Algebraic Equations satisfied by walks with given negative and positive steps that start and end at the x-axis and never go above the x-axis By Shalosh B. Ekhad In this book we will present algebraic equations satisfied by the ordinary g\ enerating function of the sequences enumerating ALL families of walks consisting of at least one negative step, at least on\ e positive steps where the size of each step is between 1 and, 3 and that start and end at the x-axis, but never go above the x-axis We also supply the first, 40, terms of the enumerating sequence --------------------------------------------------------- Proposition No., 1 Let a(n) be the number of sequences of length n, in the alphabet, {-1, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 0, 35357670, 0, 129644790, 0, 477638700, 0, 1767263190, 0, 6564120420] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 2 2 f t - f + 1 = 0 and in Maple notation f^2*t^2-f+1 = 0 ---------------------------------------- This ends Proposition, 1, that took, 0.107, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 2 Let a(n) be the number of sequences of length n, in the alphabet, {-1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 0, 0, 8414640, 0, 0, 50067108, 0, 0, 300830572, 0] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 3 3 f t - f + 1 = 0 and in Maple notation f^3*t^3-f+1 = 0 ---------------------------------------- This ends Proposition, 2, that took, 0.112, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 3 Let a(n) be the number of sequences of length n, in the alphabet, {-1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 0, 420732, 0, 0, 0, 3362260, 0, 0, 0, 27343888] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 4 4 f t - f + 1 = 0 and in Maple notation f^4*t^4-f+1 = 0 ---------------------------------------- This ends Proposition, 3, that took, 0.120, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 4 Let a(n) be the number of sequences of length n, in the alphabet, {-1, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 25408828065, 63364046190, 158229645720, 395632288590, 990419552730, 2482238709888, 6227850849066, 15641497455612, 39322596749218, 98948326105928] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 3 3 2 2 f t + f t - f + 1 = 0 and in Maple notation f^3*t^3+f^2*t^2-f+1 = 0 ---------------------------------------- This ends Proposition, 4, that took, 0.127, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 5 Let a(n) be the number of sequences of length n, in the alphabet, {-1, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 0, 16395456762, 0, 91053897066, 0, 508151297602, 0, 2848290555562, 0, 16028132445156] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 4 4 2 2 f t + f t - f + 1 = 0 and in Maple notation f^4*t^4+f^2*t^2-f+1 = 0 ---------------------------------------- This ends Proposition, 5, that took, 0.135, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 6 Let a(n) be the number of sequences of length n, in the alphabet, {-1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 133231800, 273896532, 600805296, 1305229332, 2720740792, 5843241088, 12797739672, 27206642716, 57941746476, 126405822608] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 4 4 3 3 f t + f t - f + 1 = 0 and in Maple notation f^4*t^4+f^3*t^3-f+1 = 0 ---------------------------------------- This ends Proposition, 6, that took, 0.147, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 7 Let a(n) be the number of sequences of length n, in the alphabet, {-1, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 274867141845, 744093631842, 2017066320624, 5474900965050, 14878450339822, 40479971557162, 110253945275970, 300605644859552, 820399033872096, 2241084167717824] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 4 4 3 3 2 2 f t + f t + f t - f + 1 = 0 and in Maple notation f^4*t^4+f^3*t^3+f^2*t^2-f+1 = 0 ---------------------------------------- This ends Proposition, 7, that took, 0.159, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 8 Let a(n) be the number of sequences of length n, in the alphabet, {-2, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 0, 0, 8414640, 0, 0, 50067108, 0, 0, 300830572, 0] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 3 3 f t - f + 1 = 0 and in Maple notation f^3*t^3-f+1 = 0 ---------------------------------------- This ends Proposition, 8, that took, 0.163, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 9 Let a(n) be the number of sequences of length n, in the alphabet, {-2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 0, 2, 0, 0, 0, 0, 23, 0, 0, 0, 0, 377, 0, 0, 0, 0, 7229, 0, 0, 0, 0, 151491, 0, 0, 0, 0, 3361598, 0, 0, 0, 0, 77635093, 0, 0, 0, 0, 1846620581] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 7 5 6 5 5 5 f t + f t - f t + 2 f t - f + 1 = 0 and in Maple notation f^10*t^10+f^7*t^5-f^6*t^5+2*f^5*t^5-f+1 = 0 ---------------------------------------- This ends Proposition, 9, that took, 0.196, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 10 Let a(n) be the number of sequences of length n, in the alphabet, {-2, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 645265199252, 1813615497166, 5107394107927, 14386545035342, 40621735594210, 114720169872202, 324560293765296, 918870098708832, 2604241833793991, 7388579097551618] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 6 6 4 5 4 4 3 2 3 2 2 f t - t f - t f + (t + 2 t ) f - f t - f + 1 = 0 and in Maple notation f^6*t^6-t^4*f^5-t^4*f^4+(t^3+2*t^2)*f^3-f^2*t^2-f+1 = 0 ---------------------------------------- This ends Proposition, 10, that took, 0.220, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 11 Let a(n) be the number of sequences of length n, in the alphabet, {-2, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 3, 2, 3, 23, 59, 74, 178, 753, 1859, 3299, 8937, 29884, 73955, 160368 , 445889, 1334825, 3371535, 8167687, 22732271, 64550448, 166944853, 