Enumeration of The Number of Dyck Paths of Semi-Length n obeying various restrictions By Shalosh B. Ekhad The number of Dyck paths with semi-length n is famously the Catalan numbers\ . In this article we will explicitly enumerate Dyck paths with four kinds of \ restictions (i) the heights of the peaks are not allowed to take certain values (ii) the heights of the valleys are not allowed to take certain values (iii) the upward runs can't have certain values (iv) the downward runs can't have certain values All the above restictions will be subsets of, {1, 2} ------------------------------------------------------------ Theorem Number, 1 Let a(n) be the number of Dyck paths of semi-length n To make it crystal clear here is such a path of semi-length 8 5+ H H + HH H H HH + H HH HH H + HH H H HH 4+ H H H H H + HHH H HH HH H + HH H HH H H HH + H HH H HH H H 3+ H H H H + HH HH + H H + HH HH 2+ H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 2 P(x) x - P(x) + 1 = 0 The sequence a(n) satisfies the linear recurrence 2 (2 n - 1) a(n - 1) a(n) = -------------------- n + 1 subject to the initial conditions [a(1) = 1] Just for fun, a(1000), equals 2046105521468021692642519982997827217179245642339057975844538099572176010191\ 891863964968026156453752449015750569428595097318163634370154637380666882\ 886375203359653243390929717431080443509007504772912973142253209352126946\ 839844796747697638537600100637918819326569730982083021538057087711176285\ 777909275869648636874856805956580057673173655666887003493944650164153396\ 910927037406301799052584663611016897272893305532116292143271037140718751\ 625839812072682464343153792956281748582435751481498598087586998603921577\ 523657477775758899987954012641033870640665444651660246024318184109046864\ 244732001962029120 For the sake of the OEIS here are the first, 100, terms. [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, 18367353072152, 69533550916004, 263747951750360, 1002242216651368, 3814986502092304, 14544636039226909, 55534064877048198, 212336130412243110, 812944042149730764, 3116285494907301262, 11959798385860453492, 45950804324621742364, 176733862787006701400, 680425371729975800390, 2622127042276492108820, 10113918591637898134020, 39044429911904443959240, 150853479205085351660700, 583300119592996693088040, 2257117854077248073253720, 8740328711533173390046320, 33868773757191046886429490, 131327898242169365477991900, 509552245179617138054608572, 1978261657756160653623774456, 7684785670514316385230816156, 29869166945772625950142417512, 116157871455782434250553845880, 451959718027953471447609509424, 1759414616608818870992479875972, 6852456927844873497549658464312, 26700952856774851904245220912664, 104088460289122304033498318812080, 405944995127576985730643443367112, 1583850964596120042686772779038896, 6182127958584855650487080847216336, 24139737743045626825711458546273312, 94295850558771979787935384946380125, 368479169875816659479009042713546950, 1440418573150919668872489894243865350, 5632681584560312734993915705849145100, 22033725021956517463358552614056949950, 86218923998960285726185640663701108500, 337485502510215975556783793455058624700, 1321422108420282270489942177190229544600, 5175569924646105559418940193995065716350, 20276890389709399862928998568254641025700, 79463489365077377841208237632349268884500, 311496878311103321137536291518809134027240, 1221395654430378811828760722007962130791020, 4790408930363303911328386208394864461024520, 18793142726809884575211361279087545193250040, 73745243611532458459690151854647329239335600, 289450081175264899454283846029490767264392230, 1136359577947336271931632877004667456667613940, 4462290049988320482463241297506133183499654740, 17526585015616776834735140517915655636396234280, 68854441132780194707888052034668647142985206100, 270557451039395118028642463289168566420671280440, 1063353702922273835973036658043476458723103404520, 4180080073556524734514695828170907458428751314320, 16435314834665426797069144960762886143367590394940, 64633260585762914370496637486146181462681535261000, 254224158304000796523953440778841647086547372026600, 1000134600800354781929399250536541864362461089950800, 3935312233584004685417853572763349509774031680023800, 15487357822491889407128326963778343232013931127835600, 60960876535340415751462563580829648891969728907438000, 239993345518077005168915776623476723006280827488229600, 944973797977428207852605870454939596837230758234904050, 3721443204405954385563870541379246659709506697378694300, 14657929356129575437016877846657032761712954950899755100, 57743358069601357782187700608042856334020731624756611000, 227508830794229349661819540395688853956041682601541047340, 896519947090131496687170070074100632420837521538745909320] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 2 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H H H + H HH HHH HHH + HH H HH H H HH + H HH H HH HH H 3+ H H H H H + HH H H HH + H H HH H + HH HHH HH 2+ H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 2 2 1 + (x + x) P(x) + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence 2 (n - 1) a(n - 1) 3 (n - 1) a(n - 2) a(n) = ------------------ + ------------------ n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 1] Just for fun, a(1000), equals 1530036790419022830094068760617849229908556040379774601663141696520714432010\ 505204701121726795033893282006994307845948795123386656757478216103147793\ 598816422588288234800268711441603480859153836164038767267854510780820952\ 045840116183968093556741601444043981217670681866042373596290010701135538\ 838865417082061461269201435105172255513381628729705648108431316186711898\ 379048365060433595579837015920602409701854829405120869376654505085317762\ 1901271877999637563068542637350933457 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 3, 6, 15, 36, 91, 232, 603, 1585, 4213, 11298, 30537, 83097, 227475, 625992, 1730787, 4805595, 13393689, 37458330, 105089229, 295673994, 834086421, 2358641376, 6684761125, 18985057351, 54022715451, 154000562758, 439742222071, 1257643249140, 3602118427251, 10331450919456, 29671013856627, 85317692667643, 245613376802185, 707854577312178, 2042162142208813, 5897493615536452, 17047255430494497, 49320944483427000, 142816973618414817, 413887836110423787, 1200394300050487935, 3484084625456932134, 10119592485062548155, 29412628894558563849, 85543870540455597789, 248952602654003411640, 724947137834104063053, 2112261618875209962525, 6157879192714893166503, 17961708307164474879078, 52418979494565497284659, 153054402342387832806318, 447107296039753836151995, 1306709599137475613111808, 3820682066516442811470123, 11176109832763802046866481, 32705601410484246215745189, 95747934502509579263312730, 280418620117853741708024169, 821578511126260089293299449, 2407969409295125052827266131, 7060047856454817331912175136, 20706869494801128932088054675, 60752886013114747805209321971, 178303876727715987264460117881, 523470228309211183146132538770, 1537292873088850037153655419037, 4515968752463518800863883219540, 13270012942708831885430137279857, 39004474517326916925520791131352, 114677148186439520720469807592881, 337252779926837166106515094143507, 992081497804863208806272348440575, 2919102839611001046292805620867782, 8591299535354652688143556684853307, 25291409899803750802286391976502211, 74471367333926485356188553908612137, 219333623772940705482181740240713080, 646127582088681087104424825527569497, 1903821367984370378973123565306390657, 5610824882652789101159297569379125339, 16539320523461975633674292210615157006, 48763733724885023891037362712600616695, 143801214724243338894845383828477461126, 424143211957453170823189308473526283071, 1251252511019022217034672218022874172864, 3691969061033252211450145056818715608239, 10895571836533928224569430533625487349525, 32160232558185513877392752232595140415729, 94943198059462752589589672745512131329394, 280338312871514003720592178984834743192165, 827891507006386759684746014201909924531418, 2445317582470051292788355811433410180110685, 7223813569919277908209909876381273600472448, 21343507566294190307011455122677671120241053, 63071286725499290051879587077009179781061461, 186407292265725088628262974383000850686750119] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 3 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH HH H HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-+*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 2 2 1 + P(x) x + (-x + x) P(x) - P(x) = 0 The sequence a(n) satisfies the linear recurrence 3 2 6 (787 n - 494) a(n - 1) 3 (1554 n - 925 n - 1297 n + 494) a(n - 2) a(n) = ------------------------ + -------------------------------------------- (n + 2) (518 n + 741) (518 n + 741) (n + 2) (n + 1) 3 2 2 (1036 n - 11877 n + 20213 n - 3576) a(n - 3) + ------------------------------------------------ (518 n + 741) (n + 2) (n + 1) 2 (n - 3) (5698 n + 36483 n - 36982) a(n - 4) + -------------------------------------------- (518 n + 741) (n + 2) (n + 1) 46 (518 n - 325) (n - 3) (n - 4) a(n - 5) + ----------------------------------------- (518 n + 741) (n + 2) (n + 1) subject to the initial conditions [a(1) = 1, a(2) = 1, a(3) = 2, a(4) = 6, a(5) = 17] Just for fun, a(1000), equals 1942638687787092634075059070839736201209022077282946815319037275653135842929\ 978071198907945657903781172617466363083969224160153246854962037619385875\ 192569342225163855512591691485048587058019571412725666535401525169319932\ 852940177353450721537703813480194459471273198171052012041742824260866597\ 199220430130414454153445911235037447684813910308642456487574707373973037\ 276160821493383066116278973777900945532710463532561160938805314663555118\ 339753742178503987938837722892959983615596644831529369044770946424540593\ 258068493526378634959570733961 For the sake of the OEIS here are the first, 100, terms. [1, 1, 1, 2, 6, 17, 46, 128, 372, 1109, 3349, 10221, 31527, 98178, 308179, 973911, 3096044, 9894393, 31770247, 102444145, 331594081, 1077022622, 3509197080, 11466710630, 37567784437, 123380796192, 406120349756, 1339571374103, 4427077704043, 14657212174545, 48609031351173, 161460865003722, 537105965807404, 1789190378916633, 5967904354732863, 19930752893117029, 66639631673845377, 223059824945930906, 747420469638736654, 2506919354940523274, 8416394534562275811, 28281491440851685236, 95115345845065123216, 320150194302022177713, 1078439633123050270009, 3635480814252528228147, 12264160996801189321670, 41400988028286624177559, 139851833561420188254209, 472714241540768044463088, 1598790005917078894380294, 5410484616453450209084881, 18319907822529254477956157, 62064561502420172514647235, 210372420268090214349596878, 713428575983969700667466980, 2420589227918231406542698699, 8216621938018388176822662971, 27903598584399296257747196715, 94801512792362867499229775636, 322219801408067247729427624654, 1095633138464539157194002219629, 3726905870873674851438742776654, 12682259694557529324760955138096, 43172250297927157466472517692604, 147017036918999832986390216017401, 500819129565295097124372277542255, 1706632276262636234330249927828757, 5817555515344652723969052274855585, 19837117765438137202457229641539242, 67662829147973169471601347696638294, 230861512329460824108516612679165086, 787914574556023607795027232940304227, 2689858931357235585645670820603855488, 9185426878574482655601935099075111524, 31375126323522571901608286578321312301, 107197553528773808298644304964862882709, 366348644161261571053552102947752288843, 1252309890606893996404962284901053156800, 4281873475622593649515090503337074602167, 14643943307369843047960278107739379399735, 50093563145634211504327251581044504587836, 171396940335846602791651151474323337431774, 586569028025374600641597039179964964387995, 2007834265323917253757049899333039302267879, 6874278057545250970049343695391148664820261, 23540448429773142466077904573335189361841146, 80628531015006211846328708884948573750017576, 276214953782495272087071953560836234984992169, 946429263242491965516134301386838627925692721, 3243469661931429236639328461811178056663012609, 11117585032466893531095732927653115924601993220, 38114331737181906074426303272412309451211211938, 130689812483582935654207641601137877350234682175, 448197194055487645330311359376192797256643126113, 1537336552721870275224625219617262952734884104961, 5273994732336445044772404080123129241981031636609, 18095887648452573511691589103306868215325395824288, 62099505131703685747837432631051954585141482824802, 213139069826767902014578530159155728561892598849585, 731649942179954970102764822696005646954551565253641] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 4 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH + HH H + H HH + HH H 4+ H HH HH + H HH HH H + HH HH HH HH 3+ H H H H + HH H HH HH + HH HH H HH 2+ HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 2 1 + P(x) x + P(x) x + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence 2 (518 n - 1031 n + 780) a(n - 1) a(n) = -1/195 -------------------------------- n + 2 3 2 (1036 n - 4335 n + 4367 n - 1170) a(n - 2) - 1/195 -------------------------------------------- (n + 2) (n + 1) 3 2 (2590 n - 11115 n + 11939 n - 1812) a(n - 3) + 1/195 ---------------------------------------------- (n + 2) (n + 1) 2 (n - 3) (1036 n - 1865 n + 1074) a(n - 4) + 1/65 ------------------------------------------ (n + 2) (n + 1) 161 (74 n - 55) (n - 3) (n - 4) a(n - 5) + --- ------------------------------------ 195 (n + 2) (n + 1) subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 1456286415952438964652364581682787386720721652723427223868652268704217615734\ 013661511014559584083968554107289545538396109682528052843351355027899510\ 336875839145124807012715538480932028100745003225030603205050905923754789\ 120913436417646081847745149534148691270052401108733311860953962982834165\ 