)^3*P^6 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 2 3 4 4 4 5 6 6 1 - Q - t Q + t (2 + t) Q - t Q - t Q + t Q = 0 and in Maple input format 1-Q-t^2*Q^2+t^2*(2+t)*Q^3-t^4*Q^4-t^4*Q^5+t^6*Q^6 = 0 Theorem Number, 22, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-2, 0, 1, 2}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 3, 10, 36, 133, 501, 1911, 7352, 28466, 110749, 432503, 1694130, 6652269, 26174306, 103162294, 407188481, 1609196840, 6366333545, 25210206977, 99912681474, 396260120938, 1572607862333, 6244688766609, 24809935488100, 98614926298342, 392141197189684, 1559940874527078, 6207614980707830, 24710369522689881, 98392025435036917, 391882978051814482] and the first 30 terms of B(n) are [1, 1, 2, 5, 13, 38, 116, 368, 1203, 4016, 13642, 46987, 163696, 575816, 2042175, 7294299, 26215927, 94736708, 344015468, 1254647606, 4593682529, 16878510120, 62215957762, 230007985382, 852612196852, 3168359595108, 11800740083576, 44045606325107, 164721039237571, 617148978777583, 2316181581852586] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 1 + (-1 + 10 t) P + t (-9 + 41 t) P + t (101 t - 33 t + 2) P 2 4 2 2 5 + t (-9 + 41 t) (4 t - 1) P + t (-1 + 10 t) (4 t - 1) P 3 3 6 + t (4 t - 1) P = 0 and in Maple input format 1+(-1+10*t)*P+t*(-9+41*t)*P^2+t*(101*t^2-33*t+2)*P^3+t^2*(-9+41*t)*(4*t-1)*P^4+ t^2*(-1+10*t)*(4*t-1)^2*P^5+t^3*(4*t-1)^3*P^6 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 2 3 4 4 4 5 6 6 1 + (-1 + t) Q - t Q - t (-2 + t) Q - t Q + t (-1 + t) Q + t Q = 0 and in Maple input format 1+(-1+t)*Q-t^2*Q^2-t^2*(-2+t)*Q^3-t^4*Q^4+t^4*(-1+t)*Q^5+t^6*Q^6 = 0 Theorem Number, 23, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-1, 1, 2}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 2, 6, 17, 50, 148, 439, 1309, 3906, 11676, 34932, 104574, 313227, 938504, 2812795, 8431950, 25280562, 75805312, 227328420, 681774766, 2044814604, 6133206076, 18396603346, 55182439178, 165529257635, 496543393370, 1489520882040, 4468292879465, 13404211413915, 40210980975180, 120628839014610] and the first 30 terms of B(n) are [1, 0, 1, 1, 2, 5, 8, 21, 42, 96, 222, 495, 1177, 2717, 6435, 15288, 36374, 87516, 210494, 509694, 1237736, 3014882, 7370860, 18059899, 44379535, 109298070, 269766655, 667224480, 1653266565, 4103910930, 10203669285] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 1 + (-1 + 5 t) P + 4 t (-1 + 3 t) P + t (-1 + 3 t) P = 0 and in Maple input format 1+(-1+5*t)*P+4*t*(-1+3*t)*P^2+t*(-1+3*t)^2*P^3 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 3 3 1 - Q + t Q + t Q = 0 and in Maple input format 1-Q+t^2*Q^2+t^3*Q^3 = 0 Theorem Number, 24, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-1, 0, 1, 2}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 3, 11, 42, 163, 639, 2520, 9976, 39595, 157441, 626859, 2498318, 9964308, 39764292, 158756360, 634044576, 2532959099, 10121210373, 40449588625, 161680869046, 646330342699, 2584001013295, 10331554755044, 41311180250504, 165193747735636, 660601776520476, 2641817108038628, 10565254405283016, 42254128565943032, 168992905541421688, 675890572986846224] and the first 30 terms of B(n) are [1, 1, 2, 5, 13, 36, 104, 309, 939, 2905, 9118, 28964, 92940, 300808, 980864, 3219205, 10626023, 35252867, 117485454, 393133485, 1320357501, 4449298136, 15038769672, 50973266380, 173214422068, 589998043276, 2014026871496, 6889055189032, 23608722350440, 81049178840528, 278700885572096] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 1 + (-1 + 6 t) P + 4 t (4 t - 1) P + t (4 t - 1) P = 0 and in Maple input format 1+(-1+6*t)*P+4*t*(4*t-1)*P^2+t*(4*t-1)^2*P^3 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 3 3 1 + (-1 + t) Q + t Q + t Q = 0 and in Maple input format 1+(-1+t)*Q+t^2*Q^2+t^3*Q^3 = 0 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, 1, 2} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, 0, 1, 2} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -1, 1, 2} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -1, 0, 1, 2} Theorem Number, 25, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-2, -1, 1, 2}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 2, 7, 23, 83, 299, 1107, 4122, 15523, 58769, 223848, 856085, 3286687, 12656513, 48871469, 189145479, 733547091, 2849962925, 11090427510, 43219527353, 168645172164, 658834266936, 2576566240218, 10086236606187, 39518897727591, 154966364793099, 608134677522622, 2388182454435185, 9384669282073545, 36900709434538019, 145176798536003323] and the first 30 terms of B(n) are [1, 0, 2, 2, 11, 24, 93, 272, 971, 3194, 11293, 39148, 139687, 497756, 1798002, 6517194, 23807731, 87336870, 322082967, 1192381270, 4431889344, 16527495396, 61831374003, 231973133544, 872598922407, 3290312724374, 12434632908623, 47089829065940, 178672856753641, 679155439400068, 2585880086336653] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 2 2 4 1 + (4 t - 1) P + 3 t (4 t - 1) P + t (4 t - 1) P + t (4 t - 1) P = 0 and in Maple input format 1+(4*t-1)*P+3*t*(4*t-1)*P^2+t*(4*t-1)^2*P^3+t^2*(4*t-1)^2*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 3 4 4 1 + (-2 t - 1) Q + t (2 + 3 t) Q - t (1 + 2 t) Q + t Q = 0 and in Maple input format 1+(-2*t-1)*Q+t*(2+3*t)*Q^2-t^2*(1+2*t)*Q^3+t^4*Q^4 = 0 Theorem Number, 26, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-2, -1, 0, 1, 2}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 3, 12, 51, 226, 1025, 4724, 22022, 103550, 490191, 2333057, 11153428, 53517672, 257600811, 1243270807, 6014482911, 29155412470, 141587439283, 688697290541, 3354726880350, 16362475911945, 79900712888835, 390585409850935, 1911195106959460, 9360145073678716, 45879404384126585, 225052186125474359, 1104726156237052170, 5426380342980842891, 26670460472470002517, 131158947643816147077] and the first 30 terms of B(n) are [1, 1, 3, 9, 32, 120, 473, 1925, 8034, 34188, 147787, 647141, 2864508, 12796238, 57615322, 261197436, 1191268350, 5462080688, 25162978925, 116414836445, 540648963645, 2519574506595, 11779011525030, 55225888341334, 259612579655392, 1223396051745310, 5778116086462293, 27347124593409513, 129681868681425643, 616072123886855885, 2931681447103047687] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 2 2 4 1 + (-1 + 5 t) P + 3 t (-1 + 5 t) P + t (-1 + 5 t) P + t (-1 + 5 t) P = 0 and in Maple input format 1+(-1+5*t)*P+3*t*(-1+5*t)*P^2+t*(-1+5*t)^2*P^3+t^2*(-1+5*t)^2*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 3 4 4 1 + (-1 - t) Q + t (2 + t) Q - t (1 + t) Q + t Q = 0 and in Maple input format 1+(-1-t)*Q+t*(2+t)*Q^2-t^2*(1+t)*Q^3+t^4*Q^4 = 0 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, -1, 1, 2} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, -1, 0, 1, 2} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, 1, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, 0, 1, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, 1, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, 0, 1, 3} Theorem Number, 27, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-1, 1, 3}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 2, 6, 17, 51, 150, 450, 1339, 4017, 12005, 36015, 107838, 323514, 969563, 2908689, 8721270, 26163810, 78467292, 235401876, 706081630, 2118244890, 6354087055, 19062261165, 57183355002, 171550065006, 514631838304, 1543895514912, 4631587315721, 13894761947163, 41683745034324, 125051235102972] and the first 30 terms of B(n) are [1, 0, 1, 0, 3, 0, 11, 0, 46, 0, 207, 0, 979, 0, 4797, 0, 24138, 0, 123998, 0, 647615, 0, 3428493, 0, 18356714, 0, 99229015, 0, 540807165, 0, 2968468275] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (6 t - 1) P + 7 t (-1 + 3 t) P + 4 t (-1 + 3 t) P + t (-1 + 3 t) P = 0 and in Maple input format 1+(6*t-1)*P+7*t*(-1+3*t)*P^2+4*t*(-1+3*t)^2*P^3+t*(-1+3*t)^3*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 4 4 -Q + 1 + t Q + t Q = 0 and in Maple input format -Q+1+t^2*Q^2+t^4*Q^4 = 0 Theorem Number, 28, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-1, 0, 1, 3}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 3, 11, 42, 164, 646, 2558, 10160, 40436, 161151, 642851, 2566141, 10248569, 40945149, 163628365, 654038101, 2614658117, 10453907565, 41800612493, 167154495572, 668464217966, 2673361979471, 10691849676015, 42762221211352, 171032064797402, 684073473739905, 2736115021940149, 10943874775746150, 43773579890780948, 175088014508550553, 700331307527249537] and the first 30 terms of B(n) are [1, 1, 2, 4, 10, 26, 72, 204, 593, 1753, 5263, 15995, 49127, 152231, 475359, 1494287, 4724903, 15017767, 47954400, 153764322, 494892393, 1598241869, 5177492708, 16820048006, 54785449703, 178873019471, 585312014446, 1919212203652, 6305054573239, 20750506952675, 68405627426506] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (7 t - 1) P + 7 t (-1 + 4 t) P + 4 t (-1 + 4 t) P + t (-1 + 4 t) P = 0 and in Maple input format 1+(7*t-1)*P+7*t*(-1+4*t)*P^2+4*t*(-1+4*t)^2*P^3+t*(-1+4*t)^3*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 4 4 1 + (t - 1) Q + t Q + t Q = 0 and in Maple input format 1+(t-1)*Q+t^2*Q^2+t^4*Q^4 = 0 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, 1, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, 0, 1, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -1, 1, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -1, 0, 1, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, -1, 1, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, -1, 0, 1, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, -1, 1, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, -1, 0, 1, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, 0, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, 0, 2, 3} Theorem Number, 29, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-1, 2, 3}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 2, 6, 18, 53, 158, 474, 1419, 4250, 12746, 38226, 114633, 343844, 1031455, 3094079, 9281691, 27844345, 83531075, 250588465, 751757847, 2255258037, 6765734382, 20297127564, 60891242244, 182673390242, 548019447399, 1644056975997, 4932167935001, 14796497046108, 44389477616049, 133168404754107] and the first 30 terms of B(n) are [1, 0, 0, 1, 1, 0, 3, 7, 4, 12, 45, 55, 77, 286, 546, 728, 1960, 4760, 7548, 15504, 39729, 75582, 140448, 336490, 723327, 1366200, 2992990, 6758895, 13522275, 28094040, 63183315] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (7 t - 1) P + 9 t (-1 + 3 t) P + 5 t (-1 + 3 t) P + t (-1 + 3 t) P = 0 and in Maple input format 1+(7*t-1)*P+9*t*(-1+3*t)*P^2+5*t*(-1+3*t)^2*P^3+t*(-1+3*t)^3*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 4 4 3 3 1 - Q + t Q + t Q = 0 and in Maple input format 1-Q+t^4*Q^4+t^3*Q^3 = 0 Theorem Number, 30, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-1, 0, 2, 3}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 3, 11, 43, 170, 674, 2680, 10681, 42625, 170229, 680153, 2718466, 10867802, 43453849, 173765059, 694913179, 2779220842, 11115608095, 44458645731, 177823284893, 711259293370, 2844935410543, 11379434644351, 45516809538886, 182064418728856, 728249096482120, 2912970222866704, 11651800921280025, 46606958746279004, 186427083338156614, 745706022669230400] and the first 30 terms of B(n) are [1, 1, 1, 2, 6, 16, 39, 99, 271, 763, 2146, 6062, 17359, 50337, 147057, 431874, 1275273, 3786649, 11298031, 33846202, 101762937, 306997821, 929038518, 2819426688, 8578433304, 26163061776, 79970186791, 244938841096, 751646959402, 2310683396056, 7115199919151] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (-1 + 8 t) P + 9 t (-1 + 4 t) P + 5 t (-1 + 4 t) P + t (-1 + 4 t) P = 0 and in Maple input format 1+(-1+8*t)*P+9*t*(-1+4*t)*P^2+5*t*(-1+4*t)^2*P^3+t*(-1+4*t)^3*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 3 3 4 4 1 + (t - 1) Q + t Q + t Q = 0 and in Maple input format 1+(t-1)*Q+t^3*Q^3+t^4*Q^4 = 0 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, 0, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -1, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -1, 0, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, -1, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, -1, 0, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, -1, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, -1, 0, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, 1, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, 0, 1, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, 1, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, 0, 1, 2, 3} Theorem Number, 31, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-1, 1, 2, 3}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 3, 12, 47, 187, 745, 2975, 11886, 47516, 189990, 759792, 3038736, 12153899, 48612912, 194444927, 777762353, 3111005004, 12443904514, 49775318244, 199100487406, 796399889530, 3185594123645, 12742362131739, 50969410412196, 203877540288382, 815509890780225, 3262038840424330, 13048153425298685, 52192608502945190, 208770420029267135, 835081642442080460] and the first 30 terms of B(n) are [1, 0, 1, 1, 3, 5, 14, 28, 74, 168, 432, 1045, 2684, 6721, 17355, 44408, 115502, 299812, 785570, 2060094, 5434475, 14362841, 38114760, 101360402, 270373303, 722696570, 1936398635, 5198249550, 13982513625, 37674988080, 101685303765] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (-1 + 9 t) P + 10 t (-1 + 4 t) P + 5 t (-1 + 4 t) P + t (-1 + 4 t) P = 0 and in Maple input format 1+(-1+9*t)*P+10*t*(-1+4*t)*P^2+5*t*(-1+4*t)^2*P^3+t*(-1+4*t)^3*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 4 4 3 3 2 2 -Q + t Q + t Q + t Q + 1 = 0 and in Maple input format -Q+t^4*Q^4+t^3*Q^3+t^2*Q^2+1 = 0 Theorem Number, 32, : Let A(n) be the total number of walks, starting at 0 with exactly n steps drawn from the set of steps, {-1, 0, 1, 2, 3}, always staying in the half-line x>=0 and ending anywhere, and let B(n) be the number of such walks that also end \ at 0. For the sake of the OEIS, the first 30 terms of A(n), strarting at n=0 are [1, 4, 19, 93, 460, 2286, 11389, 56820, 283707, 1417270, 7082203, 35397217, 176939609, 884539875, 4422156325, 22108902955, 110537972445, 552666946226, 2763254048143, 13815984862445, 69078910747420, 345390940474305, 1726941777835520, 8634662515911130, 43173145722633525, 215865126684615801, 1079323456806695659, 5396609395848818469, 26983018333444332713, 134914987442359969710, 674574557326669901686] and the first 30 terms of B(n) are [1, 1, 2, 5, 14, 41, 125, 393, 1265, 4147, 13798, 46476, 158170, 543050, 1878670, 6542330, 22915999, 80682987, 285378270, 1013564805, 3613262795, 12924536005, 46373266470, 166856922125, 601928551824, 2176616383346, 7888184659826, 28645799759632, 104224861693855, 379885129946864, 1386926469714491] Let : infinity ----- \ n P(t) = ) A[n] t / ----- n = 0 infinity ----- \ n Q(t) = ) B[n] t / ----- n = 0 Then P=P(t) satisfies the following algebraic equation 2 2 3 3 4 1 + (-1 + 10 t) P + 10 t (-1 + 5 t) P + 5 t (-1 + 5 t) P + t (-1 + 5 t) P = 0 and in Maple input format 1+(-1+10*t)*P+10*t*(-1+5*t)*P^2+5*t*(-1+5*t)^2*P^3+t*(-1+5*t)^3*P^4 = 0 and Q=Q(t) satisfies the following algebraic equation 2 2 3 3 4 4 1 + (t - 1) Q + t Q + t Q + t Q = 0 and in Maple input format 1+(t-1)*Q+t^2*Q^2+t^3*Q^3+t^4*Q^4 = 0 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, 1, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, 0, 1, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -1, 1, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -1, 0, 1, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, -1, 1, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-2, -1, 0, 1, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, -1, 1, 2, 3} 70, terms do not suffice to find algebraic equations for the generating func\ tions when the step-set is, {-3, -2, -1, 0, 1, 2, 3} This ends this webbook that took, 129950.654, to generate