429281385, 1189787311, 3299504856, 8708248080, 23118437489, 63845014804, 175463878127, 470269479575, 1270311652558, 3501884445317, 9604857045847, 26027895342456, 71002490056153, 195692892371919, 537321155970160, 1467430337299719] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 9 9 5 7 5 6 5 4 5 4 4 3 3 f t + f t + t f - t f + (2 t + 2 t ) f + t f + t f - f + 1 = 0 and in Maple notation f^10*t^10+f^9*t^9+t^5*f^7-t^5*f^6+(2*t^5+2*t^4)*f^5+t^4*f^4+t^3*f^3-f+1 = 0 ---------------------------------------- This ends Proposition, 11, that took, 0.293, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 12 Let a(n) be the number of sequences of length n, in the alphabet, {-2, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 2, 5, 17, 14, 103, 65, 544, 515, 2671, 4333, 12920, 32888, 66569, 225063, 389929, 1426875, 2581052, 8652846, 18130991, 51937472, 127733905, 318505753, 879213643, 2034543521, 5892047281, 13539791786, 38764350879, 92547902870, 253609842517, 638716733669, 1670011621498, 4398787899731, 11151727980457, 30093346643625, 75616292374270, 204712934528781] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 7 8 7 5 7 6 5 6 5 4 5 4 4 f t - t f + (-t + t ) f + (t - t ) f + (2 t - t ) f - t f 2 3 2 2 + 2 t f - f t - f + 1 = 0 and in Maple notation f^10*t^10-t^7*f^8+(-t^7+t^5)*f^7+(t^6-t^5)*f^6+(2*t^5-t^4)*f^5-t^4*f^4+2*t^2*f^ 3-f^2*t^2-f+1 = 0 ---------------------------------------- This ends Proposition, 12, that took, 0.350, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 13 Let a(n) be the number of sequences of length n, in the alphabet, {-2, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 5, 9, 31, 78, 248, 705, 2196, 6632, 20780, 64709, 204902, 650000, 2080483, 6683564, 21593311, 70024903, 228022074, 744976876, 2441850778, 8026618762, 26455041139, 87405982153, 289438774174, 960462359139, 3193366842536 , 10636635056279, 35489063311272, 118596791583351, 396914141297320, 1330230442462987, 4464042344334714, 14999217181926990, 50456596364848778, 169921812232536963, 572844723715864685, 1933116776188266041, 6529668152176835624] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 9 9 7 8 7 5 7 6 5 6 5 4 5 t f + t f - t f + (-t + t ) f + (-2 t - t ) f + (2 t + t ) f 3 2 3 2 2 + (t + 2 t ) f - t f - f + 1 = 0 and in Maple notation t^10*f^10+t^9*f^9-t^7*f^8+(-t^7+t^5)*f^7+(-2*t^6-t^5)*f^6+(2*t^5+t^4)*f^5+(t^3+ 2*t^2)*f^3-t^2*f^2-f+1 = 0 ---------------------------------------- This ends Proposition, 13, that took, 0.417, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 14 Let a(n) be the number of sequences of length n, in the alphabet, {-3, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 0, 420732, 0, 0, 0, 3362260, 0, 0, 0, 27343888] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 4 4 f t - f + 1 = 0 and in Maple notation f^4*t^4-f+1 = 0 ---------------------------------------- This ends Proposition, 14, that took, 0.424, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 15 Let a(n) be the number of sequences of length n, in the alphabet, {-3, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 0, 2, 0, 0, 0, 0, 23, 0, 0, 0, 0, 377, 0, 0, 0, 0, 7229, 0, 0, 0, 0, 151491, 0, 0, 0, 0, 3361598, 0, 0, 0, 0, 77635093, 0, 0, 0, 0, 1846620581] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 7 5 6 5 5 5 f t + f t - f t + 2 f t - f + 1 = 0 and in Maple notation f^10*t^10+f^7*t^5-f^6*t^5+2*f^5*t^5-f+1 = 0 ---------------------------------------- This ends Proposition, 15, that took, 0.457, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 16 Let a(n) be the number of sequences of length n, in the alphabet, {-3, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 2, 1, 2, 17, 17, 43, 220, 322, 877, 3495, 6513, 18246, 63069, 137364, 389520, 1240075, 2986569, 8518188, 25878573, 66493272, 190276431, 563345305, 1509236554, 4329167366, 12645267502, 34810974533, 100065738510, 290410780163, 813932210810, 2344530239608, 6787557305833, 19254309739598, 55576193661986, 160849076903780, 460095808260232, 1330726621028529, 3854609838686679, 11091289883698738] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 8 8 5 7 6 5 6 5 5 4 3 4 3 3 f t - f t + t f + (-t - t ) f + 2 t f + (t + 3 t ) f - t f - f + 1 = 0 and in Maple notation f^10*t^10-f^8*t^8+t^5*f^7+(-t^6-t^5)*f^6+2*t^5*f^5+(t^4+3*t^3)*f^4-t^3*f^3-f+1 = 0 ---------------------------------------- This ends Proposition, 16, that took, 0.513, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 17 Let a(n) be the number of sequences of length n, in the alphabet, {-3, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 3, 0, 16, 0, 100, 0, 655, 0, 4465, 0, 31599, 0, 230390, 0, 1717910, 0 , 13034753, 0, 100308732, 0, 781057488, 0, 6142515700, 0, 48719605150, 0, 389274014325, 0, 3130375135624, 0, 25315962247754, 0, 205765906922296, 0, 1679968849194124, 0, 13771490153093158] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 20 20 18 19 18 18 16 17 16 14 16 14 15 t f - t f + t f - 3 t f + (t + 3 t ) f - 3 t f 14 12 14 12 10 13 12 10 12 + (5 t + 3 t ) f + (-3 t - 3 t ) f + (2 t + 3 t ) f 10 8 11 10 8 6 10 8 6 9 + (-6 t - t ) f + (-4 t + 6 t + t ) f + (-6 t - t ) f 8 6 8 6 4 7 6 4 6 4 5 + (2 t + 3 t ) f + (-3 t - 3 t ) f + (5 t + 3 t ) f - 3 t f 4 2 4 3 2 2 2 + (t + 3 t ) f - 3 f t + f t - f + 1 = 0 and in Maple notation t^20*f^20-t^18*f^19+t^18*f^18-3*t^16*f^17+(t^16+3*t^14)*f^16-3*t^14*f^15+(5*t^ 14+3*t^12)*f^14+(-3*t^12-3*t^10)*f^13+(2*t^12+3*t^10)*f^12+(-6*t^10-t^8)*f^11+( -4*t^10+6*t^8+t^6)*f^10+(-6*t^8-t^6)*f^9+(2*t^8+3*t^6)*f^8+(-3*t^6-3*t^4)*f^7+( 5*t^6+3*t^4)*f^6-3*t^4*f^5+(t^4+3*t^2)*f^4-3*f^3*t^2+f^2*t^2-f+1 = 0 ---------------------------------------- This ends Proposition, 17, that took, 0.746, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 18 Let a(n) be the number of sequences of length n, in the alphabet, {-3, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 2, 5, 22, 14, 164, 65, 1030, 657, 5868, 7463, 31765, 73575, 173849 , 631556, 1053086, 4877803, 7526655, 34948691, 61382672, 239407864, 524309309, 1621763388, 4415274965, 11255289437, 35813332389, 82153463817, 279458110861, 633479334487, 2118070224696, 5075741777630, 15824514104397, 41275863623366, 118428013850973, 334523141763061, 899513350738990, 2678023253678681] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 20 20 18 19 18 18 16 17 14 16 15 14 15 t f - t f + t f - 3 t f + 3 t f + (2 t - 3 t ) f 14 13 12 14 13 12 11 10 13 + (3 t - t + 3 t ) f + (t - 3 t + t - 3 t ) f 12 10 12 11 10 8 11 + (3 t + 3 t ) f + (t - 6 t - t ) f 10 9 8 6 10 9 8 6 9 8 6 8 + (t - 6 t + 6 t + t ) f + (t - 6 t - t ) f + (3 t + 3 t ) f 7 6 5 4 7 6 5 4 6 5 4 5 + (t - 3 t + t - 3 t ) f + (3 t - t + 3 t ) f + (2 t - 3 t ) f 2 4 2 3 2 2 + 3 t f - 3 t f + t f - f + 1 = 0 and in Maple notation t^20*f^20-t^18*f^19+t^18*f^18-3*t^16*f^17+3*t^14*f^16+(2*t^15-3*t^14)*f^15+(3*t ^14-t^13+3*t^12)*f^14+(t^13-3*t^12+t^11-3*t^10)*f^13+(3*t^12+3*t^10)*f^12+(t^11 -6*t^10-t^8)*f^11+(t^10-6*t^9+6*t^8+t^6)*f^10+(t^9-6*t^8-t^6)*f^9+(3*t^8+3*t^6) *f^8+(t^7-3*t^6+t^5-3*t^4)*f^7+(3*t^6-t^5+3*t^4)*f^6+(2*t^5-3*t^4)*f^5+3*t^2*f^ 4-3*t^2*f^3+t^2*f^2-f+1 = 0 ---------------------------------------- This ends Proposition, 18, that took, 1.159, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 19 Let a(n) be the number of sequences of length n, in the alphabet, {-3, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 3, 16, 33, 115, 390, 1087, 4060, 12555, 42953, 148067, 492739, 1735298, 5944320, 20744252, 72905575, 254998049, 903660769, 3195209422, 11355589072, 40507136044, 144620988953, 518478617875, 1861257943227, 6697455408050, 24152234870325, 87226107921628, 315651869078757, 1143924927595869, 4151936835886485, 15091681888691404, 54925223488389666, 200157938880285184, 730258764785275647, 2667274260621421838, 9752675597285646950, 35695329641773808896, 130773052695581564343] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 20 20 18 19 18 18 17 16 17 16 15 14 16 t f - t f + t f + (-t - 3 t ) f + (t + 3 t + 3 t ) f 15 14 15 14 13 12 14 + (-2 t - 3 t ) f + (4 t + t + 3 t ) f 13 12 11 10 13 12 11 10 12 + (-5 t - 3 t - 2 t - 3 t ) f + (5 t + 5 t + 3 t ) f 11 10 9 8 11 10 9 8 6 10 + (-4 t - 6 t - t - t ) f + (t + 6 t + 6 t + t ) f 9 8 7 6 9 8 7 6 8 + (-4 t - 6 t - t - t ) f + (5 t + 5 t + 3 t ) f 7 6 5 4 7 6 5 4 6 + (-5 t - 3 t - 2 t - 3 t ) f + (4 t + t + 3 t ) f 5 4 5 4 3 2 4 3 2 3 2 2 + (-2 t - 3 t ) f + (t + 3 t + 3 t ) f + (-t - 3 t ) f + t f - f + 1 = 0 and in Maple notation t^20*f^20-t^18*f^19+t^18*f^18+(-t^17-3*t^16)*f^17+(t^16+3*t^15+3*t^14)*f^16+(-2 *t^15-3*t^14)*f^15+(4*t^14+t^13+3*t^12)*f^14+(-5*t^13-3*t^12-2*t^11-3*t^10)*f^ 13+(5*t^12+5*t^11+3*t^10)*f^12+(-4*t^11-6*t^10-t^9-t^8)*f^11+(t^10+6*t^9+6*t^8+ t^6)*f^10+(-4*t^9-6*t^8-t^7-t^6)*f^9+(5*t^8+5*t^7+3*t^6)*f^8+(-5*t^7-3*t^6-2*t^ 5-3*t^4)*f^7+(4*t^6+t^5+3*t^4)*f^6+(-2*t^5-3*t^4)*f^5+(t^4+3*t^3+3*t^2)*f^4+(-t ^3-3*t^2)*f^3+t^2*f^2-f+1 = 0 ---------------------------------------- This ends Proposition, 19, that took, 1.606, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 20 Let a(n) be the number of sequences of length n, in the alphabet, {-2, -1, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 25408828065, 63364046190, 158229645720, 395632288590, 990419552730, 2482238709888, 6227850849066, 15641497455612, 39322596749218, 98948326105928] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 3 3 2 2 f t + f t - f + 1 = 0 and in Maple notation f^3*t^3+f^2*t^2-f+1 = 0 ---------------------------------------- This ends Proposition, 20, that took, 1.613, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 21 Let a(n) be the number of sequences of length n, in the alphabet, {-2, -1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 645265199252, 1813615497166, 5107394107927, 14386545035342, 40621735594210, 114720169872202, 324560293765296, 918870098708832, 2604241833793991, 7388579097551618] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 6 6 4 5 4 4 3 2 3 2 2 f t - t f - t f + (t + 2 t ) f - f t - f + 1 = 0 and in Maple notation f^6*t^6-t^4*f^5-t^4*f^4+(t^3+2*t^2)*f^3-f^2*t^2-f+1 = 0 ---------------------------------------- This ends Proposition, 21, that took, 1.637, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 22 Let a(n) be the number of sequences of length n, in the alphabet, {-2, -1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 2, 1, 2, 17, 17, 43, 220, 322, 877, 3495, 6513, 18246, 63069, 137364, 389520, 1240075, 2986569, 8518188, 25878573, 66493272, 190276431, 563345305, 1509236554, 4329167366, 12645267502, 34810974533, 100065738510, 290410780163, 813932210810, 2344530239608, 6787557305833, 19254309739598, 55576193661986, 160849076903780, 460095808260232, 1330726621028529, 3854609838686679, 11091289883698738] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 8 8 5 7 6 5 6 5 5 4 3 4 3 3 t f - t f + t f + (-t - t ) f + 2 t f + (t + 3 t ) f - t f - f + 1 = 0 and in Maple notation t^10*f^10-t^8*f^8+t^5*f^7+(-t^6-t^5)*f^6+2*t^5*f^5+(t^4+3*t^3)*f^4-t^3*f^3-f+1 = 0 ---------------------------------------- This ends Proposition, 22, that took, 1.693, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 23 Let a(n) be the number of sequences of length n, in the alphabet, {-2, -1, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 9861191391746256, 37660870323158835, 144029959800495438, 551546279543420059, 2114684919809270434, 8117356580480783638, 31193334574672753772, 119994768635233629431, 462054434301743595662, 1780873197452044558004] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 5 5 4 3 4 3 2 3 2 2 f t + (-t - t ) f + (t + t ) f + (t + t) f + (-t - 1) f + 1 = 0 and in Maple notation f^5*t^5+(-t^4-t^3)*f^4+(t^3+t^2)*f^3+(t^2+t)*f^2+(-t-1)*f+1 = 0 ---------------------------------------- This ends Proposition, 23, that took, 1.719, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 24 Let a(n) be the number of sequences of