142352139116820596104418788220221329115699403613078628786955744338071295\ 785210326056171164845618097 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 4, 8, 13, 31, 71, 144, 318, 729, 1611, 3604, 8249, 18803, 42907, 98858, 228474, 528735, 1228800, 2865180, 6693712, 15676941, 36807239, 86584783, 204060509, 481823778, 1139565120, 2699329341, 6403500057, 15211830451, 36183117255, 86171536894, 205459894230, 490417795075, 1171809537516, 2802705573178, 6709741472212, 16077633792535, 38557470978465, 92543928713413, 222292252187223, 534346494991678, 1285375448785399, 3094086155360823, 7452774657572624, 17962853486458385, 43320517724198861, 104535253974750259, 252390609561764423, 609699521485999914, 1473608895007574938, 3563397472304118410, 8620933920694737411, 20866285270840567272, 50527739882573522706, 122405798503806040965, 296657476936886314309, 719256672330233630085, 1744545018773625965860, 4232966162296687137816, 10274640808649742568633, 24948409910226791411907, 60599469176019760272535, 147244830551438483529790, 357892285312932720768838, 870164978202711215226483, 2116332778149031726542808, 5148677470688468794716894, 12529484860432553111063488, 30499528076281485551927755, 74262935424150986850220725, 180870207357836188170341069, 440630810522439231975963839, 1073724026979876073732588658, 2617084008072054452326466527, 6380388743696091681799439061, 15558886191823487315141504508, 37949779254079097680017924283, 92584221613943667202286169191, 225922433438019608173638548805, 551409374869368431174631180753, 1346105758407373798214652723522, 3286793196606678831626181940722, 8026971507652658919500628768874, 19607185640982275440523729181033, 47902823795394726062462438796736, 117054324948067108888357885549374, 286083328086281443692788683942327, 699317934466646636669713105276839, 1709748244857899463255482766619759, 4180839106559551650627420507840427, 10225086357877339275440446600065760, 25011587559313816213376886296261184, 61190615947546730766871192612064833, 149725673741988363612845353038008205, 366415846156516055604065564079552535, 896845046441850691487935128438196159] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 5 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ H HH HH + HH H HH HH + H HH H H + HH H HH HH 4+ H HH H H + H HH H H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 2 2 1 + (x + x) P(x) + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence 2 (n - 1) a(n - 1) 3 (n - 1) a(n - 2) a(n) = ------------------ + ------------------ n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 1] Just for fun, a(1000), equals 1530036790419022830094068760617849229908556040379774601663141696520714432010\ 505204701121726795033893282006994307845948795123386656757478216103147793\ 598816422588288234800268711441603480859153836164038767267854510780820952\ 045840116183968093556741601444043981217670681866042373596290010701135538\ 838865417082061461269201435105172255513381628729705648108431316186711898\ 379048365060433595579837015920602409701854829405120869376654505085317762\ 1901271877999637563068542637350933457 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 3, 6, 15, 36, 91, 232, 603, 1585, 4213, 11298, 30537, 83097, 227475, 625992, 1730787, 4805595, 13393689, 37458330, 105089229, 295673994, 834086421, 2358641376, 6684761125, 18985057351, 54022715451, 154000562758, 439742222071, 1257643249140, 3602118427251, 10331450919456, 29671013856627, 85317692667643, 245613376802185, 707854577312178, 2042162142208813, 5897493615536452, 17047255430494497, 49320944483427000, 142816973618414817, 413887836110423787, 1200394300050487935, 3484084625456932134, 10119592485062548155, 29412628894558563849, 85543870540455597789, 248952602654003411640, 724947137834104063053, 2112261618875209962525, 6157879192714893166503, 17961708307164474879078, 52418979494565497284659, 153054402342387832806318, 447107296039753836151995, 1306709599137475613111808, 3820682066516442811470123, 11176109832763802046866481, 32705601410484246215745189, 95747934502509579263312730, 280418620117853741708024169, 821578511126260089293299449, 2407969409295125052827266131, 7060047856454817331912175136, 20706869494801128932088054675, 60752886013114747805209321971, 178303876727715987264460117881, 523470228309211183146132538770, 1537292873088850037153655419037, 4515968752463518800863883219540, 13270012942708831885430137279857, 39004474517326916925520791131352, 114677148186439520720469807592881, 337252779926837166106515094143507, 992081497804863208806272348440575, 2919102839611001046292805620867782, 8591299535354652688143556684853307, 25291409899803750802286391976502211, 74471367333926485356188553908612137, 219333623772940705482181740240713080, 646127582088681087104424825527569497, 1903821367984370378973123565306390657, 5610824882652789101159297569379125339, 16539320523461975633674292210615157006, 48763733724885023891037362712600616695, 143801214724243338894845383828477461126, 424143211957453170823189308473526283071, 1251252511019022217034672218022874172864, 3691969061033252211450145056818715608239, 10895571836533928224569430533625487349525, 32160232558185513877392752232595140415729, 94943198059462752589589672745512131329394, 280338312871514003720592178984834743192165, 827891507006386759684746014201909924531418, 2445317582470051292788355811433410180110685, 7223813569919277908209909876381273600472448, 21343507566294190307011455122677671120241053, 63071286725499290051879587077009179781061461, 186407292265725088628262974383000850686750119] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 6 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 2 2 1 + P(x) x + (x - x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 1) a(n - 1) (n - 2) a(n - 2) (2 n - 7) a(n - 3) a(n) = ------------------ + ---------------- + ------------------ n + 1 n + 1 n + 1 (n - 5) a(n - 4) - ---------------- n + 1 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2] Just for fun, a(1000), equals 1258976834224262773199644936372839877301103060006870706162324567937209163404\ 664302285780840930405187497908781943435725250693212008816722364247013884\ 666083868760309087497764707542210251550367470907064283210333730955232698\ 389440175341142704919913139331694533974031860259334845579975521445364654\ 700207290194856330574815120730458306294903984090244990972375467298717489\ 01445379751297218627473874798563437753301835209165 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 2, 4, 8, 17, 37, 82, 185, 423, 978, 2283, 5373, 12735, 30372, 72832, 175502, 424748, 1032004, 2516347, 6155441, 15101701, 37150472, 91618049, 226460893, 560954047, 1392251012, 3461824644, 8622571758, 21511212261, 53745962199, 134474581374, 336908488839, 845139060165, 2122553644686, 5336735929371, 13432403613621, 33843022209066, 85349327734485, 215440028338359, 544288586926914, 1376230297675914, 3482537223611046, 8819175375714063, 22349794473772659, 56678600914995057, 143830921235537742, 365225623668676437, 927972354829010775, 2359192024476568203, 6001174121892988758, 15273713134056377698, 38893747432145085266, 99090832134641995427, 252579381177903040849, 644118340220292169786, 1643348924746923013481, 4194532932723720267271, 10710773165730370402070, 27361217667381195152609, 69923263927774760117419, 178761583832906815958299, 457180542019634361749654, 1169653910683020997823700, 2993493968182857335738916, 7663836950023084292126586, 19627124209913879819201256, 50281185027971273570344779, 128851301008215990676245297, 330295607482296149639113771, 846922848867278127081934118, 2172243398314031502060434813, 5573055540747246795936497203, 14301951559375317288722742625, 36712267090479571354186761752, 94262318866766131085885820862, 242087967735412291153757221292, 621890217530867044998372625244, 1597927417599990976164331285618, 4106772441264045401019924649921, 10557037252659735639822884541089, 27144318295978988020876731613899, 69808615378820015816460193046634, 179568484819409906464233459965565, 461998012612770916903282585499931, 1188877068859680412470053314034196, 3059980617900905437254279674385261, 7877408686568953921404087246339411, 20282861001228149602530202549462410, 52234134723235412099021791134645474, 134541797507827311283829108795938674, 346605946314513254492433135097630809, 893077485129793636878895057273178901, 2301521609537728551186835553085928773, 5932151623905973421624468114595077244, 15292526112023196667544094397358322057, 39428894200253097844359818441341768857, 101675651191856047918093722573856681533, 262231577763896699071580185616362344474] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 7 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H H + HH H H HH + H HH HH H + HH H H HH 4+ H H H H + H HH HH H + HH H H HH + H HH H H 3+ H H H H + HH H HH HH + H HH H H + HH HHH HH 2+ H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 3 2 2 2 1 + (x + x + x + x) P(x) + (-x - x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (n - 2) a(n - 1) 2 (n - 2) a(n - 2) (4 n - 11) a(n - 3) a(n) = ---------------- + ------------------ + ------------------- n + 1 n + 1 n + 1 (8 n - 25) a(n - 4) 6 (n - 4) a(n - 5) (5 n - 22) a(n - 6) + ------------------- + ------------------ + ------------------- n + 1 n + 1 n + 1 3 (n - 5) a(n - 7) + ------------------ n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 2, a(5) = 3, a(6) = 7, a(7) = 17] Just for fun, a(1000), equals 5032496365637955067683347870950409710915701764522282276774675157243603802582\ 866298211231210653006394070833138976334822500759759891785781976829184637\ 643038378788330330417356109675556672422365107611262498459448932887962130\ 643216278920681957917488069930040634733745054315448220108085618003032954\ 450492248812276239145688825121717358769928635223051275485462691699729085\ 004111315387526723342034239842070602836639078569636160428330200510115437\ 8 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 2, 3, 7, 17, 39, 91, 219, 533, 1307, 3234, 8067, 20255, 51150, 129839, 331109, 847876, 2179289, 5620427, 14540036, 37721539, 98116305, 255818920, 668475637, 1750371384, 4592023450, 12068440795, 31770262976, 83765896945, 221181546661, 584827643203, 1548351907603, 4104332864117, 10892248356011, 28937953375916, 76960653386603, 204879046734386, 545925151025906, 1455981438846053, 3886400469418666, 10382245378672359, 27756918405827983, 74263029726249001, 198829910977439625, 532703401910546531, 1428143266813608445, 3831150646223428804, 10283645508579576861, 27619465812286909488, 74220560789749220334, 199556083749746908195, 536820382086347568072, 1444801763827645841740, 3890408061198204485087, 10480511016119495728047, 28246414132024260752912, 76160755522727900890180, 205437651850071834885351, 554375257812010095795798, 1496570004169021588945828, 4041609047370900210653163, 10918690306434774952578648, 29508078135961103546885935, 79773895103635321195662633, 215737548410098076340647126, 583621812616151827778503173, 1579334324213728450307002233, 4275132085798309736678255810, 11575884594778904645167342803, 31353387126118713173448037627, 84944829539009789690078850478, 230201612728140081149731098573, 624015753012404624623636602384, 1691981062322825431740014538160, 4588865052096950827456946241043, 12448646109079949902773787405256, 33778729419959874702068588413132, 91678238541123561975081610471607, 248879128104875104185719943760541, 675783451463072308953369784249936, 1835359604551371265619822938995004, 4985710857337519919668459945842445, 13546376824511567847839151376237576, 36813514224468113012736613431442804, 100063905775460344278497620851740509, 272039349875194325299625521607283254, 739721482329877908140333154428499903, 2011801570464323989823815200231900135, 5472436183516459408824867590415304632, 14888577978323084494016869894352271445, 40513618989439805975549944975329369435, 110261163534686841700263792854738579832, 300134742948778688458516646666561272035, 817110224328981701161007905911583958297, 2224919190647836747971327271445212007300, 6059205160189587412826547467523630270925, 16503783917191051549681811499901147080856, 44958988651672462155755561208723586036494] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 8 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH + HH H + H HH + HH H 4+ H HH + H HH + HH HH 3+ H H H + HH H HH HH + HH HH H HH 2+ HH HH H HH + H H HH H + HH HH H HH + H H HH H 1+ HH H HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 1 + (x + x) P(x) + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 1) a(n - 1) (2 n - 1) a(n - 2) 3 (2 n - 5) a(n - 3) a(n) = ------------------ - ------------------ + -------------------- n + 1 n + 1 n + 1 (n - 2) a(n - 4) 2 (2 n - 7) a(n - 5) - ---------------- + -------------------- n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 3134627918704846396589045582068293464668313386749891925759492579301366817419\ 266879466277019895299966556024437351904982229786354769359885950645415854\ 487762568754124509622513897418551096745213564746196714896494879980931898\ 708502918937008296833152741264657083164783324054454272242916859440693717\ 64365001116217342900388877265443604205502181219138765202835544779530 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 42, 84, 170, 354, 740, 1549, 3269, 6945, 14811, 31711, 68177, 147091, 318313, 690837, 1503351, 3279445, 7169907, 15708485, 34482475, 75830981, 167042763, 368548926, 814341362, 1801867812, 3992172298, 8855912464, 19668236110, 43729885814, 97330067640, 216844435346, 483571453768, 1079354856828, 2411240775366, 5391038526802, 12062679137620, 27010935099196, 60526554883902, 135721815227704, 304535849636366, 683755276072670, 1536121565708096, 3453050076522008, 7766456476976834, 17477390474177854, 39350957376284872, 88644320911742304, 199781711595784890, 450466438264400446, 1016164744759318568, 2293264805607373804, 5177575503207130558, 