On the Generating Functions enumerating walks confined to the Non-Negative H\ alf Line With step-sets that are subsets of , {-2, -1, 0, 1, 2} 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 {-2, -1, 0, 1, 2} 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, {-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, 2, : 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 + (-1 + 4 t) P + t (-2 + 5 t) P + t (-1 + 3 t) P = 0 and in Maple input format 1+(-1+4*t)*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 + (-1 + t) Q + t Q = 0 and in Maple input format 1+(-1+t)*Q+t^3*Q^3 = 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, {-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, 4, : 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 + (-1 + t) Q + t Q = 0 and in Maple input format 1+(-1+t)*Q+t^2*Q^2 = 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, {-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, 6, : 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 + (-1 + 4 t) P + 2 t (-1 + 3 t) P + t (-1 + 4 t) P = 0 and in Maple input format 1+(-1+4*t)*P+2*t*(-1+3*t)*P^2+t^2*(-1+4*t)*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 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, {-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, 8, : 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 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, {-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 + (6 t - 1) 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 (6 t - 1) (-1 + 3 t) P 3 3 6 + t (-1 + 3 t) P = 0 and in Maple input format 1+(6*t-1)*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*(6*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, 10, : 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 (57 t + 2 - 22 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*(57*t^2+2-22*t)*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 + (-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, 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, 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) (3 t - 1) P + t (9 t - 1) (3 t - 1) P 3 3 6 + t (3 t - 1) 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)*(3*t-1)*P^4+t^2 *(9*t-1)*(3*t-1)^2*P^5+t^3*(3*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 - 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, 12, : 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 (2 - 33 t + 101 t ) 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*(2-33*t+101*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, 13, : 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 (3 t - 1) P + t (3 t - 1) P = 0 and in Maple input format 1+(-1+5*t)*P+4*t*(3*t-1)*P^2+t*(3*t-1)^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, 14, : 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 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, {-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 + (-1 - 2 t) Q + t (2 + 3 t) Q - t (1 + 2 t) Q + t Q = 0 and in Maple input format 1+(-1-2*t)*Q+t*(2+3*t)*Q^2-t^2*(1+2*t)*Q^3+t^4*Q^4 = 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, {-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 This ends this webbook that took, 670.268, to generate