length n, in the alphabet, {-2, -1, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 3, 6, 21, 66, 206, 694, 2343, 8006, 27865, 97842, 346560, 1238017, 4451859, 16104105, 58569206, 214013423, 785324563, 2892811352, 10692822131, 39649034086, 147443120646, 549749019862, 2054764213960, 7697272862049, 28894655660026, 108677590661657, 409493420065062, 1545562470596778, 5842680517890137, 22119801344728755, 83860065166879578, 318345575635570632, 1209984597470883971, 4604353717435642583, 17540344670612420506, 66890292116966006476, 255340921774236374052, 975637911204231539435] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 9 9 8 8 7 5 7 6 5 6 5 4 5 t f + t f - t f + (-2 t + t ) f + (-t - t ) f + (3 t + 2 t ) f 4 3 4 2 2 + (2 t + 3 t ) f + t f - f + 1 = 0 and in Maple notation t^10*f^10+t^9*f^9-t^8*f^8+(-2*t^7+t^5)*f^7+(-t^6-t^5)*f^6+(3*t^5+2*t^4)*f^5+(2* t^4+3*t^3)*f^4+t^2*f^2-f+1 = 0 ---------------------------------------- This ends Proposition, 24, that took, 1.788, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 25 Let a(n) be the number of sequences of length n, in the alphabet, {-2, -1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 3, 3, 23, 48, 155, 612, 1609, 6255, 20608, 67954, 250621, 837858, 2997773, 10682234, 37447731, 135767710, 484626014, 1747304695, 6345838687, 22949010094, 83737139716, 305552771525, 1117272519230, 4101926037674, 15064915295407, 55488468018578, 204729501895013, 756350275118646, 2800027056148971, 10377918836812794, 38521413439022433, 143184883556986540, 532814894354798905, 1985211915360494824, 7404609038051674655, 27647122999848204744, 103334287176052280814, 386579072024085298844] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 8 7 8 5 7 5 6 4 5 3 4 2 3 2 2 t f + (-t - t ) f + t f - 2 t f - t f + 3 t f + 2 t f - t f - f + 1 = 0 and in Maple notation t^10*f^10+(-t^8-t^7)*f^8+t^5*f^7-2*t^5*f^6-t^4*f^5+3*t^3*f^4+2*t^2*f^3-t^2*f^2- f+1 = 0 ---------------------------------------- This ends Proposition, 25, that took, 1.897, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 26 Let a(n) be the number of sequences of length n, in the alphabet, {-2, -1, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 4, 15, 54, 197, 778, 3046, 12378, 50688, 210821, 885836, 3755794, 16053550, 69077136, 299051044, 1301497997, 5691174700, 24991961429, 110168793923, 487328507125, 2162490768266, 9623634039899, 42941087514502, 192072611056724, 861064794485586, 3868232998947027, 17411324425937991, 78511976428487851, 354628109413966644, 1604341160570722942, 7268823842760461184 , 32978945219886339519, 149823281943148384308, 681490024904553949414, 3103471368624092952988, 14148708406910701942775, 64571602595986047193440, 294984276632934745080426, 1348861148942366519449640] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 9 9 8 7 8 7 5 7 6 5 6 t f + t f + (-t - t ) f + (-2 t + t ) f + (-3 t - 2 t ) f 5 4 5 4 3 4 3 2 3 + (t + t ) f + (2 t + 3 t ) f + (t + 2 t ) f - f + 1 = 0 and in Maple notation t^10*f^10+t^9*f^9+(-t^8-t^7)*f^8+(-2*t^7+t^5)*f^7+(-3*t^6-2*t^5)*f^6+(t^5+t^4)* f^5+(2*t^4+3*t^3)*f^4+(t^3+2*t^2)*f^3-f+1 = 0 ---------------------------------------- This ends Proposition, 26, that took, 1.976, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 27 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -1, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 0, 16395456762, 0, 91053897066, 0, 508151297602, 0, 2848290555562, 0, 16028132445156] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 4 4 2 2 f t + f t - f + 1 = 0 and in Maple notation f^4*t^4+f^2*t^2-f+1 = 0 ---------------------------------------- This ends Proposition, 27, that took, 1.984, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 28 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 3, 2, 3, 23, 59, 74, 178, 753, 1859, 3299, 8937, 29884, 73955, 160368 , 445889, 1334825, 3371535, 8167687, 22732271, 64550448, 166944853, 429281385, 1189787311, 3299504856, 8708248080, 23118437489, 63845014804, 175463878127, 470269479575, 1270311652558, 3501884445317, 9604857045847, 26027895342456, 71002490056153, 195692892371919, 537321155970160, 1467430337299719] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 9 9 5 7 5 6 5 4 5 4 4 3 3 f t + f t + t f - t f + (2 t + 2 t ) f + t f + t f - f + 1 = 0 and in Maple notation f^10*t^10+f^9*t^9+t^5*f^7-t^5*f^6+(2*t^5+2*t^4)*f^5+t^4*f^4+t^3*f^3-f+1 = 0 ---------------------------------------- This ends Proposition, 28, that took, 2.039, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 29 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 3, 0, 16, 0, 100, 0, 655, 0, 4465, 0, 31599, 0, 230390, 0, 1717910, 0 , 13034753, 0, 100308732, 0, 781057488, 0, 6142515700, 0, 48719605150, 0, 389274014325, 0, 3130375135624, 0, 25315962247754, 0, 205765906922296, 0, 1679968849194124, 0, 13771490153093158] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 20 20 18 19 18 18 16 17 16 14 16 14 15 t f - t f + t f - 3 t f + (t + 3 t ) f - 3 t f 14 12 14 12 10 13 12 10 12 + (5 t + 3 t ) f + (-3 t - 3 t ) f + (2 t + 3 t ) f 10 8 11 10 8 6 10 8 6 9 + (-6 t - t ) f + (-4 t + 6 t + t ) f + (-6 t - t ) f 8 6 8 6 4 7 6 4 6 4 5 + (2 t + 3 t ) f + (-3 t - 3 t ) f + (5 t + 3 t ) f - 3 t f 4 2 4 2 3 2 2 + (t + 3 t ) f - 3 t f + t f - f + 1 = 0 and in Maple notation t^20*f^20-t^18*f^19+t^18*f^18-3*t^16*f^17+(t^16+3*t^14)*f^16-3*t^14*f^15+(5*t^ 14+3*t^12)*f^14+(-3*t^12-3*t^10)*f^13+(2*t^12+3*t^10)*f^12+(-6*t^10-t^8)*f^11+( -4*t^10+6*t^8+t^6)*f^10+(-6*t^8-t^6)*f^9+(2*t^8+3*t^6)*f^8+(-3*t^6-3*t^4)*f^7+( 5*t^6+3*t^4)*f^6-3*t^4*f^5+(t^4+3*t^2)*f^4-3*t^2*f^3+t^2*f^2-f+1 = 0 ---------------------------------------- This ends Proposition, 29, that took, 2.269, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 30 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -1, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 3, 6, 21, 66, 206, 694, 2343, 8006, 27865, 97842, 346560, 1238017, 4451859, 16104105, 58569206, 