11694317639185843274, 26423715558659122260, 59728010995453379421, 135058602016778295285, 305507289981338348257, 691307640696528081979, 1564829833276593856647, 3543272754560789200681, 8025639916478510155339, 18183959683820504446497, 41212333247384488368873, 93431281455612601484243, 211875294974960472423745, 480604617462452614426815, 1090465639820281676643073, 2474853003558648387670983, 5618201029263006344110809, 12757123617698163352573711, 28974313940315478259062979, 65822724585319538752564981, 149567876677718197286698671, 339936780414026450535924453, 772774892529317395366017603, 1757116730487566683312107729, 3996122690354078859729209647, 9090033908542877936842438557, 20681339112467629598326301035, 47062642985297219558085765929, 107116551464339266855499450015, 243847122183568901454753070765, 555210537090801909689698475727, 1264372572489361483075032316117, 2879838126594606020044275873923, 6560472630207720552639437125041, 14947710059210582396858604974621, 34063179534729044501719950750975, 77636370512116064921945474111629, 176975591352231844422902101233579, 403485801251068480867972873270661, 920043891167408188870554973636539] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 9 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H H + HH HH 4+ H H + H H + HH HH + H H 3+ H H H H + HH H HH HH H HH + H HH H H HH H + HH HHH HHH HH 2+ H H H H + H H + HH HH + H H 1+ H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-+*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 2 2 1 + P(x) x + (-x + x) P(x) - P(x) = 0 The sequence a(n) satisfies the linear recurrence 3 2 6 (787 n - 494) a(n - 1) 3 (1554 n - 925 n - 1297 n + 494) a(n - 2) a(n) = ------------------------ + -------------------------------------------- (n + 2) (518 n + 741) (518 n + 741) (n + 2) (n + 1) 3 2 2 (1036 n - 11877 n + 20213 n - 3576) a(n - 3) + ------------------------------------------------ (518 n + 741) (n + 2) (n + 1) 2 (n - 3) (5698 n + 36483 n - 36982) a(n - 4) + -------------------------------------------- (518 n + 741) (n + 2) (n + 1) 46 (518 n - 325) (n - 3) (n - 4) a(n - 5) + ----------------------------------------- (518 n + 741) (n + 2) (n + 1) subject to the initial conditions [a(1) = 1, a(2) = 1, a(3) = 2, a(4) = 6, a(5) = 17] Just for fun, a(1000), equals 1942638687787092634075059070839736201209022077282946815319037275653135842929\ 978071198907945657903781172617466363083969224160153246854962037619385875\ 192569342225163855512591691485048587058019571412725666535401525169319932\ 852940177353450721537703813480194459471273198171052012041742824260866597\ 199220430130414454153445911235037447684813910308642456487574707373973037\ 276160821493383066116278973777900945532710463532561160938805314663555118\ 339753742178503987938837722892959983615596644831529369044770946424540593\ 258068493526378634959570733961 For the sake of the OEIS here are the first, 100, terms. [1, 1, 1, 2, 6, 17, 46, 128, 372, 1109, 3349, 10221, 31527, 98178, 308179, 973911, 3096044, 9894393, 31770247, 102444145, 331594081, 1077022622, 3509197080, 11466710630, 37567784437, 123380796192, 406120349756, 1339571374103, 4427077704043, 14657212174545, 48609031351173, 161460865003722, 537105965807404, 1789190378916633, 5967904354732863, 19930752893117029, 66639631673845377, 223059824945930906, 747420469638736654, 2506919354940523274, 8416394534562275811, 28281491440851685236, 95115345845065123216, 320150194302022177713, 1078439633123050270009, 3635480814252528228147, 12264160996801189321670, 41400988028286624177559, 139851833561420188254209, 472714241540768044463088, 1598790005917078894380294, 5410484616453450209084881, 18319907822529254477956157, 62064561502420172514647235, 210372420268090214349596878, 713428575983969700667466980, 2420589227918231406542698699, 8216621938018388176822662971, 27903598584399296257747196715, 94801512792362867499229775636, 322219801408067247729427624654, 1095633138464539157194002219629, 3726905870873674851438742776654, 12682259694557529324760955138096, 43172250297927157466472517692604, 147017036918999832986390216017401, 500819129565295097124372277542255, 1706632276262636234330249927828757, 5817555515344652723969052274855585, 19837117765438137202457229641539242, 67662829147973169471601347696638294, 230861512329460824108516612679165086, 787914574556023607795027232940304227, 2689858931357235585645670820603855488, 9185426878574482655601935099075111524, 31375126323522571901608286578321312301, 107197553528773808298644304964862882709, 366348644161261571053552102947752288843, 1252309890606893996404962284901053156800, 4281873475622593649515090503337074602167, 14643943307369843047960278107739379399735, 50093563145634211504327251581044504587836, 171396940335846602791651151474323337431774, 586569028025374600641597039179964964387995, 2007834265323917253757049899333039302267879, 6874278057545250970049343695391148664820261, 23540448429773142466077904573335189361841146, 80628531015006211846328708884948573750017576, 276214953782495272087071953560836234984992169, 946429263242491965516134301386838627925692721, 3243469661931429236639328461811178056663012609, 11117585032466893531095732927653115924601993220, 38114331737181906074426303272412309451211211938, 130689812483582935654207641601137877350234682175, 448197194055487645330311359376192797256643126113, 1537336552721870275224625219617262952734884104961, 5273994732336445044772404080123129241981031636609, 18095887648452573511691589103306868215325395824288, 62099505131703685747837432631051954585141482824802, 213139069826767902014578530159155728561892598849585, 731649942179954970102764822696005646954551565253641] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 10 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH H + HH H HH H + H HH H H + HH HHH HH 4+ H H H HH + H H HH H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 3 2 2 2 1 + (x + x + x + x) P(x) + (-x - x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (n - 2) a(n - 1) 2 (n - 2) a(n - 2) (4 n - 11) a(n - 3) a(n) = ---------------- + ------------------ + ------------------- n + 1 n + 1 n + 1 (8 n - 25) a(n - 4) 6 (n - 4) a(n - 5) (5 n - 22) a(n - 6) + ------------------- + ------------------ + ------------------- n + 1 n + 1 n + 1 3 (n - 5) a(n - 7) + ------------------ n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 2, a(5) = 3, a(6) = 7, a(7) = 17] Just for fun, a(1000), equals 5032496365637955067683347870950409710915701764522282276774675157243603802582\ 866298211231210653006394070833138976334822500759759891785781976829184637\ 643038378788330330417356109675556672422365107611262498459448932887962130\ 643216278920681957917488069930040634733745054315448220108085618003032954\ 450492248812276239145688825121717358769928635223051275485462691699729085\ 004111315387526723342034239842070602836639078569636160428330200510115437\ 8 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 2, 3, 7, 17, 39, 91, 219, 533, 1307, 3234, 8067, 20255, 51150, 129839, 331109, 847876, 2179289, 5620427, 14540036, 37721539, 98116305, 255818920, 668475637, 1750371384, 4592023450, 12068440795, 31770262976, 83765896945, 221181546661, 584827643203, 1548351907603, 4104332864117, 10892248356011, 28937953375916, 76960653386603, 204879046734386, 545925151025906, 1455981438846053, 3886400469418666, 10382245378672359, 27756918405827983, 74263029726249001, 198829910977439625, 532703401910546531, 1428143266813608445, 3831150646223428804, 10283645508579576861, 27619465812286909488, 74220560789749220334, 199556083749746908195, 536820382086347568072, 1444801763827645841740, 3890408061198204485087, 10480511016119495728047, 28246414132024260752912, 76160755522727900890180, 205437651850071834885351, 554375257812010095795798, 1496570004169021588945828, 4041609047370900210653163, 10918690306434774952578648, 29508078135961103546885935, 79773895103635321195662633, 215737548410098076340647126, 583621812616151827778503173, 1579334324213728450307002233, 4275132085798309736678255810, 11575884594778904645167342803, 31353387126118713173448037627, 84944829539009789690078850478, 230201612728140081149731098573, 624015753012404624623636602384, 1691981062322825431740014538160, 4588865052096950827456946241043, 12448646109079949902773787405256, 33778729419959874702068588413132, 91678238541123561975081610471607, 248879128104875104185719943760541, 675783451463072308953369784249936, 1835359604551371265619822938995004, 4985710857337519919668459945842445, 13546376824511567847839151376237576, 36813514224468113012736613431442804, 100063905775460344278497620851740509, 272039349875194325299625521607283254, 739721482329877908140333154428499903, 2011801570464323989823815200231900135, 5472436183516459408824867590415304632, 14888577978323084494016869894352271445, 40513618989439805975549944975329369435, 110261163534686841700263792854738579832, 300134742948778688458516646666561272035, 817110224328981701161007905911583958297, 2224919190647836747971327271445212007300, 6059205160189587412826547467523630270925, 16503783917191051549681811499901147080856, 44958988651672462155755561208723586036494] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 11 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 8 7 6 4 3 5 4 3 2 1 + P(x) x + (x - x - x ) P(x) + (-x + 2 x - x + x) P(x) 3 2 + (x - x - 1) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 5, 11, 24, 57, 144, 372, 975, 2599, 7034, 19255, 53180, 148003, 414669, 1168583, 3309965, 9417619, 26903482, 77135132, 221882675, 640167987, 1852051736, 5371604821, 15615701322, 45493557800, 132801218479, 388380892804, 1137793101374, 3338640756508, 9811444976478, 28874410514343, 85089055499271, 251062242039930, 741661480428401, 2193404109916285, 6493732624209176, 19244612927417149, 57087494918009214, 169499896222369916, 503704245748823670, 1498103831591091720, 4459148690324974746, 13282824473905760724, 39595247125992327380, 118112658824000350935, 352564073724228762263, 1053067267819047647894, 3147310248208346318691, 9411914397201715689446, 28161873328356415853499, 84310374304806162800388, 252538536047080810063920, 756821047479734870004282, 2269184594955084652062778, 6806911847737199160327252, 20428057745042393973886164, 61332972248531115276166721, 184223316897519623345142493, 553570340754024262883587076, 1664075129179954456113236918, 5004255670510395419647992850, 15054533179282941656336544480, 45305547064101101002014555081, 136391407800832118701427545506, 410742526317450555973886691132, 1237357027459076679860600273061, 3728712980383529042117290872416, 11239771653000251283063632954872, 33891191647300845836892120270716, 102221764972718918420287191970977, 308406563867691955783949538208132, 930731064375205610352860394179535, 2809583584687925692759767734567619, 8483474785894820513981510960380932, 25622216546905410445960913957344384, 77404797856905999765156293712962098, 233896963757697561811498319225808515, 706942629003345757182913931191141168, 2137194760439596020395041505716809732, 6462520419710446535873907396397948970, 19545883025630706093090159307574417520, 59129189778381423499055502557443282788, 178912066929076132529225458012038924442, 541459893156594813899117463704826359276, 1639003438384079332823732087668677809459, 4962246944540204164864398045428068552078, 15026571786218811427963335269362683689306, 45511655441436972602494360433292184459387, 137868406725038587813500458989008504213356, 417719314653591198430239389885464003185415, 1265844513126222100424179966483016753211211, 3836635671525142162469091325441479452161765, 11630371363301668270248776998500012831643793, 35262078868245586492590345377534907667989780, 106928157055023767988446407693621821087682601, 324298294910417419773262407966647764772501890, 983703647690565263058909078056959212444156902, 2984348874248612535224147399385619233085165606] ------------------------------------------------------------ Theorem Number, 12 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 3 2 3 1 + P(x) x + (-x + x) P(x) + (x - x - 1) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 45, 90, 187, 401, 851, 1815, 3925, 8517, 18528, 40542, 89084, 196294, 433960, 962393, 2139725, 4768821, 10653090, 23847852, 53488512, 120189110, 270525714, 609872556, 1376944183, 3113165351, 7047902436, 15975594178, 36254691983, 82367121385, 187327128656, 426463015182, 971794420430, 2216459060360, 5059629805741, 11559371030889, 26429562274813, 60474319746292, 138472393707020, 317287830492994, 727493771194568, 1669087821473347, 3831706461817534, 8801529240735440, 20228664934673007, 46516855554565603, 107023436481333016, 246356874740823900, 567361687949009896, 1307246629320351536, 3013355124126816199, 6949151353923786831, 16032292599544117909, 37002915021440054034, 85437203538695002412, 197343801364604072373, 455995108480664878088, 1054028117510157798185, 2437220586009809344922, 5637462008726047404420, 13044104958896714457266, 30191362314092522717088, 69901256387316040143189, 161889000866928364911673, 375038792667449550811129, 869076638050116790409663, 2014464785302280486552292, 4670655362261462738855867, 10832019860018976585731355, 25127635081038417757157497, 58304423258510204427627370, 135318255823699614530347478, 314133072780421992235087867, 729408461607772465334189902, 1694046655330387389384834123, 3935274500031711929656356229, 9143606614506172522351127413, 21249596815155851886618287769, 49393797017310482802102109124, 114836688005465996914314854101, 267038248650994935249780214935, 621082017773923851102098581060, 1444791837130910209523805410400, 3361558483284669762056309772047, 7822641317082942993733966553444, 18207145325605714380902195077507, 42384238894815170678689367097517, 98682343013046975787558512672747, 229797628412303496855469928034966, 535206276701024419566301617271889, 1246708738543803536060775647603284, 