214013423, 785324563, 2892811352, 10692822131, 39649034086, 147443120646, 549749019862, 2054764213960, 7697272862049, 28894655660026, 108677590661657, 409493420065062, 1545562470596778, 5842680517890137, 22119801344728755, 83860065166879578, 318345575635570632, 1209984597470883971, 4604353717435642583, 17540344670612420506, 66890292116966006476, 255340921774236374052, 975637911204231539435] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 9 9 8 8 7 5 7 6 5 6 5 4 5 t f + t f - t f + (-2 t + t ) f + (-t - t ) f + (3 t + 2 t ) f 4 3 4 2 2 + (2 t + 3 t ) f + t f - f + 1 = 0 and in Maple notation t^10*f^10+t^9*f^9-t^8*f^8+(-2*t^7+t^5)*f^7+(-t^6-t^5)*f^6+(3*t^5+2*t^4)*f^5+(2* t^4+3*t^3)*f^4+t^2*f^2-f+1 = 0 ---------------------------------------- This ends Proposition, 30, that took, 2.337, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 31 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -1, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 0, 13, 0, 120, 0, 1288, 0, 15046, 0, 185658, 0, 2380720, 0, 31411376, 0, 423660504, 0, 5814905977, 0, 80956085304, 0, 1140478875656, 0, 16227516683124, 0, 232870988052180, 0, 3366482778363616, 0, 48981220255732960, 0, 716707681487535144, 0, 10539913681632290532, 0, 155697664218428455520, 0, 2309297999296926348448] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 14 14 12 13 12 12 10 11 10 8 10 8 9 8 8 t f - t f + t f - t f + (-t + 2 t ) f + t f - t f 6 4 7 6 6 4 5 4 2 4 2 3 2 2 + (2 t - t ) f - t f + t f + (-t + 2 t ) f - t f + t f - f + 1 = 0 and in Maple notation t^14*f^14-t^12*f^13+t^12*f^12-t^10*f^11+(-t^10+2*t^8)*f^10+t^8*f^9-t^8*f^8+(2*t ^6-t^4)*f^7-t^6*f^6+t^4*f^5+(-t^4+2*t^2)*f^4-t^2*f^3+t^2*f^2-f+1 = 0 ---------------------------------------- This ends Proposition, 31, that took, 2.532, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 32 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 6, 15, 52, 156, 528, 1765, 6114, 21236, 74776, 265291, 949595, 3422399, 12410903, 45247919, 165752707, 609808036, 2252176796, 8347161607, 31035445149, 115729542652, 432703786186, 1621835578983, 6092706372588, 22936590083350, 86516207497332, 326932457744595, 1237536547815069, 4691914691117079, 17815222622885223, 67739562218872488, 257909787594052190, 983184309422800372, 3752440379470237568, 14337614322431343300, 54840178875567735030, 209969537892201797328, 804687981679200752808] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 20 20 18 19 18 18 17 16 17 16 14 16 t f - t f + t f + (t - 3 t ) f + (2 t + 3 t ) f 15 14 15 14 13 12 14 + (5 t - t ) f + (9 t + t + 3 t ) f 13 12 11 10 13 12 10 12 + (2 t - 4 t + t - 3 t ) f + (3 t + 4 t ) f 11 10 8 11 10 9 8 6 10 + (-3 t - 11 t - t ) f + (-6 t - 10 t + 6 t + t ) f 9 8 6 9 8 6 8 + (-3 t - 11 t - t ) f + (3 t + 4 t ) f 7 6 5 4 7 6 5 4 6 5 4 5 + (2 t - 4 t + t - 3 t ) f + (9 t + t + 3 t ) f + (5 t - t ) f 4 2 4 3 2 3 2 2 + (2 t + 3 t ) f + (t - 3 t ) f + t f - f + 1 = 0 and in Maple notation t^20*f^20-t^18*f^19+t^18*f^18+(t^17-3*t^16)*f^17+(2*t^16+3*t^14)*f^16+(5*t^15-t ^14)*f^15+(9*t^14+t^13+3*t^12)*f^14+(2*t^13-4*t^12+t^11-3*t^10)*f^13+(3*t^12+4* t^10)*f^12+(-3*t^11-11*t^10-t^8)*f^11+(-6*t^10-10*t^9+6*t^8+t^6)*f^10+(-3*t^9-\ 11*t^8-t^6)*f^9+(3*t^8+4*t^6)*f^8+(2*t^7-4*t^6+t^5-3*t^4)*f^7+(9*t^6+t^5+3*t^4) *f^6+(5*t^5-t^4)*f^5+(2*t^4+3*t^2)*f^4+(t^3-3*t^2)*f^3+t^2*f^2-f+1 = 0 ---------------------------------------- This ends Proposition, 32, that took, 3.171, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 33 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -1, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 3, 16, 48, 208, 778, 3305, 13499, 57999, 247426, 1080038, 4725641, 20929207, 93118686, 417432294, 1879871543, 8510737402, 38686261748, 176564376942, 808602162394, 3715180084791, 17119059401564, 79095591109170, 366346002995878, 1700682157965819, 7911704506752742, 36878195675123781, 172211459271608956, 805555550951269942, 3774174295168643551, 17709186843741843257, 83212029736653279793, 391515337546259034843, 1844394597724020439367, 8699040178460691523419, 41074630787462631404420, 194149125687057441136325, 918614252117154387538888, 4350560194974760150572558] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 20 20 18 19 18 18 16 17 16 15 14 16 t f - t f + 2 t f - 3 t f + (2 t + 3 t + 3 t ) f 15 15 14 13 12 14 + 2 t f + (3 t + 3 t + 3 t ) f 13 12 11 10 13 11 10 12 + (-t + t - 2 t - 3 t ) f + (5 t + 2 t ) f 11 10 9 8 11 10 9 8 6 10 + (-4 t - t - t - t ) f + (-6 t + 2 t + 2 t + t ) f 9 8 7 6 9 7 6 8 + (-4 t - t - t - t ) f + (5 t + 2 t ) f 7 6 5 4 7 6 5 4 6 5 5 + (-t + t - 2 t - 3 t ) f + (3 t + 3 t + 3 t ) f + 2 t f 4 3 2 4 2 3 2 2 + (2 t + 3 t + 3 t ) f - 3 t f + 2 t f - f + 1 = 0 and in Maple notation t^20*f^20-t^18*f^19+2*t^18*f^18-3*t^16*f^17+(2*t^16+3*t^15+3*t^14)*f^16+2*t^15* f^15+(3*t^14+3*t^13+3*t^12)*f^14+(-t^13+t^12-2*t^11-3*t^10)*f^13+(5*t^11+2*t^10 )*f^12+(-4*t^11-t^10-t^9-t^8)*f^11+(-6*t^10+2*t^9+2*t^8+t^6)*f^10+(-4*t^9-t^8-t ^7-t^6)*f^9+(5*t^7+2*t^6)*f^8+(-t^7+t^6-2*t^5-3*t^4)*f^7+(3*t^6+3*t^5+3*t^4)*f^ 6+2*t^5*f^5+(2*t^4+3*t^3+3*t^2)*f^4-3*t^2*f^3+2*t^2*f^2-f+1 = 0 ---------------------------------------- This ends Proposition, 33, that took, 3.798, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 34 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -2, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 133231800, 273896532, 600805296, 1305229332, 2720740792, 5843241088, 12797739672, 27206642716, 57941746476, 126405822608] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 4 4 3 3 f t + f t - f + 1 = 0 and in Maple notation f^4*t^4+f^3*t^3-f+1 = 0 ---------------------------------------- This ends Proposition, 34, that took, 3.809, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 35 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -2, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 2, 5, 17, 14, 103, 65, 544, 515, 2671, 4333, 12920, 32888, 66569, 225063, 389929, 1426875, 2581052, 8652846, 18130991, 51937472, 127733905, 318505753, 879213643, 2034543521, 5892047281, 13539791786, 38764350879, 92547902870, 253609842517, 638716733669, 1670011621498, 4398787899731, 11151727980457, 30093346643625, 75616292374270, 204712934528781] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 7 8 7 5 7 6 5 6 5 4 5 4 4 f t - t f + (-t + t ) f + (t - t ) f + (2 t - t ) f - t f 2 3 2 2 + 2 t f - f t - f + 1 = 0 and in Maple notation f^10*t^10-t^7*f^8+(-t^7+t^5)*f^7+(t^6-t^5)*f^6+(2*t^5-t^4)*f^5-t^4*f^4+2*t^2*f^ 3-f^2*t^2-f+1 = 0 ---------------------------------------- This ends Proposition, 35, that took, 3.860, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 36 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 2, 5, 22, 14, 164, 65, 1030, 657, 5868, 7463, 31765, 73575, 173849 , 631556, 1053086, 4877803, 7526655, 34948691, 61382672, 239407864, 524309309, 1621763388, 4415274965, 11255289437, 35813332389, 82153463817, 279458110861, 633479334487, 2118070224696, 5075741777630, 15824514104397, 41275863623366, 118428013850973, 334523141763061, 899513350738990, 2678023253678681] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 20 20 18 19 18 18 16 17 14 16 15 14 15 t f - t f + t f - 3 t f + 3 t f + (2 t - 3 t ) f 14 13 12 14 13 12 11 10 13 + (3 t - t + 3 t ) f + (t - 3 t + t - 3 t ) f 12 10 12 11 10 8 11 + (3 t + 3 t ) f + (t - 6 t - t ) f 10 9 8 6 10 9 8 6 9 8 6 8 + (t - 6 t + 6 t + t ) f + (t - 6 t - t ) f + (3 t + 3 t ) f 7 6 5 4 7 6 5 4 6 5 4 5 + (t - 3 t + t - 3 t ) f + (3 t - t + 3 t ) f + (2 t - 3 t ) f 2 4 2 3 2 2 + 3 t f - 3 t f + f t - f + 1 = 0 and in Maple notation t^20*f^20-t^18*f^19+t^18*f^18-3*t^16*f^17+3*t^14*f^16+(2*t^15-3*t^14)*f^15+(3*t ^14-t^13+3*t^12)*f^14+(t^13-3*t^12+t^11-3*t^10)*f^13+(3*t^12+3*t^10)*f^12+(t^11 -6*t^10-t^8)*f^11+(t^10-6*t^9+6*t^8+t^6)*f^10+(t^9-6*t^8-t^6)*f^9+(3*t^8+3*t^6) *f^8+(t^7-3*t^6+t^5-3*t^4)*f^7+(3*t^6-t^5+3*t^4)*f^6+(2*t^5-3*t^4)*f^5+3*t^2*f^ 4-3*t^2*f^3+f^2*t^2-f+1 = 0 ---------------------------------------- This ends Proposition, 36, that took, 4.194, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 37 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -2, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 3, 3, 23, 48, 155, 612, 1609, 6255, 20608, 67954, 250621, 837858, 2997773, 10682234, 37447731, 135767710, 484626014, 1747304695, 6345838687, 22949010094, 83737139716, 305552771525, 1117272519230, 4101926037674, 15064915295407, 55488468018578, 204729501895013, 756350275118646, 2800027056148971, 10377918836812794, 38521413439022433, 143184883556986540, 532814894354798905, 1985211915360494824, 7404609038051674655, 27647122999848204744, 103334287176052280814, 386579072024085298844] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 8 7 8 5 7 5 6 4 5 3 4 2 3 2 2 t f + (-t - t ) f + t f - 2 t f - t f + 3 t f + 2 t f - t f - f + 1 = 0 and in Maple notation t^10*f^10+(-t^8-t^7)*f^8+t^5*f^7-2*t^5*f^6-t^4*f^5+3*t^3*f^4+2*t^2*f^3-t^2*f^2- f+1 = 0 ---------------------------------------- This ends Proposition, 37, that took, 4.267, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 38 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -2, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 6, 15, 52, 156, 528, 1765, 6114, 21236, 74776, 265291, 949595, 3422399, 12410903, 45247919, 165752707, 609808036, 2252176796, 8347161607, 31035445149, 115729542652, 432703786186, 1621835578983, 6092706372588, 22936590083350, 86516207497332, 326932457744595, 1237536547815069, 4691914691117079, 17815222622885223, 67739562218872488, 257909787594052190, 983184309422800372, 3752440379470237568, 14337614322431343300, 54840178875567735030, 209969537892201797328, 804687981679200752808] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 20 20 18 19 18 18 17 16 17 16 14 16 t f - t f + t f + (t - 3 t ) f + (2 t + 3 t ) f 15 14 15 14 13 12 14 + (5 t - t ) f + (9 t + t + 3 t ) f 13 12 11 10 13 12 10 12 + (2 t - 4 t + t - 3 t ) f + (3 t + 4 t ) f 11 10 8 11 10 9 8 6 10 + (-3 t - 11 t - t ) f + (-6 t - 10 t + 6 t + t ) f 9 8 6 9 8 6 8 + (-3 t - 11 t - t ) f + (3 t + 4 t ) f 7 6 5 4 7 6 5 4 6 5 4 5 + (2 t - 4 t + t - 3 t ) f + (9 t + t + 3 t ) f + (5 t - t ) f 4 2 4 3 2 3 2 2 + (2 t + 3 t ) f + (t - 3 t ) f + f t - f + 1 = 0 and in Maple notation t^20*f^20-t^18*f^19+t^18*f^18+(t^17-3*t^16)*f^17+(2*t^16+3*t^14)*f^16+(5*t^15-t ^14)*f^15+(9*t^14+t^13+3*t^12)*f^14+(2*t^13-4*t^12+t^11-3*t^10)*f^13+(3*t^12+4* t^10)*f^12+(-3*t^11-11*t^10-t^8)*f^11+(-6*t^10-10*t^9+6*t^8+t^6)*f^10+(-3*t^9-\ 11*t^8-t^6)*f^9+(3*t^8+4*t^6)*f^8+(2*t^7-4*t^6+t^5-3*t^4)*f^7+(9*t^6+t^5+3*t^4) *f^6+(5*t^5-t^4)*f^5+(2*t^4+3*t^2)*f^4+(t^3-3*t^2)*f^3+f^2*t^2-f+1 = 0 ---------------------------------------- This ends Proposition, 38, that took, 4.805, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 39 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -2, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 0, 11, 4, 86, 78, 805, 1192, 8452, 16990, 96270, 236808, 1164643, 3284046, 14743066, 45644056, 193172365, 637821100, 2598848478, 8972601330, 35688679563, 127118293564, 498084862421, 1813552391314, 7042085802066, 26046219527470, 100619518600486, 376413085980710, 1450310346991644, 5471244629599182, 21059012766071621, 79947499571174084, 307715177621912926, 1173886029775191684, 4520932319331403266, 17312819399699279000, 66739472841683131101, 256368332246429186536, 989408338157591223177] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 14 14 13 12 13 12 11 12 11 10 11 t f + (-t - t ) f + (t + t ) f + (-t - t ) f 10 8 10 9 8 7 9 8 7 6 8 + (t + 2 t ) f + (2 t - 5 t - t ) f + (-2 t + 3 t + 2 t ) f 7 6 4 7 6 5 4 6 5 4 3 5 + (2 t - 3 t - t ) f + (-2 t + 3 t + 2 t ) f + (2 t - 5 t - t ) f 4 2 4 3 2 3 2 2 + (t + 2 t ) f + (-t - t ) f + (t + t) f + (-1 - t) f + 1 = 0 and in Maple notation t^14*f^14+(-t^13-t^12)*f^13+(t^12+t^11)*f^12+(-t^11-t^10)*f^11+(t^10+2*t^8)*f^ 