2904528122457933390105864083112466, 6767863837948236105079296085122986] ------------------------------------------------------------ Theorem Number, 13 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 7+ HHH + H HH + HH HH 6+ H H + H H + HH HH 5+ HH HH + H H + HH HH 4+ H H + HH HH 3+ H HH HH + HH H HH HH + HH HH HH HH 2+ HH HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 2 1 + P(x) x + P(x) x + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence 2 (518 n - 1031 n + 780) a(n - 1) a(n) = -1/195 -------------------------------- n + 2 3 2 (1036 n - 4335 n + 4367 n - 1170) a(n - 2) - 1/195 -------------------------------------------- (n + 2) (n + 1) 3 2 (2590 n - 11115 n + 11939 n - 1812) a(n - 3) + 1/195 ---------------------------------------------- (n + 2) (n + 1) 2 (n - 3) (1036 n - 1865 n + 1074) a(n - 4) + 1/65 ------------------------------------------ (n + 2) (n + 1) 161 (74 n - 55) (n - 3) (n - 4) a(n - 5) + --- ------------------------------------ 195 (n + 2) (n + 1) subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 1456286415952438964652364581682787386720721652723427223868652268704217615734\ 013661511014559584083968554107289545538396109682528052843351355027899510\ 336875839145124807012715538480932028100745003225030603205050905923754789\ 120913436417646081847745149534148691270052401108733311860953962982834165\ 142352139116820596104418788220221329115699403613078628786955744338071295\ 785210326056171164845618097 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 4, 8, 13, 31, 71, 144, 318, 729, 1611, 3604, 8249, 18803, 42907, 98858, 228474, 528735, 1228800, 2865180, 6693712, 15676941, 36807239, 86584783, 204060509, 481823778, 1139565120, 2699329341, 6403500057, 15211830451, 36183117255, 86171536894, 205459894230, 490417795075, 1171809537516, 2802705573178, 6709741472212, 16077633792535, 38557470978465, 92543928713413, 222292252187223, 534346494991678, 1285375448785399, 3094086155360823, 7452774657572624, 17962853486458385, 43320517724198861, 104535253974750259, 252390609561764423, 609699521485999914, 1473608895007574938, 3563397472304118410, 8620933920694737411, 20866285270840567272, 50527739882573522706, 122405798503806040965, 296657476936886314309, 719256672330233630085, 1744545018773625965860, 4232966162296687137816, 10274640808649742568633, 24948409910226791411907, 60599469176019760272535, 147244830551438483529790, 357892285312932720768838, 870164978202711215226483, 2116332778149031726542808, 5148677470688468794716894, 12529484860432553111063488, 30499528076281485551927755, 74262935424150986850220725, 180870207357836188170341069, 440630810522439231975963839, 1073724026979876073732588658, 2617084008072054452326466527, 6380388743696091681799439061, 15558886191823487315141504508, 37949779254079097680017924283, 92584221613943667202286169191, 225922433438019608173638548805, 551409374869368431174631180753, 1346105758407373798214652723522, 3286793196606678831626181940722, 8026971507652658919500628768874, 19607185640982275440523729181033, 47902823795394726062462438796736, 117054324948067108888357885549374, 286083328086281443692788683942327, 699317934466646636669713105276839, 1709748244857899463255482766619759, 4180839106559551650627420507840427, 10225086357877339275440446600065760, 25011587559313816213376886296261184, 61190615947546730766871192612064833, 149725673741988363612845353038008205, 366415846156516055604065564079552535, 896845046441850691487935128438196159] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 14 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H H + H HH HHH + HH H H HH + H HH HH H 3+ H H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH H H 1+ H H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 1 + (x + x) P(x) + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 1) a(n - 1) (2 n - 1) a(n - 2) 3 (2 n - 5) a(n - 3) a(n) = ------------------ - ------------------ + -------------------- n + 1 n + 1 n + 1 (n - 2) a(n - 4) 2 (2 n - 7) a(n - 5) - ---------------- + -------------------- n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 3134627918704846396589045582068293464668313386749891925759492579301366817419\ 266879466277019895299966556024437351904982229786354769359885950645415854\ 487762568754124509622513897418551096745213564746196714896494879980931898\ 708502918937008296833152741264657083164783324054454272242916859440693717\ 64365001116217342900388877265443604205502181219138765202835544779530 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 42, 84, 170, 354, 740, 1549, 3269, 6945, 14811, 31711, 68177, 147091, 318313, 690837, 1503351, 3279445, 7169907, 15708485, 34482475, 75830981, 167042763, 368548926, 814341362, 1801867812, 3992172298, 8855912464, 19668236110, 43729885814, 97330067640, 216844435346, 483571453768, 1079354856828, 2411240775366, 5391038526802, 12062679137620, 27010935099196, 60526554883902, 135721815227704, 304535849636366, 683755276072670, 1536121565708096, 3453050076522008, 7766456476976834, 17477390474177854, 39350957376284872, 88644320911742304, 199781711595784890, 450466438264400446, 1016164744759318568, 2293264805607373804, 5177575503207130558, 11694317639185843274, 26423715558659122260, 59728010995453379421, 135058602016778295285, 305507289981338348257, 691307640696528081979, 1564829833276593856647, 3543272754560789200681, 8025639916478510155339, 18183959683820504446497, 41212333247384488368873, 93431281455612601484243, 211875294974960472423745, 480604617462452614426815, 1090465639820281676643073, 2474853003558648387670983, 5618201029263006344110809, 12757123617698163352573711, 28974313940315478259062979, 65822724585319538752564981, 149567876677718197286698671, 339936780414026450535924453, 772774892529317395366017603, 1757116730487566683312107729, 3996122690354078859729209647, 9090033908542877936842438557, 20681339112467629598326301035, 47062642985297219558085765929, 107116551464339266855499450015, 243847122183568901454753070765, 555210537090801909689698475727, 1264372572489361483075032316117, 2879838126594606020044275873923, 6560472630207720552639437125041, 14947710059210582396858604974621, 34063179534729044501719950750975, 77636370512116064921945474111629, 176975591352231844422902101233579, 403485801251068480867972873270661, 920043891167408188870554973636539] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 15 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 3 2 3 1 + P(x) x + (-x + x) P(x) + (x - x - 1) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 45, 90, 187, 401, 851, 1815, 3925, 8517, 18528, 40542, 89084, 196294, 433960, 962393, 2139725, 4768821, 10653090, 23847852, 53488512, 120189110, 270525714, 609872556, 1376944183, 3113165351, 7047902436, 15975594178, 36254691983, 82367121385, 187327128656, 426463015182, 971794420430, 2216459060360, 5059629805741, 11559371030889, 26429562274813, 60474319746292, 138472393707020, 317287830492994, 727493771194568, 1669087821473347, 3831706461817534, 8801529240735440, 20228664934673007, 46516855554565603, 107023436481333016, 246356874740823900, 567361687949009896, 1307246629320351536, 3013355124126816199, 6949151353923786831, 16032292599544117909, 37002915021440054034, 85437203538695002412, 197343801364604072373, 455995108480664878088, 1054028117510157798185, 2437220586009809344922, 5637462008726047404420, 13044104958896714457266, 30191362314092522717088, 69901256387316040143189, 161889000866928364911673, 375038792667449550811129, 869076638050116790409663, 2014464785302280486552292, 4670655362261462738855867, 10832019860018976585731355, 25127635081038417757157497, 58304423258510204427627370, 135318255823699614530347478, 314133072780421992235087867, 729408461607772465334189902, 1694046655330387389384834123, 3935274500031711929656356229, 9143606614506172522351127413, 21249596815155851886618287769, 49393797017310482802102109124, 114836688005465996914314854101, 267038248650994935249780214935, 621082017773923851102098581060, 1444791837130910209523805410400, 3361558483284669762056309772047, 7822641317082942993733966553444, 18207145325605714380902195077507, 42384238894815170678689367097517, 98682343013046975787558512672747, 229797628412303496855469928034966, 535206276701024419566301617271889, 1246708738543803536060775647603284, 2904528122457933390105864083112466, 6767863837948236105079296085122986] ------------------------------------------------------------ Theorem Number, 16 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H H + H HH HHH + HH H H HH + H HH HH H 3+ H H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH H H 1+ H H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 2 3 1 + P(x) x + (x - x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 1) a(n - 1) (n - 2) a(n - 2) (2 n - 7) a(n - 3) a(n) = ------------------ - ---------------- + ------------------ n + 1 n + 1 n + 1 2 (n - 5) a(n - 4) (n - 8) a(n - 6) + ------------------ - ---------------- n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 2] Just for fun, a(1000), equals 1176950714989472039879867097164787467731168548390670636801992136416332972086\ 898277348698087993919571204091110820742448542481636855988358314899690938\ 672843649138648682133884841687298042373689270639308856986357452480963423\ 383401026848945091071625435678738959251923453512607097256262920947479103\ 356615910327226686348752585420237437 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 22, 42, 80, 152, 292, 568, 1112, 2185, 4313, 8557, 17050, 34089, 68370, 137542, 277475, 561185, 1137595, 2311014, 4704235, 9593662, 19598920, 40103635, 82185653, 168666493, 346613232, 713200114, 1469254621, 3030218948, 6256281188, 12930039374, 26748697772, 55386529370, 114785051382, 238083048103, 494216315763, 1026681547651, 2134372036796, 4440242721757, 9243424565624, 19254704030249, 40133535117994, 83701671288887, 174665494666782, 364684302692317, 761824952311410, 1592257031239222, 3329531677118927, 6965586177249102, 14579064797995464, 30527584089316653, 63949861857983311, 134018617814709631, 280972131660117384, 589289169477022354, 1236390172104441711, 2595012857019532078, 5448483097227962922, 11443510685976418890, 24042863051171641274, 50530333059247995344, 106231480858589059892, 223401249061635751536, 469943442677589917028, 988848424941723500999, 2081299761445939780379, 4381845845937552096915, 9227711560622784969822, 19437589516267668693817, 40954291792943394506568, 86310235142767979004036, 181940086112454163969993, 383614626529901470057969, 809021619679751516526841, 1706557543555582414238756, 3600603520716009828550165, 7598380174796313007709308, 16038212465609702505254728, 33859308082950214419570804, 71496599403133792320146486, 150999659234389445825596123, 318968628398739129595019281, 673906439126292155812743214, 1424062795029526232414783635, 3009781004697296226116225432, 6362316258145714947705909656, 13451435620129537321656044592, 28444187117982772743078140610, 60157315325649798906283809598, 127248293230730230066048509796, 269204737445574334829247116243, 569612221084905567519367331211] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 17 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} To make it crystal clear here is such a path of semi-length 8 3+ H H + H H H H + HH HH HH HH 2.5+ H H H H + H H H H + H H H H + H HH H 2+ H H H H + HH H H + H H H H 1.5+ HH HH HH HH + H H H H + H H H H 1+ H H H H H H + HH H H H H H H + H H H H H H H H + H H H H H H H H 0.5+ H H H H H H H H + H H H H H H H H +H HH HH HH H -*-+-+-+-+*-+-+-+-++-+-+-+-+*-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 2 1 + (x + 2 x - 3 x + 1) P(x) + (3 x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence (7 n - 13) a(n - 1) 2 (7 n - 16) a(n - 2) (7 n - 19) a(n - 3) a(n) = ------------------- - --------------------- + ------------------- n - 1 n - 1 n - 1 2 (2 n - 5) a(n - 4) + -------------------- n - 1 subject to the initial conditions [a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 9] Just for fun, a(1000), equals 3289346391626539101813155177572627650945910687625728029306154349919885008424\ 562012779719925592344491191388079707430456180678738568333584620067184234\ 513521592384389380963000499371452185839175905026267522285034926888559531\ 903736341262514195381952881729901941796423040191760323852471411203274149\ 569108108677836711233758950633967492428868260969935951989860441372979505\ 029895048630530489968962629868140382200656574544165511219873615499568672\ 766576537672005908506328820762696090709573416126643680531141550140136744\ 304118233012639564415569873853310741766620633869235964458061462241620851\ 75145927784775161 For the sake of the OEIS here are the first, 100, terms. [1, 1, 2, 4, 9, 22, 58, 163, 483, 1494, 4783, 15740, 52956, 181391, 630533, 2218761, 7888266, 28291588, 102237141, 371884771, 1360527143, 5002837936, 18479695171, 68539149518, 255137783916, 952914971191, 3569834343547, 13410481705795, 50506214531252, 190661396754623, 721309231245930, 2734340902348660, 10384729349891801, 39508833052959062, 150556764433793122, 574604028257812551, 2196133765002623111, 8404921968965752371, 32207692734494651832, 123567904825171570007, 474617269824566355386, 1824933678818837074702, 7024125634258699052147, 27061810866951754785671, 104356677565423473137257, 402775764533193562448982, 1555852247194778679374801, 6014766389029773513342902, 23270147619699743008876442, 90093831591035950206998211, 349054331386847956567890747, 1353257428538389364335407949, 5249831449006596883287218598, 20378710829070480349853765305, 79152379106432483868793868282, 307607755667932328273940701518, 1196099515989815008181990112109, 4653345274135966755121774940981, 18112685962605211418277062361519, 70536220680328737640650868003150, 274818405130762127363775023157511, 1071215083196600681122389714471826, 