10+(2*t^9-5*t^8-t^7)*f^9+(-2*t^8+3*t^7+2*t^6)*f^8+(2*t^7-3*t^6-t^4)*f^7+(-2*t^6 +3*t^5+2*t^4)*f^6+(2*t^5-5*t^4-t^3)*f^5+(t^4+2*t^2)*f^4+(-t^3-t^2)*f^3+(t^2+t)* f^2+(-1-t)*f+1 = 0 ---------------------------------------- This ends Proposition, 39, that took, 5.208, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 40 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -2, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 3, 15, 52, 196, 848, 3285, 14647, 60702, 270321, 1175419, 5255914, 23491263, 106039951, 481526629, 2196502989, 10079226967, 46405794075, 214644323424, 995842152796, 4636334700470, 21645379432275, 101333050920532, 475554054349281, 2236867470931726, 10543910516031653, 49798007373922185, 235624174511043642, 1116777689178292522, 5301626970881411615, 25205850251590771049, 120006708503224545684, 572117331555550458916, 2730911696937623440681, 13050977933834611429924, 62440110843797303986797, 299050203169880371941497, 1433708348090695408846840, 6880058216300684749855770] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 20 20 18 19 16 17 16 15 14 16 15 14 15 t f - t f - t f + (2 t + 3 t + 3 t ) f + (t - 4 t ) f 14 13 12 14 13 12 11 10 13 + (2 t + t + 2 t ) f + (-3 t - 2 t - t - 3 t ) f 12 11 10 12 11 10 9 8 11 + (t + 7 t + 8 t ) f + (-4 t - 7 t - t - 2 t ) f 10 9 8 6 10 9 8 7 6 9 + (t - 2 t + 5 t + t ) f + (-4 t - 7 t - t - 2 t ) f 8 7 6 8 7 6 5 4 7 + (t + 7 t + 8 t ) f + (-3 t - 2 t - t - 3 t ) f 6 5 4 6 5 4 5 4 3 2 4 3 2 + (2 t + t + 2 t ) f + (t - 4 t ) f + (2 t + 3 t + 3 t ) f - f t - f + 1 = 0 and in Maple notation t^20*f^20-t^18*f^19-t^16*f^17+(2*t^16+3*t^15+3*t^14)*f^16+(t^15-4*t^14)*f^15+(2 *t^14+t^13+2*t^12)*f^14+(-3*t^13-2*t^12-t^11-3*t^10)*f^13+(t^12+7*t^11+8*t^10)* f^12+(-4*t^11-7*t^10-t^9-2*t^8)*f^11+(t^10-2*t^9+5*t^8+t^6)*f^10+(-4*t^9-7*t^8- t^7-2*t^6)*f^9+(t^8+7*t^7+8*t^6)*f^8+(-3*t^7-2*t^6-t^5-3*t^4)*f^7+(2*t^6+t^5+2* t^4)*f^6+(t^5-4*t^4)*f^5+(2*t^4+3*t^3+3*t^2)*f^4-f^3*t^2-f+1 = 0 ---------------------------------------- This ends Proposition, 40, that took, 5.887, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 41 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -2, -1, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [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, 274867141845, 744093631842, 2017066320624, 5474900965050, 14878450339822, 40479971557162, 110253945275970, 300605644859552, 820399033872096, 2241084167717824] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 4 4 3 3 2 2 f t + f t + f t - f + 1 = 0 and in Maple notation f^4*t^4+f^3*t^3+f^2*t^2-f+1 = 0 ---------------------------------------- This ends Proposition, 41, that took, 5.899, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 42 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -2, -1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 5, 9, 31, 78, 248, 705, 2196, 6632, 20780, 64709, 204902, 650000, 2080483, 6683564, 21593311, 70024903, 228022074, 744976876, 2441850778, 8026618762, 26455041139, 87405982153, 289438774174, 960462359139, 3193366842536 , 10636635056279, 35489063311272, 118596791583351, 396914141297320, 1330230442462987, 4464042344334714, 14999217181926990, 50456596364848778, 169921812232536963, 572844723715864685, 1933116776188266041, 6529668152176835624] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 9 9 7 8 7 5 7 6 5 6 5 4 5 t f + t f - t f + (-t + t ) f + (-2 t - t ) f + (2 t + t ) f 3 2 3 2 2 + (t + 2 t ) f - t f - f + 1 = 0 and in Maple notation t^10*f^10+t^9*f^9-t^7*f^8+(-t^7+t^5)*f^7+(-2*t^6-t^5)*f^6+(2*t^5+t^4)*f^5+(t^3+ 2*t^2)*f^3-t^2*f^2-f+1 = 0 ---------------------------------------- This ends Proposition, 42, that took, 5.963, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 43 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -2, -1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 3, 16, 33, 115, 390, 1087, 4060, 12555, 42953, 148067, 492739, 1735298, 5944320, 20744252, 72905575, 254998049, 903660769, 3195209422, 11355589072, 40507136044, 144620988953, 518478617875, 1861257943227, 6697455408050, 24152234870325, 87226107921628, 315651869078757, 1143924927595869, 4151936835886485, 15091681888691404, 54925223488389666, 200157938880285184, 730258764785275647, 2667274260621421838, 9752675597285646950, 35695329641773808896, 130773052695581564343] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 20 20 18 19 18 18 17 16 17 16 15 14 16 t f - t f + t f + (-t - 3 t ) f + (t + 3 t + 3 t ) f 15 14 15 14 13 12 14 + (-2 t - 3 t ) f + (4 t + t + 3 t ) f 13 12 11 10 13 12 11 10 12 + (-5 t - 3 t - 2 t - 3 t ) f + (5 t + 5 t + 3 t ) f 11 10 9 8 11 10 9 8 6 10 + (-4 t - 6 t - t - t ) f + (t + 6 t + 6 t + t ) f 9 8 7 6 9 8 7 6 8 + (-4 t - 6 t - t - t ) f + (5 t + 5 t + 3 t ) f 7 6 5 4 7 6 5 4 6 + (-5 t - 3 t - 2 t - 3 t ) f + (4 t + t + 3 t ) f 5 4 5 4 3 2 4 3 2 3 2 2 + (-2 t - 3 t ) f + (t + 3 t + 3 t ) f + (-t - 3 t ) f + t f - f + 1 = 0 and in Maple notation t^20*f^20-t^18*f^19+t^18*f^18+(-t^17-3*t^16)*f^17+(t^16+3*t^15+3*t^14)*f^16+(-2 *t^15-3*t^14)*f^15+(4*t^14+t^13+3*t^12)*f^14+(-5*t^13-3*t^12-2*t^11-3*t^10)*f^ 13+(5*t^12+5*t^11+3*t^10)*f^12+(-4*t^11-6*t^10-t^9-t^8)*f^11+(t^10+6*t^9+6*t^8+ t^6)*f^10+(-4*t^9-6*t^8-t^7-t^6)*f^9+(5*t^8+5*t^7+3*t^6)*f^8+(-5*t^7-3*t^6-2*t^ 5-3*t^4)*f^7+(4*t^6+t^5+3*t^4)*f^6+(-2*t^5-3*t^4)*f^5+(t^4+3*t^3+3*t^2)*f^4+(-t ^3-3*t^2)*f^3+t^2*f^2-f+1 = 0 ---------------------------------------- This ends Proposition, 43, that took, 6.360, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 44 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -2, -1, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 4, 15, 54, 197, 778, 3046, 12378, 50688, 210821, 885836, 3755794, 16053550, 69077136, 299051044, 1301497997, 5691174700, 24991961429, 110168793923, 487328507125, 2162490768266, 9623634039899, 42941087514502, 192072611056724, 861064794485586, 3868232998947027, 17411324425937991, 78511976428487851, 354628109413966644, 1604341160570722942, 7268823842760461184 , 