4177323183244069093685741008136202, 16296848936793099441935749419523779, 63604423103740186283024835074100419, 248338098813162270659452650321497622, 969987771171029999449381574110315199, 3790101265933944939618704480857715284, 14814671741206925284291813349880162932, 57927549942538373437547202127871692979, 226582129078227610530624915866698146289, 886562118118615134989272323449529514677, 3470016654677134080958962113677550103074, 13585913523362049921786656972261958849929, 53208035532322666331381772934235887854662, 208445752260644142910821280376855464645554, 816832150505028367285449929808641793404649, 3201792565891852961532086177745940693859197, 12553676574768161918442923601638547491147259, 49233755168825636040190509798702664485568774, 193137363402581189781843747869732287019888207, 757840984470589034800991146439643751861772044, 2974374049385115360730728310785431473727584884, 11676592865799907305271600189113407813938977097, 45849774529775678994287493317245372378896223759, 180076204941122301719476603297392517178068338299, 707409923937410757893225686357741965383144870722, 2779581290322485827219473017204671948137505116329, 10923927891708038398667186903772046854896908583564, 42940526005207159580738533951311940846400161042252, 168827401527645830488158392515333238343950696104299, 663901382984815930428284363518445433570702229671429, 2611243420694303752657197022109899747540266225714237, 10272412328169638252044717519540824646909475201719354, 40418206582627380727519801114662572244091122051893759, 159059428206188977677275334863625816582883878433966332, 626062804643219538493657461465488203619840743484621724, 2464625148912081487251507784009490002286894109768942799, 9704133613649570792653803635613114442747623661282312917, 38215017094610905301981615724011907881762196971724186529, 150515516977222850615573432724842861091525181107595601954] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 18 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H + HH + HH HH + H H + H H 3+ H H + H H + H HH + H H + H H 2+ H H H H + H H HH HH + H H H H H H + HH H HH HH H HH + H H H H H H 1+ H H H H H H + H H H H H H + H H H H H H + HH HH HH HH HH HH +H HH HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-*+-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 3 2 2 2 1 + (x + x - x - x + 1) P(x) + (x + x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence 3 (n - 2) a(n - 1) 2 (n - 4) a(n - 2) 6 (n - 3) a(n - 3) a(n) = ------------------ + ------------------ - ------------------ n - 1 n - 1 n - 1 2 (n - 4) a(n - 4) (5 n - 17) a(n - 5) 3 (n - 4) a(n - 6) - ------------------ + ------------------- + ------------------ n - 1 n - 1 n - 1 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 4, a(6) = 8] Just for fun, a(1000), equals 3763397943187744689557058298717036661701274707108865401916940111974607872077\ 658288045952433378143909073181283205238953536359626707954324027248931203\ 293142036206158332931880545284748241890280047855652032379329135254532027\ 418033699746548607596676475393378221638914474356962386199724649823596587\ 328530039058456194196361496424523167984070089076512580890702546756287815\ 032033874024553898640073004897040106444624152147473336171196216009277955\ 755465619716781799721328921010420649 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 2, 4, 8, 18, 41, 98, 240, 602, 1538, 3990, 10484, 27845, 74636, 201641, 548520, 1501147, 4130169, 11417537, 31697323, 88335563, 247032739, 693016665, 1949776933, 5500153093, 15553283123, 44080461094, 125191586993, 356241890080, 1015542517685, 2899897830889, 8293768547892, 23755451317723, 68136244293124, 195686735756336, 562704829653673, 1619967823913525, 4668866393038729, 13470098409287510, 38901041194789990, 112450505301046743, 325350499795352311, 942134675221160158, 2730415764681587458, 7919228360313768631, 22985836060498323909, 66764735360530456928, 194057426731047586386, 564413570865224960203, 1642620166663351240445, 4783420332146312182912, 13937710312803494005796, 40633684407300598033641, 118526042023024400779817, 345911977622328522715727, 1010028323307390500414420, 2950597469676623473580492, 8623579438992579475081299, 25215028507019728783868248, 73759693396276342547566305, 215854080907620693861830681, 631941720978248049121784338, 1850818211226686502653504084, 5422694078322422657486457016, 15893733753413182801895787348, 46600585250786617171607185753, 136680562222566567550660388647, 401021482382448314332554769888, 1176981830637742252320304842085, 3455485266944898602230379995380, 10148026678529994810685008317507, 29811490327627053532556549200311, 87601537368610156299641801091562, 257491138454528754065273560703219, 757063086930258548848658706013351, 2226475475420250467994521359008779, 6549630223943234361577342672535706, 19271953848944351261094294070549004, 56720754945695480096258405339418098, 166979380329006582686255781263536965, 491683542308681667998212844009195570, 1448131419704670211913722491631048010, 4266063776811701283206969539871248671, 12570178524515842724568644540915080142, 37046572745428956862697268525140367578, 109205610321775427616856875957768924740, 321980889215005203257661276943201041326, 949514174948532341005286932136918262543, 2800638604072808547641255591093610467486, 8262187851536834686256841645171148545066, 24378872289013286125654724617628842666447, 71946711756248222025227206505790090246738, 212366188207300131383966258865261445645756, 626953350753974988907463750838598419202540, 1851223774793914375545886271767185599658694, 5467073315060779063517258382870052587610843, 16148108703282747519769980332748062103004374, 47704326029009105434055253692071275392467484] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 19 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ H HH HH + HH H HH H + H HH H H + HH HHH HH 4+ H H H + H H + HH HH 3+ H H H + HH HH HH HH + HH HH H HH 2+ HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 3 -1 + (x - 3 x + 6 x - 7 x + 7 x - 4 x + 1) P(x) 4 3 2 2 2 + (-3 x + 5 x - 9 x + 8 x - 3) P(x) + (2 x - 4 x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 5, 12, 29, 74, 199, 554, 1580, 4599, 13629, 41007, 124950, 384808, 1196010, 3747027, 11821256, 37523229, 119754660, 384047582, 1236976421, 3999780023, 12979240458, 42253474511, 137960509242, 451669185680, 1482402326444, 4876508223150, 16075907075992, 53100657695590, 175720809546355, 582495048581828, 1934011326455707, 6431021929933012, 21414877019549000, 71405175936331293, 238390696364328014, 796827183622750392, 2666406700941573674, 8932035228863731542, 29951087039643233174, 100528914233734415521, 337726663331342107359, 1135577613895114232222, 3821447399021126861689, 12870108500369910092496, 43377502675993578865587, 146305498040330514857658, 493806920018551613984013, 1667790945150358726863700, 5636405421129196150426245, 19060222673631623909309669, 64492388315933639743827121, 218340300664204189679262424, 739596870103919458535559424, 2506589512129250439904929232, 8499436166773644383255497931, 28834203095531408174101316777, 97865435244147141643724580885, 332312929827145883877564896974, 1128898818275654051949507939112, 3836598428445000055724268993497, 13044132578449189905222612800257, 44366572070761163512125173414595, 150960354073285919729699123970279, 513843778173627271234045629372531, 1749667435722230424981844473882405, 5959796016071331726800894082927865, 20307396272010013338048407694338611, 69218109534357021809344240032151246, 236006375173418561456532818720534102, 804937742592748279543019011219757331, 2746196546755413745801079650875838825, 9371909379169374468353399703869545521, 31992501744803496108426994087772295442, 109241748360914014259847741134534865867, 373117994182316404320539025462855575157, 1274728742041129800013073990663875472910, 4356126017486989008745040937927098462749, 14889884127520776439199122505070866766943, 50908193937206048172852892645702958973655, 174095247164727931352474741478183236910534, 595506430413343673847234697689138746032752, 2037435488027457279179217158357126956782541, 6972311652168179652233080636088514515646073, 23865083446147062045271868163117891384795476, 81703408011380205708376672572522315362235932, 279773335179699265305006937822281020199174023, 958207041437321309580759936460430793948328530, 3282443900489104842728265442779956311398980686, 11246523018497041223979591658610452594266227547, 38540771747913170357268601376105839730333735506, 132099727203511387502602637549167246847556748004, 452857008965978701094742813830497985574565158089, 1552731118955646347123917939554121904818246830295, 5324830582291895640932544203205749100761200265373, 18263673804706504903253280721851321222705981935308, 62652987313369405584985020744074808260772456463574, 214963759678046363792947428718327637564925230657702] ------------------------------------------------------------ Theorem Number, 20 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 3 4 3 2 2 -1 + (x + 2 x - x + x - x + 1) P(x) + (-2 x + x - 2 x + 2 x - 3) P(x) 2 + (x - x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 46, 94, 199, 439, 961, 2113, 4728, 10638, 24004, 54567, 124738, 286100, 658902, 1523561, 3533694, 8219807, 19175241, 44844787, 105117745, 246938893, 581271855, 1370801044, 3238379640, 7662884761, 18160172211, 43099390904, 102426035307, 243726755578, 580655580178, 1384932709103, 3306794777897, 7903693118041, 18909278995761, 45281554377074, 108530027712894, 260341203346223, 625005317744160, 1501608816996828, 3610336668014877, 8686460115399371, 20913632597618146, 50384354111881549, 121459160713524172, 292969666722436726, 707070780696631056, 1707424831884850449, 4125246497394886642, 9971935177734506502, 24116945055530422878, 58354041165562962555, 141259692927959905224, 342103485342872695337, 828862025018976189145, 2009029841925232601585, 4871515952686337473281, 11817073539321134031521, 28675998351756531656835, 69612131191902560435320, 169045763869085582882160, 410650213529812223368770, 997892674377742909288876, 2425690847473912236264926, 5898246126299524653958956, 14346379729918995608354591, 34905194678328946279956726, 84949845555910352980918162, 206802853409472003358290619, 503580166605875837883509918, 1226579626343965688411938881, 2988373791468938272294751313, 7282546274134082391128998885, 17751619375203540121720680747, 43280914162805330098090017988, 105549463086185917149368841094, 257462696364279602325677747884, 628158000160191822477935767084, 1532912888447769419915804392627, 3741604135787909331331389618256, 9134564337311934498324607449420, 22305163745068650394954160327031, 54476421551688686284866768832516, 133074664291962641673573493873036, 325135188262952797925515855558143, 794534125103762472134490936108882, 1941955965184663382273487480183164, 4747257141764020796262302049059161, 11607028692460250300264994515896655, 28383941376456748393597046710009226, 69421845442964953819449619451462120, 169820428896576356561050206615538112, 415482369434004891765163304919548259] ------------------------------------------------------------ Theorem Number, 21 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 3 2 2 2 1 + (x + x - x - x + 1) P(x) + (x + x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence 3 (n - 2) a(n - 1) 2 (n - 4) a(n - 2) 6 (n - 3) a(n - 3) a(n) = ------------------ + ------------------ - ------------------ n - 1 n - 1 n - 1 2 (n - 4) a(n - 4) (5 n - 17) a(n - 5) 3 (n - 4) a(n - 6) - ------------------ + ------------------- + ------------------ n - 1 n - 1 n - 1 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 4, a(6) = 8] Just for fun, a(1000), equals 3763397943187744689557058298717036661701274707108865401916940111974607872077\ 658288045952433378143909073181283205238953536359626707954324027248931203\ 293142036206158332931880545284748241890280047855652032379329135254532027\ 418033699746548607596676475393378221638914474356962386199724649823596587\ 328530039058456194196361496424523167984070089076512580890702546756287815\ 032033874024553898640073004897040106444624152147473336171196216009277955\ 755465619716781799721328921010420649 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 2, 4, 8, 18, 41, 98, 240, 602, 1538, 3990, 10484, 27845, 74636, 201641, 548520, 1501147, 4130169, 11417537, 31697323, 88335563, 247032739, 693016665, 1949776933, 5500153093, 15553283123, 44080461094, 125191586993, 356241890080, 1015542517685, 2899897830889, 8293768547892, 23755451317723, 68136244293124, 195686735756336, 562704829653673, 1619967823913525, 4668866393038729, 13470098409287510, 38901041194789990, 112450505301046743, 325350499795352311, 942134675221160158, 2730415764681587458, 7919228360313768631, 22985836060498323909, 66764735360530456928, 194057426731047586386, 564413570865224960203, 1642620166663351240445, 4783420332146312182912, 13937710312803494005796, 40633684407300598033641, 118526042023024400779817, 345911977622328522715727, 1010028323307390500414420, 2950597469676623473580492, 8623579438992579475081299, 25215028507019728783868248, 73759693396276342547566305, 215854080907620693861830681, 631941720978248049121784338, 1850818211226686502653504084, 5422694078322422657486457016, 15893733753413182801895787348, 46600585250786617171607185753, 136680562222566567550660388647, 401021482382448314332554769888, 1176981830637742252320304842085, 3455485266944898602230379995380, 10148026678529994810685008317507, 29811490327627053532556549200311, 87601537368610156299641801091562, 257491138454528754065273560703219, 757063086930258548848658706013351, 2226475475420250467994521359008779, 6549630223943234361577342672535706, 19271953848944351261094294070549004, 56720754945695480096258405339418098, 166979380329006582686255781263536965, 491683542308681667998212844009195570, 1448131419704670211913722491631048010, 4266063776811701283206969539871248671, 