32978945219886339519, 149823281943148384308, 681490024904553949414, 3103471368624092952988, 14148708406910701942775, 64571602595986047193440, 294984276632934745080426, 1348861148942366519449640] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 10 10 9 9 8 7 8 7 5 7 6 5 6 t f + t f + (-t - t ) f + (-2 t + t ) f + (-3 t - 2 t ) f 5 4 5 4 3 4 3 2 3 + (t + t ) f + (2 t + 3 t ) f + (t + 2 t ) f - f + 1 = 0 and in Maple notation t^10*f^10+t^9*f^9+(-t^8-t^7)*f^8+(-2*t^7+t^5)*f^7+(-3*t^6-2*t^5)*f^6+(t^5+t^4)* f^5+(2*t^4+3*t^3)*f^4+(t^3+2*t^2)*f^3-f+1 = 0 ---------------------------------------- This ends Proposition, 44, that took, 6.442, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 45 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -2, -1, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 3, 16, 48, 208, 778, 3305, 13499, 57999, 247426, 1080038, 4725641, 20929207, 93118686, 417432294, 1879871543, 8510737402, 38686261748, 176564376942, 808602162394, 3715180084791, 17119059401564, 79095591109170, 366346002995878, 1700682157965819, 7911704506752742, 36878195675123781, 172211459271608956, 805555550951269942, 3774174295168643551, 17709186843741843257, 83212029736653279793, 391515337546259034843, 1844394597724020439367, 8699040178460691523419, 41074630787462631404420, 194149125687057441136325, 918614252117154387538888, 4350560194974760150572558] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 20 20 18 19 18 18 16 17 16 15 14 16 t f - t f + 2 t f - 3 t f + (2 t + 3 t + 3 t ) f 15 15 14 13 12 14 + 2 t f + (3 t + 3 t + 3 t ) f 13 12 11 10 13 11 10 12 + (-t + t - 2 t - 3 t ) f + (5 t + 2 t ) f 11 10 9 8 11 10 9 8 6 10 + (-4 t - t - t - t ) f + (-6 t + 2 t + 2 t + t ) f 9 8 7 6 9 7 6 8 + (-4 t - t - t - t ) f + (5 t + 2 t ) f 7 6 5 4 7 6 5 4 6 5 5 + (-t + t - 2 t - 3 t ) f + (3 t + 3 t + 3 t ) f + 2 t f 4 3 2 4 2 3 2 2 + (2 t + 3 t + 3 t ) f - 3 t f + 2 t f - f + 1 = 0 and in Maple notation t^20*f^20-t^18*f^19+2*t^18*f^18-3*t^16*f^17+(2*t^16+3*t^15+3*t^14)*f^16+2*t^15* f^15+(3*t^14+3*t^13+3*t^12)*f^14+(-t^13+t^12-2*t^11-3*t^10)*f^13+(5*t^11+2*t^10 )*f^12+(-4*t^11-t^10-t^9-t^8)*f^11+(-6*t^10+2*t^9+2*t^8+t^6)*f^10+(-4*t^9-t^8-t ^7-t^6)*f^9+(5*t^7+2*t^6)*f^8+(-t^7+t^6-2*t^5-3*t^4)*f^7+(3*t^6+3*t^5+3*t^4)*f^ 6+2*t^5*f^5+(2*t^4+3*t^3+3*t^2)*f^4-3*t^2*f^3+2*t^2*f^2-f+1 = 0 ---------------------------------------- This ends Proposition, 45, that took, 7.072, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 46 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -2, -1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 3, 15, 52, 196, 848, 3285, 14647, 60702, 270321, 1175419, 5255914, 23491263, 106039951, 481526629, 2196502989, 10079226967, 46405794075, 214644323424, 995842152796, 4636334700470, 21645379432275, 101333050920532, 475554054349281, 2236867470931726, 10543910516031653, 49798007373922185, 235624174511043642, 1116777689178292522, 5301626970881411615, 25205850251590771049, 120006708503224545684, 572117331555550458916, 2730911696937623440681, 13050977933834611429924, 62440110843797303986797, 299050203169880371941497, 1433708348090695408846840, 6880058216300684749855770] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 20 20 18 19 16 17 16 15 14 16 15 14 15 t f - t f - t f + (2 t + 3 t + 3 t ) f + (t - 4 t ) f 14 13 12 14 13 12 11 10 13 + (2 t + t + 2 t ) f + (-3 t - 2 t - t - 3 t ) f 12 11 10 12 11 10 9 8 11 + (t + 7 t + 8 t ) f + (-4 t - 7 t - t - 2 t ) f 10 9 8 6 10 9 8 7 6 9 + (t - 2 t + 5 t + t ) f + (-4 t - 7 t - t - 2 t ) f 8 7 6 8 7 6 5 4 7 + (t + 7 t + 8 t ) f + (-3 t - 2 t - t - 3 t ) f 6 5 4 6 5 4 5 4 3 2 4 2 3 + (2 t + t + 2 t ) f + (t - 4 t ) f + (2 t + 3 t + 3 t ) f - t f - f + 1 = 0 and in Maple notation t^20*f^20-t^18*f^19-t^16*f^17+(2*t^16+3*t^15+3*t^14)*f^16+(t^15-4*t^14)*f^15+(2 *t^14+t^13+2*t^12)*f^14+(-3*t^13-2*t^12-t^11-3*t^10)*f^13+(t^12+7*t^11+8*t^10)* f^12+(-4*t^11-7*t^10-t^9-2*t^8)*f^11+(t^10-2*t^9+5*t^8+t^6)*f^10+(-4*t^9-7*t^8- t^7-2*t^6)*f^9+(t^8+7*t^7+8*t^6)*f^8+(-3*t^7-2*t^6-t^5-3*t^4)*f^7+(2*t^6+t^5+2* t^4)*f^6+(t^5-4*t^4)*f^5+(2*t^4+3*t^3+3*t^2)*f^4-t^2*f^3-f+1 = 0 ---------------------------------------- This ends Proposition, 46, that took, 7.793, seconds to generate. --------------------------------- --------------------------------------------------------- Proposition No., 47 Let a(n) be the number of sequences of length n, in the alphabet, {-3, -2, -1, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 3, 6, 35, 138, 689, 3272, 16522, 83792, 434749, 2278888, 12093271, 64741330 , 349470487, 1899418046, 10387322922, 57111322368, 315523027610, 1750681516380, 9751416039535, 54507046599094, 305650440453943, 1718956630038438, 9693209009913658, 54794959143735984, 310457570693809237, 1762705520682665544, 10027857107877385345, 57151686288411033894, 326279642607630891545, 1865707051569388661592, 10684308742180810054918, 61271675090362129009048, 351842023188483741065950, 2022914958249315372328220, 11644465032923181133034754 , 67103614092340792585084972, 387107295639502088469299294, 2235385554839773350530747032, 12920742642773813918875894076] The ordinary generating function, f=f(t) infinity ----- \ n f(t) = ) a(n) t / ----- n = 0 satisfies the algebraic equation 14 14 13 12 13 12 11 12 10 9 8 10 t f + (-t - t ) f + (t + t ) f + (2 t + 4 t + 2 t ) f 9 8 7 9 8 7 6 8 6 4 7 + (-2 t - 3 t - t ) f + (2 t + 3 t + t ) f + (3 t - t ) f 6 5 4 6 5 4 3 5 4 3 2 4 + (2 t + 3 t + t ) f + (-2 t - 3 t - t ) f + (2 t + 4 t + 2 t ) f 2 2 + (t + t) f + (-t - 1) f + 1 = 0 and in Maple notation t^14*f^14+(-t^13-t^12)*f^13+(t^12+t^11)*f^12+(2*t^10+4*t^9+2*t^8)*f^10+(-2*t^9-\ 3*t^8-t^7)*f^9+(2*t^8+3*t^7+t^6)*f^8+(3*t^6-t^4)*f^7+(2*t^6+3*t^5+t^4)*f^6+(-2* t^5-3*t^4-t^3)*f^5+(2*t^4+4*t^3+2*t^2)*f^4+(t^2+t)*f^2+(-t-1)*f+1 = 0 ---------------------------------------- This ends Proposition, 47, that took, 8.280, seconds to generate. --------------------------------- -------------------------------------