12570178524515842724568644540915080142, 37046572745428956862697268525140367578, 109205610321775427616856875957768924740, 321980889215005203257661276943201041326, 949514174948532341005286932136918262543, 2800638604072808547641255591093610467486, 8262187851536834686256841645171148545066, 24378872289013286125654724617628842666447, 71946711756248222025227206505790090246738, 212366188207300131383966258865261445645756, 626953350753974988907463750838598419202540, 1851223774793914375545886271767185599658694, 5467073315060779063517258382870052587610843, 16148108703282747519769980332748062103004374, 47704326029009105434055253692071275392467484] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 22 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 2+ H H H H + HH HH HH HH + HH HH HH HH + H H H H H H H H + H H H H H H H H 1.5+ H HH H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H 1+ H H H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H 0.5+ HH H H H H H H HH + H H H H H H H H + H H H H H H H H +H HH HH HH H +H HH HH HH H -*-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-*+-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 5 4 2 2 3 2 1 + (x + x - x - x + 1) P(x) + (x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 1, 1, 2, 3, 6, 11, 22, 45, 95, 205, 452, 1013, 2303, 5299, 12318, 28887, 68261, 162378, 388524, 934445, 2257825, 5477978, 13340342, 32597052, 79895596, 196374894, 483913063, 1195304610, 2958976037, 7339834801, 18241094639, 45412924028, 113245700219, 282834394783, 707407654401, 1771727090682, 4443029765522, 11155412690573, 28040604020532, 70559904175088, 177735079073901, 448136569131253, 1130964218420963, 2856730577032510, 7221937035815530, 18271971340356949, 46264602062392922, 117227479522291455, 297244335207374079, 754203530022169492, 1914883646655804475, 4864787119569857793, 12366373073388985492, 31453425461785677412, 80044458790641774203, 203809045620843784774, 519201725396429756541, 1323309312702391048197, 3374362078593316366604, 8608350819607942123095, 21970475292914355350454, 56097532741804671821420, 143293600571198079874492, 366170004247708309164512, 936065320726018453315109, 2393821227622028733131447, 6123989383217901770489560, 15672183956044108406708589, 40121062224202163233919784, 102744544659869498024526596, 263199097281447721431070670, 674443076084351358917512888, 1728771776052830935789889303, 4432592643567142135509390624, 11368476318425902647780179864, 29165378348002562991663729555, 74842906904770854698923268980, 192109239699919575705173539470, 493239045378109676774564459254, 1266704647942467831838666714549, 3253862825667795067086975460886, 8360387768269617869541470016906, 21485939561519119323971592020157, 55230701932447147705776562477159, 142004689399176019947282705368515, 365189625557095661180664282333447, 939346000239253860837716511164830, 2416696053162063667458685953035157, 6218787279970599797897622870739574, 16005697704745345705632449082412180, 41202810360147745584041893709115887, 106086600734727185860399731461973554, 273195725269569579262914425143253384, 703663787334094074112217219749705993, 1812728489668564198964352359374610211, 4670624475445511892818963507039155536, 12036222263015752748284469824403616053, 31022513337880797655272396083908134269, 79971237524146989974840114770529202975] ------------------------------------------------------------ Theorem Number, 23 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH H + HH H H HH + H HH HH H + HH H H HH 4+ H HH HH H + H HH HH H + HH HH H HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - x + x - x + 1) P(x) + (x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 4, 8, 18, 40, 91, 213, 506, 1218, 2966, 7294, 18090, 45194, 113632, 287327, 730186, 1863977, 4777509, 12289898, 31720176, 82117774, 213178108, 554823435, 1447397817, 3784127064, 9913343789, 26018984599, 68410290554, 180162323411, 475196423227, 1255186628492, 3319955888745, 8792478337054, 23313863133954, 61888878237389, 164467529111164, 437514750895099, 1165005085521436, 3105022797156006, 8282937176309562, 22114068273505488, 59088226972484393, 158003152307405371, 422813155834784436, 1132232400449265104, 3033990842200935686, 8135281602930567564, 21827264601898788794, 58598143522316010039, 157404179282151204174, 423045542218716765139, 1137594743452912452613, 3060618745844589852386, 8238424391588288975038, 22186297926687272423033, 59775628331378116451380, 161122179240285688242086, 434482162221497135042578, 1172108773956891793300694, 3163278407606545361590941, 8540335279906181472947846, 23066152471490359501137078, 62320812139929346207300967, 168439506971919225588258595, 455410615205746441651620508, 1231703823760505986269633728, 3332339148450456918147838167, 9018366051490575879544837616, 24413972116862043819738815721, 66111540013999155652084562422, 179077417530677201597463612360, 485205493746336496557728723796, 1315008887370019158511357609032, 3564894162691250059222924187153, 9666662859580349116822421353996, 26218968527010913350495426276500, 71131335168680718796826444485856, 193023408967072905414753719183441, 523914112753491222458732852061593, 1422357723794427722247327153795571, 3862369444640190578023693432554655, 10490416497562008857897222598208662, 28498590066657605402946101068227090, 77436123391842858541274538722111900, 210451173947131125254485079572740558, 572064013100223785974710999433555658, 1555326255238638414011723574401111725, 4229413184640997453434597390339399864, 11503199764674649726828795101413864073, 31292149913413969843695821586747884491, 85139020625938221899250912882028413646, 231684389965423872121630053321756963050, 630577096684214752553581697741642950844, 1716529914048561172329410428733040296684, 4673420308399441399562494693207807848635, 12725860269271441688238739378854893163811] ------------------------------------------------------------ Theorem Number, 24 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - 2 x + x - x + 1) P(x) + (2 x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 23, 46, 90, 178, 362, 744, 1535, 3194, 6704, 14152, 30019, 63993, 137027, 294528, 635254, 1374563, 2982983, 6490689, 14157809, 30951925, 67809639, 148847546, 327326203, 721035228, 1590820794, 3515055239, 7777682719, 17232166570, 38226855740, 84899762419, 188767040056, 420148088468, 936077556199, 2087530911028, 4659567865103, 10409537599216, 23274106872900, 52077999630392, 116616556556144, 261322452488208, 585990624975113, 1314890839669959, 2952311871586680, 6632807182343025, 14910247851249936, 33536291670621966, 75470769540757993, 169928986936543073, 382800909986262158, 862755122454412830, 1945377625304332347, 4388490288087053044, 9904091476130975332, 22361254774054848157, 50507256400673637226, 114125267802978496554, 257973369424416665508, 583348170337435269883, 1319580231355477362189, 2986029400547774283515, 6759245279299394441018, 15305372653077265127430, 34667865012474818872923, 78549578510974131598343, 178028874433432077107813, 403611252116127504636989, 915290323330934693322979, 2076222803712862095242661, 4710915866189376517310722, 10691779547594570193232569, 24271966712323362836221071, 55114699810857040680904084, 125179934207220209125653142, 284383356087736137374477620, 646209443923063259545590287, 1468722246098351522200998572, 3338881124332810099290521494, 7591977293065388173220549042, 17266301921829442269815327549, 39276443078065295093655855985, 89361682067950853306373719225, 203355033871187599510752721774, 462850820082071205035423702170, 1053677818530795805415739939561, 2399128776874249888355418021047, 5463569785243751528140091257717, 12444429345978153227988825135555, 28349632447886106488535958107141, 64594015240375900277795861639819, 147200074451998200230485801692760, 335500536871502998738039665095175] ------------------------------------------------------------ Theorem Number, 25 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H H + HH H HH H H HH + H HH H H HH H + HH HHH HHH HH 2+ H H H H + H H + HH HH + H H 1+ H H H + HH HH H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-*+-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 3 -1 + (x - 3 x + 6 x - 7 x + 7 x - 4 x + 1) P(x) 4 3 2 2 2 + (-3 x + 5 x - 9 x + 8 x - 3) P(x) + (2 x - 4 x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 5, 12, 29, 74, 199, 554, 1580, 4599, 13629, 41007, 124950, 384808, 1196010, 3747027, 11821256, 37523229, 119754660, 384047582, 1236976421, 3999780023, 12979240458, 42253474511, 137960509242, 451669185680, 1482402326444, 4876508223150, 16075907075992, 53100657695590, 175720809546355, 582495048581828, 1934011326455707, 6431021929933012, 21414877019549000, 71405175936331293, 238390696364328014, 796827183622750392, 2666406700941573674, 8932035228863731542, 29951087039643233174, 100528914233734415521, 337726663331342107359, 1135577613895114232222, 3821447399021126861689, 12870108500369910092496, 43377502675993578865587, 146305498040330514857658, 493806920018551613984013, 1667790945150358726863700, 5636405421129196150426245, 19060222673631623909309669, 64492388315933639743827121, 218340300664204189679262424, 739596870103919458535559424, 2506589512129250439904929232, 8499436166773644383255497931, 28834203095531408174101316777, 97865435244147141643724580885, 332312929827145883877564896974, 1128898818275654051949507939112, 3836598428445000055724268993497, 13044132578449189905222612800257, 44366572070761163512125173414595, 150960354073285919729699123970279, 513843778173627271234045629372531, 1749667435722230424981844473882405, 5959796016071331726800894082927865, 20307396272010013338048407694338611, 69218109534357021809344240032151246, 236006375173418561456532818720534102, 804937742592748279543019011219757331, 2746196546755413745801079650875838825, 9371909379169374468353399703869545521, 31992501744803496108426994087772295442, 109241748360914014259847741134534865867, 373117994182316404320539025462855575157, 1274728742041129800013073990663875472910, 4356126017486989008745040937927098462749, 14889884127520776439199122505070866766943, 50908193937206048172852892645702958973655, 174095247164727931352474741478183236910534, 595506430413343673847234697689138746032752, 2037435488027457279179217158357126956782541, 6972311652168179652233080636088514515646073, 23865083446147062045271868163117891384795476, 81703408011380205708376672572522315362235932, 279773335179699265305006937822281020199174023, 958207041437321309580759936460430793948328530, 3282443900489104842728265442779956311398980686, 11246523018497041223979591658610452594266227547, 38540771747913170357268601376105839730333735506, 132099727203511387502602637549167246847556748004, 452857008965978701094742813830497985574565158089, 1552731118955646347123917939554121904818246830295, 5324830582291895640932544203205749100761200265373, 18263673804706504903253280721851321222705981935308, 62652987313369405584985020744074808260772456463574, 214963759678046363792947428718327637564925230657702] ------------------------------------------------------------ Theorem Number, 26 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H + HH + HH HH + H H + H H 3+ H H H H + H H H H H H + H HH H H HH H + H H H H H H + H H H HH H 2+ H H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - x + x - x + 1) P(x) + (x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 4, 8, 18, 40, 91, 213, 506, 1218, 2966, 7294, 18090, 45194, 113632, 287327, 730186, 1863977, 4777509, 12289898, 31720176, 82117774, 213178108, 554823435, 1447397817, 3784127064, 9913343789, 26018984599, 68410290554, 180162323411, 475196423227, 1255186628492, 3319955888745, 8792478337054, 23313863133954, 61888878237389, 164467529111164, 437514750895099, 1165005085521436, 3105022797156006, 8282937176309562, 22114068273505488, 59088226972484393, 158003152307405371, 422813155834784436, 1132232400449265104, 3033990842200935686, 8135281602930567564, 21827264601898788794, 58598143522316010039, 157404179282151204174, 423045542218716765139, 1137594743452912452613, 3060618745844589852386, 8238424391588288975038, 22186297926687272423033, 59775628331378116451380, 161122179240285688242086, 434482162221497135042578, 1172108773956891793300694, 3163278407606545361590941, 8540335279906181472947846, 23066152471490359501137078, 62320812139929346207300967, 168439506971919225588258595, 455410615205746441651620508, 1231703823760505986269633728, 3332339148450456918147838167, 9018366051490575879544837616, 24413972116862043819738815721, 66111540013999155652084562422, 179077417530677201597463612360, 485205493746336496557728723796, 1315008887370019158511357609032, 3564894162691250059222924187153, 9666662859580349116822421353996, 26218968527010913350495426276500, 71131335168680718796826444485856, 193023408967072905414753719183441, 523914112753491222458732852061593, 1422357723794427722247327153795571, 3862369444640190578023693432554655, 10490416497562008857897222598208662, 28498590066657605402946101068227090, 77436123391842858541274538722111900, 210451173947131125254485079572740558, 572064013100223785974710999433555658, 1555326255238638414011723574401111725, 4229413184640997453434597390339399864, 11503199764674649726828795101413864073, 31292149913413969843695821586747884491, 85139020625938221899250912882028413646, 231684389965423872121630053321756963050, 630577096684214752553581697741642950844, 1716529914048561172329410428733040296684, 4673420308399441399562494693207807848635, 12725860269271441688238739378854893163811] ------------------------------------------------------------ Theorem Number, 27 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 3+ H H H H + H H H H H H H H + HH HH HH HH HH HH HH HH 2.5+ H H H H H H H H + H H H H H H H H + H H H H H H H H + H HH H H HH H 2+ H H H H H H + H H H H + H H H H 1.5+ HH HH HH HH + H H H H + H H H H 1+ H H H H + H H H H + H H H H + H H H H 0.5+ H H H H + H H H H +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 11 8 7 6 5 4 3 2 4 1 + (x + 5 x - 4 x + x + 8 x - 13 x + 11 x - 5 x + x) P(x) 10 8 7 6 5 4 3 2 3 + (x - 2 x + 7 x - 5 x - 5 x + 15 x - 17 x + 6 x + x - 1) P(x) 7 6 5 4 3 2 2 + (-x + 3 x + 2 x - 5 x + 10 x - 2 x - 5 x + 3) P(x) 4 3 2 + (x - 2 x + x + 3 x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 5, 11, 24, 56, 137, 342, 868, 2244, 5898, 15710, 42318, 115130, 315988, 873987, 2433850, 6818817, 19207583, 54368124, 154566571, 441170093, 1263743791, 3631928823, 10469366215, 30262294237, 87697552313, 254737832828, 741557710866, 2163091745386, 6321543350471, 18506961317790, 54270237686657, 159388976479292, 468796269858875, 1380709235668826, 4071726468668022, 12022149426495012, 35537281370217828, 105161912796014943, 311516186564212982, 923693984027211180, 2741448599955929354, 8143614844929624191, 24211486350973102526, 72040380979057674053, 214518725825060434720, 639255444595557293027, 1906294432131198569606, 5688516172292896015312, 16985934111754227612671, 50751626579012467559668, 151729466496455616456714, 453878163512655310718512, 1358465365027806620823730, 4068073895590287827596524, 12188538292628170746494937, 36536669130279271273984796, 109575443745411176634589183, 328773877188688943209402226, 986902694241744874001063914, 2963724710651864454152959060, 8903929895112963035115096535, 26760861777760479738154583626, 80461361687147615051359123250, 242012764225416332274809693066, 728195035609549205333788549825, 2191850210261955076448200735234, 6599683829476151407315998952988, 19878338456197719524514453209314, 59893198378853047065859564378161, 180514199286027803237968674182642, 544224147309109545341009069931750, 1641243874197093363128061723845171, 4951009406218658131790885463642588, 14939508254280379949450573966840561, 45091792880100598379264544493548085, 136136385534215301418904256079601606, 411115095603962266818532083813502917, 1241830232360723880955925599610699965, 3752043299066990230660285310808424077, 11339073735727996386620246050598846552, 34275899331747270617690918316024634201, 103633267292098387972530198824553304851, 313405184841647748330399551839462469697, 947998184578353904948223285745480323621, 2868143962814566734042495633455499430815, 8679292946943892987023681298595543891888, 26269731274270809187344129769013473494574, 79526668456480056173967770688485742475580, 240798560242717163434488091712960480611620, 729251021974818567525999460776053339798709, 2208922400001670223713295626979935584881835, 6692099232802723883171399688028284064369134, 20277811881705502440640261639290768794228070, 61454681299114118345047991421087817478357603, 186278368760806250961881329637233439138917486, 564731374967657902503340295115976495961162145, 1712347684359215469609616649122211481378906176] ------------------------------------------------------------ Theorem Number, 28 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 10 6 5 4 3 2 3 1 + (x - 2 x + x - 2 x + 2 x - x + x - 1) P(x) 7 6 5 4 3 2 2 + (x + x - x + x - 3 x + 2 x - 2 x + 3) P(x) 4 3 2 + (x + x - x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 17, 34, 65, 133, 277, 572, 1199, 2548, 5435, 11661, 25202, 54731, 119348, 261396, 574697, 1267637, 2804766, 6223660, 13845748, 30876213, 69007994, 154549822, 346790404, 779545026, 1755254743, 3958377577, 8939866583, 20218263933, 45784748248, 103807846272, 235636648909, 535466068728, 1218071478804, 2773591163064, 6321494978078, 14420661400511, 32924586831606, 75232979162492, 172041452385538, 393713260872074, 901644278692778, 2066263403285064, 4738263632158243, 10872374445702404, 24962637101521442, 57346546860289679, 131815132773209257, 303148512760931479, 697540229550486854, 1605826655979897539, 3698588507211274869, 8522638804153427168, 19647441284817832815, 45313253580871653227, 104550383069514369365, 241324409472237417738, 557245419733284748247, 1287229693480919378873, 2974574026651753256696, 6876189561850918905880, 15900858902075293511482, 36782273871125357477166, 85113327142737810030492, 197012375003682813852835, 456165570455991041730121, 1056527571106193560850298, 2447738008468144928854371, 5672459058686943865573806, 13149125315278295774252093, 30488654546469742502130612, 70711902848157227103981151, 164042637933125322263068179, 380652001553129860482798995, 883494697499474927314001201, 2051075452619209350256289939, 4762759874731327435883409954, 11061975317641962863310400262, 25698117521092741572233415672, 59712096957248664358561521889, 138775742402184313315715377316, 322591509897645444074929806223, 750029645171417394902589537216, 1744167142214391071760524674061, 4056767007670417319587124446474, 9437403083691333062035763415892, 21958549988526521939432355828298, 51101285390525603579799232888122, 118942032240129949401856916912786, 276893431455538839876673911173260, 644706688082075915267897498545600, 1501351643055827208280357431912040, 3496809282282403489853994898965247] ------------------------------------------------------------ Theorem Number, 29 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ H HH HH + HH H HH HH + H HH H H + HH H HH HH 4+ H HH H H + H HH H H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 3 4 3 2 2 -1 + (x + 2 x - x + x - x + 1) P(x) + (-2 x + x - 2 x + 2 x - 3) P(x) 2 + (x - x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 46, 94, 199, 439, 961, 2113, 4728, 10638, 24004, 54567, 124738, 286100, 658902, 1523561, 3533694, 8219807, 19175241, 44844787, 105117745, 246938893, 581271855, 1370801044, 3238379640, 7662884761, 18160172211, 43099390904, 102426035307, 243726755578, 580655580178, 1384932709103, 3306794777897, 7903693118041, 18909278995761, 45281554377074, 108530027712894, 260341203346223, 625005317744160, 1501608816996828, 3610336668014877, 8686460115399371, 20913632597618146, 50384354111881549, 121459160713524172, 292969666722436726, 707070780696631056, 1707424831884850449, 4125246497394886642, 9971935177734506502, 24116945055530422878, 58354041165562962555, 141259692927959905224, 342103485342872695337, 828862025018976189145, 2009029841925232601585, 4871515952686337473281, 11817073539321134031521, 28675998351756531656835, 69612131191902560435320, 169045763869085582882160, 410650213529812223368770, 997892674377742909288876, 2425690847473912236264926, 5898246126299524653958956, 14346379729918995608354591, 34905194678328946279956726, 84949845555910352980918162, 206802853409472003358290619, 503580166605875837883509918, 1226579626343965688411938881, 2988373791468938272294751313, 7282546274134082391128998885, 17751619375203540121720680747, 43280914162805330098090017988, 105549463086185917149368841094, 257462696364279602325677747884, 628158000160191822477935767084, 1532912888447769419915804392627, 3741604135787909331331389618256, 9134564337311934498324607449420, 22305163745068650394954160327031, 54476421551688686284866768832516, 133074664291962641673573493873036, 325135188262952797925515855558143, 794534125103762472134490936108882, 1941955965184663382273487480183164, 4747257141764020796262302049059161, 11607028692460250300264994515896655, 28383941376456748393597046710009226, 69421845442964953819449619451462120, 169820428896576356561050206615538112, 415482369434004891765163304919548259] ------------------------------------------------------------ Theorem Number, 30 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - 2 x + x - x + 1) P(x) + (2 x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 23, 46, 90, 178, 362, 744, 1535, 3194, 6704, 14152, 30019, 63993, 137027, 294528, 635254, 1374563, 2982983, 6490689, 14157809, 30951925, 67809639, 148847546, 327326203, 721035228, 1590820794, 3515055239, 7777682719, 17232166570, 38226855740, 84899762419, 188767040056, 420148088468, 936077556199, 2087530911028, 4659567865103, 10409537599216, 23274106872900, 52077999630392, 116616556556144, 261322452488208, 585990624975113, 1314890839669959, 2952311871586680, 6632807182343025, 14910247851249936, 33536291670621966, 75470769540757993, 169928986936543073, 382800909986262158, 862755122454412830, 1945377625304332347, 4388490288087053044, 9904091476130975332, 22361254774054848157, 50507256400673637226, 114125267802978496554, 257973369424416665508, 583348170337435269883, 1319580231355477362189, 2986029400547774283515, 6759245279299394441018, 15305372653077265127430, 34667865012474818872923, 78549578510974131598343, 178028874433432077107813, 403611252116127504636989, 915290323330934693322979, 2076222803712862095242661, 4710915866189376517310722, 10691779547594570193232569, 24271966712323362836221071, 55114699810857040680904084, 125179934207220209125653142, 284383356087736137374477620, 646209443923063259545590287, 1468722246098351522200998572, 3338881124332810099290521494, 7591977293065388173220549042, 17266301921829442269815327549, 39276443078065295093655855985, 89361682067950853306373719225, 203355033871187599510752721774, 462850820082071205035423702170, 1053677818530795805415739939561, 2399128776874249888355418021047, 5463569785243751528140091257717, 12444429345978153227988825135555, 28349632447886106488535958107141, 64594015240375900277795861639819, 147200074451998200230485801692760, 335500536871502998738039665095175] ------------------------------------------------------------ Theorem Number, 31 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 10 6 5 4 3 2 3 1 + (x - 2 x + x - 2 x + 2 x - x + x - 1) P(x) 7 6 5 4 3 2 2 + (x + x - x + x - 3 x + 2 x - 2 x + 3) P(x) 4 3 2 + (x + x - x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 17, 34, 65, 133, 277, 572, 1199, 2548, 5435, 11661, 25202, 54731, 119348, 261396, 574697, 1267637, 2804766, 6223660, 13845748, 30876213, 69007994, 154549822, 346790404, 779545026, 1755254743, 3958377577, 8939866583, 20218263933, 45784748248, 103807846272, 235636648909, 535466068728, 1218071478804, 2773591163064, 6321494978078, 14420661400511, 32924586831606, 75232979162492, 172041452385538, 393713260872074, 901644278692778, 2066263403285064, 4738263632158243, 10872374445702404, 24962637101521442, 57346546860289679, 131815132773209257, 303148512760931479, 697540229550486854, 1605826655979897539, 3698588507211274869, 8522638804153427168, 19647441284817832815, 45313253580871653227, 104550383069514369365, 241324409472237417738, 557245419733284748247, 1287229693480919378873, 2974574026651753256696, 6876189561850918905880, 15900858902075293511482, 36782273871125357477166, 85113327142737810030492, 197012375003682813852835, 456165570455991041730121, 1056527571106193560850298, 2447738008468144928854371, 5672459058686943865573806, 13149125315278295774252093, 30488654546469742502130612, 70711902848157227103981151, 164042637933125322263068179, 380652001553129860482798995, 883494697499474927314001201, 2051075452619209350256289939, 4762759874731327435883409954, 11061975317641962863310400262, 25698117521092741572233415672, 59712096957248664358561521889, 138775742402184313315715377316, 322591509897645444074929806223, 750029645171417394902589537216, 1744167142214391071760524674061, 4056767007670417319587124446474, 9437403083691333062035763415892, 21958549988526521939432355828298, 51101285390525603579799232888122, 118942032240129949401856916912786, 276893431455538839876673911173260, 644706688082075915267897498545600, 1501351643055827208280357431912040, 3496809282282403489853994898965247] ------------------------------------------------------------ Theorem Number, 32 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 7 6 3 2 2 4 3 2 1 + (x + x - 2 x + x - x + 1) P(x) + (x + 2 x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 5, 9, 15, 26, 48, 89, 165, 310, 591, 1138, 2205, 4297, 8427, 16622, 32938, 65530, 130863, 262249, 527197, 1062810, 2148146, 4352276, 8837592, 17982158, 36658592, 74865246, 153145667, 313763676, 643771865, 1322679887, 2721049245, 5604608198, 11557189002, 23857809325, 49300998950, 101977645247, 211133587197, 437515330510, 907391552370, 1883409945578, 3912262282749, 8132599519471, 16917470461058, 35215450619627, 73351807432625, 152882167526053, 318831083771137, 665291972000431, 1389000105612281, 2901494411312661, 6064042668767917, 12679900670970242, 26526198371525921, 55517844071636598, 116247016822421760, 243509692791621693, 510304958692509935, 1069834525683015957, 2243733025635694956, 4707477231914068428, 9880131493978309950, 20743874289698006871, 43567745659235343423, 91534327932366425904, 192372244159797445202, 404423446349987694296, 850475692006620639552, 1789021153134275019838, 3764380495324636198963, 7923054498549552785817, 16680521298024087725246, 35127021615881690891674, 73992001969543494910782, 155896761247337062814246, 328545486973552202438004, 692559921201329410518511, 1460226000605629209724777, 3079507053993080839704110, 6495885392920695093764064, 13705321751718305123493788, 28922220684172584949140422, 61046872393071002819789051, 128879108693661861817539403, 272136603848726764672198432, 574744475893032262155788238, 1214071321413282376714153974, 2565035305811338986244912025, 5420264847804173772189630486, 11455762461878381622441607611, 24215988133284034517908107932, 51198061405255014156095915233, 108262097724071227150484889909, 228965199188889890733294634337] ------------------------------------------------------------ Theorem Number, 33 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} To make it crystal clear here is such a path of semi-length 8 4+ H H H + HH HH HH + HH HH HH HH HH HH + H H H H H H + H H H H H H 3+ H H H H + H H + H H + H H + H H 2+ H H H + H H HH + H H H H + HH HH H HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-+*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 2 3 2 3 2 (x + 6 x - 5 x + 1) P(x) + (-2 x - 7 x + 8 x - 2) P(x) + x + 2 x - 3 x + 1 = 0 The sequence a(n) satisfies the linear recurrence 3 (3 n - 8) a(n - 1) 2 (13 n - 41) a(n - 2) (23 n - 82) a(n - 3) a(n) = -------------------- - ---------------------- + -------------------- n - 2 n - 2 n - 2 2 (2 n - 7) a(n - 4) + -------------------- n - 2 subject to the initial conditions [a(1) = 1, a(2) = 2, a(3) = 5, a(4) = 13] Just for fun, a(1000), equals 2294852754289166873194745240744716696400162003604451471919349038871798708759\ 985515980497399820654715006096982414325293200892959752156601149204990805\ 687679930914845292727663372586151594857641323104223705084407972460582264\ 151906793186700469235346124125424546017351927882501324272614315282368024\ 403215649010618328967673941022252336474830518725397757832130859917751895\ 520441225253513156953902581611300814954120812156098685828373536391487812\ 780206218650802124500033660279244428482155973116014069893687991871039145\ 110748048004169328308216003607981143848666627666052791798373189437259659\ 10838216803951755 For the sake of the OEIS here are the first, 100, terms. [1, 1, 2, 5, 13, 35, 97, 276, 805, 2404, 7343, 22916, 72980, 236857, 782275, 2625265, 8938718, 30834165, 107608097, 379454447, 1350434278, 4845475311, 17512579630, 63703732426, 233063976059, 857067469749, 3166309373615, 11745982220846, 43736933794243, 163409819344676, 612415062653080, 2301627415153461, 8672479097155517, 32755202434296043, 123984146212620561, 470251102237219228, 1786931561888319841, 6802111091955394040, 25935000481117133188, 99035202677825435351, 378714703735090725951, 1450161164914390585861, 5559907771213558938638, 21341984205122852560893, 82014231825056416720796, 315503776035235853259224, 1214944984225803265071151, 4682988152685541482167655, 18066884936114900040405865, 69761879491768621433998564, 269593873446659212331819465, 1042659103588738798492691976, 4035522642951587235093153098, 15630326377535004825801568625, 60580622596891078305187048717, 234954799022066401286175096249, 911817804607232966024505466198, 3540739324334828815547899214203, 13757249611617498751298768053550, 53482451235316161394731597980646, 208028972039315837554562703280391, 809581363462187024686623479276609, 3152195528967300922441653022087165, 12279311456620186669220734310534676, 47855930704419796437254210788090117, 186591326996456188074216453601428676, 727837589857913954866271653914448127, 2840263226482229508927814379889125412, 11088117839418127296239715416601374972, 43303853832899658207759334855606819361, 169184023923463815215789120012593671511, 661227802779463285262335516176427174917, 2585216519033849852365967319406025650858, 10110953862989288005529976557149201077877, 39557812322609983187610683547128490025670, 154815012305789737915436589641595859035930, 606080723095376880287197108000408422832253, 2373452607630639311893291610124147603947255, 9297359341705524351656599534984653754497847, 36430409270011712917964437114572839186997972, 142787580419004663743200658031919670997305563, 559803330744779975316250219791823995349377736, 2195310843115125080658366559682519878840449560, 8611306228635277159582763698634403207776289761, 33787152803729175821110610212791205953989028449, 132599200819335867902133242830382746280843188399, 520516222178449202132307841211200739615490687640, 2043756204211896745456271602150084580859076559797, 8026438190091726482558414378696312179044698889266, 31529217576565327472436830267540760128882295714228, 123879017372729808518446538424138729132651799070687, 486826603799928265645549933622898954173602057439801, 1913553855490697946168587048080963283029764569087255, 7523065238081545110380588350779825825337019791677582, 29582488687247614294663723104553799984106941166448379, 116347855974749992271994825333787424917496753640924748, 457682159047523146233073763271664147749474830321693884, 1800734516219945168407651917874519112264567516084702153, 7086205568817265180220428054863762846745722603087326763, 27890381792125086129987025545179282072101239785985421313, 109791870387631469837221785325496346929480250746218198934] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 34 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 2+ H H H H + HH HH HH HH + HH HH HH HH + H H H H H H H H + H H H H H H H H 1.5+ H HH H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H 1+ H H H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H 0.5+ HH H H H H H H HH + H H H H H H H H + H H H H H H H H +H HH HH HH H +H HH HH HH H -*-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-*+-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 2 (x + 4 x + 6 x + x - 4 x - x + 1) P(x) 5 4 3 2 4 3 2 + (-2 x - 5 x - 2 x + 5 x + 2 x - 2) P(x) + x + x - x - x + 1 = 0 The sequence a(n) satisfies the linear recurrence (4 n - 11) a(n - 1) (n - 5) a(n - 2) (13 n - 50) a(n - 3) a(n) = ------------------- + ---------------- - -------------------- n - 2 n - 2 n - 2 3 (n - 5) a(n - 4) 2 (7 n - 32) a(n - 5) (11 n - 52) a(n - 6) - ------------------ + --------------------- + -------------------- n - 2 n - 2 n - 2 3 (n - 5) a(n - 7) + ------------------ n - 2 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 3, a(5) = 5, a(6) = 12, a(7) = 25] Just for fun, a(1000), equals 2892929439472754275639110962905703645191735357727095508899750650229296998699\ 650250417702312500933508160542528875502594632077456926638556488398534818\ 079656524695088294746931723795733904343776817447344284277088298632959438\ 188606625737348008403331499207787392501687009279451173947644782842916335\ 590525409631737159819076331285702848545178346291410865239808433214918383\ 281914221503672809369919922485847168730055063398763474767613956751723358\ 415883171519041713477635602814169613 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 3, 5, 12, 25, 58, 133, 317, 766, 1891, 4741, 12070, 31132, 81253, 214281, 570355, 1530619, 4137681, 11258218, 30810977, 84762419, 234280495, 650288121, 1811922049, 5066241036, 14210585700, 39976144968, 112758502436, 318834665353, 903583180747, 2566163949117, 7302124082604, 20816325699097, 59442392855164, 170011454608695, 486974888157768, 1396825475257367, 4011890812853959, 11537047874589090, 33216120587777263, 95737962218271909, 276232723988433327, 797807894315299271, 2306386750043567194, 6673543245670751681, 19326479099806588116, 56014905144409127161, 162477391986594286662, 471634639910545402699, 1370022841010420490639, 3982411331837222431784, 11583726101568771675180, 33714965843652599376280, 98188034231868635978533, 286118252811544040061832, 834207468756277346585543, 2433520415189651964513444, 7102638836095197443964402, 20740544969288284445616965, 60593857605397876664140545, 177107831170009544910107609, 517893841632468570994296682, 1515064053665165872006908549, 4434067310543872167835372653, 12982211772826628997427774566, 38024593915428031841352012403, 111415240793505685251038663461, 326575523622006641554282378168, 957583807196547533996136789550, 2808791050774407741236831189929, 8241515930178006758508408779475, 24189982354208029625605722410770, 71023278166409550987269096563282, 208592503751217119003468709661551, 612809900635087260733886986413235, 1800853439393854697704563428184329, 5293623629682496252191869130693941, 15564907984243243464692606902277257, 45777883078979090173886494418209242, 134672050453738930657164966639280471, 396286196017995203972219876472995851, 1166399487230985962929210807281724423, 3433917863777186820431620943202976729, 10111931029595215538737170868048324250, 29783614937181281767108547739284676661, 87743999090625245792470311866733184860, 258554336136611861603433787894215869383, 762041063139495898839722169297774206845, 2246440452120897232386872856999881987840, 6623680132878369046356230732069581005915, 19533932545131858458849086269537812355069, 57618767371669844191146990726871805226294, 169988801410185552666885303822533699739870, 501599289695207480435044190215096219606659, 1480375913258479345517696378084308279464613, 4369823586894576366277873359867799338612566, 12901225049094700532567305828699773777530580, 38095314014096636968163517056229082815915739] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 35 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 3+ H + H H + HH HH 2.5+ H H + H H + H H + H H 2+ H H + H H + H H 1.5+ HH HH + H H + H H 1+ H H H H H H H + HH H H H H H H H H H H + H H H H H H H H H H H H + H H H H H H H H H H H H 0.5+ H H H H H H H H H H H H + H H H H H H H H H H H H +H HH HH HH HH HH H -*-+-+-+-+*-+-+-+-+*-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-*+-+-+-+-*+-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 5 4 3 2 3 (x - 5 x + 17 x - 33 x + 48 x - 50 x + 38 x - 21 x + 7 x - 1) P(x) 8 7 6 5 4 3 2 2 + (2 x - 14 x + 36 x - 65 x + 81 x - 71 x + 46 x - 18 x + 3) P(x) 7 6 5 4 3 2 6 + (3 x - 11 x + 26 x - 40 x + 41 x - 32 x + 15 x - 3) P(x) + x 5 4 3 2 - 3 x + 6 x - 7 x + 7 x - 4 x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 5, 12, 29, 73, 191, 515, 1423, 4016, 11550, 33782, 100285, 301601, 917453, 2819032, 8739226, 27306172, 85917302, 272020626, 866042142, 2771054911, 8906471230, 28743122005, 93103701715, 302595131090, 986491638639, 3225155513120, 10571477201642, 34734525664670, 114380001028568, 377426487739339, 1247802486733972, 4132704433632150, 13710305670791170, 45555230759506051, 151588381485252350, 505117103962315635, 1685318390817555108, 5629939570533566457, 18829039535489653704, 63041766068983498846, 211290295092257519477, 708857262476170927125, 2380372322293683444246, 8000519212343233816197, 26912891103278271708156, 90605470625550145989495, 305270266739420930224776, 1029285358763534837968591, 3472917212610870719742343, 11725933706847799816131273, 39617095639065173521260768, 133933308282058583840095252, 453058801258483109746801548, 1533452885279882538913708039, 5193099587030088664738418302, 17596010355689674118628585974, 59651944220632387061740782490, 202325111110360981815515125967, 686565766543993889712912661484, 2330850700786944843575127811490, 7916621371417009688220182573159, 26899976341585164858850143207558, 91441712187753668305149533593626, 310964886869350257470014517247471, 1057906572289872839937979802873309, 3600368132007536340493979956839736, 12257588977235388766990065303733393, 41746195764971019221719546619210564, 142225619636382332708473747666868965, 484711608545868758871039594575802936, 1652454223827522842694891077792167211, 5635232514959349536452881788006391475, 19223247478353196836769619347624426025, 65594961158855736489917835304683184500, 223892451416477792097573483364152113873, 764416871676281018630000391579467925265, 2610595855682553759241158792056987908755, 8917938033037119811160946389492968441096, 30472049379779812370737771183808007567895, 104147387896357198112766868077020865346598, 356042478207032964420160366674586329912842, 1217472830482654367123710375363511208848469, 4164071572807548222532254086946394843957380, 14245445909023137909472207851417680078927899, 48745043939037526704472699105679124683015870, 166831912108575354641975997808571637444928802, 571110047846431480187447237404210280787983461, 1955466295225403890769384626670001428938599481, 6696820388761040155082887663506978766100582208, 22938908226837376211520173476618193978940347640, 78588803144148146411902589572617354596253534902, 269296414258721844770482354845230421821153368717, 922955199188638766026473549203556385751488028029, 3163800372694296765359516020827859298678154703554, 10847114705369298297448060139211994422412852426744, 37195847616260924175209839052026809176807308013863, 127569885555564875549553141747312074602015517696386] ------------------------------------------------------------ Theorem Number, 36 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H H + HHH HH H + HH H H HH + H HH H H 3+ H H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 7 6 5 4 3 3 (x - 2 x - 5 x - 3 x - 2 x + 3 x + x - 1) P(x) 7 6 5 4 3 2 2 + (3 x + 8 x + 5 x + 6 x - 7 x + x - 3 x + 3) P(x) 7 6 5 4 3 2 6 4 3 + (-x - 4 x - 2 x - 6 x + 5 x - 2 x + 3 x - 3) P(x) + x + 2 x - x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 43, 88, 181, 389, 845, 1831, 4021, 8945, 19980, 44876, 101560, 231010, 527584, 1210453, 2788476, 6445177, 14945402, 34762125, 81073108, 189553176, 444229775, 1043335656, 2455327148, 5789089278, 13673287432, 32347923587, 76645650172, 181868152606, 432131947999, 1028094409010, 2448927381166, 5840049815756, 13942114199763, 33318698172213, 79702566442900, 190836157036246, 457334328621662, 1096917010956494, 2633078585793776, 6325404794268179, 15206619342001349, 36583326860847321, 88069812063175602, 212154504630779797, 511384369072275705, 1233394476770162448, 2976499939131468798, 7187032770060977636, 17362981676250949732, 41968309776585627857, 101492313014921186269, 245556679552748148628, 594388103914626682818, 1439399709664903706555, 3487218741318126496572, 8451957378207794626600, 20493205067305036075159, 49708582948760009479704, 120619210443382870051297, 292792615812251132529932, 710980136125717425588637, 1727046021860570120941582, 4196575501209940797812052, 10200621702455464613659962, 24802455440798284389096014, 60324716048583750707955995, 146765765674535755148007651, 357173774922625517418331537, 869473358467539692845081275, 2117150070857988843015817831, 5156588999541444162109803583, 12562783839142431859361805544, 30613913629937142427400197707, 74620580935239731515082935125, 181929247599493983737210197721, 443657637304923176383168028461, 1082162432233326872644453144656, 2640180374641018514672900497678, 6442717456740989437135783799985, 15725218378158726422687662180370, 38389658112166505912696121858576, 93738863949937324569728179979473, 228934372538591942169266458710404, 559224587509088449210338239175827, 1366292159965124659559109224182072, 3338728826968157284819565578655591, 8160132627279040287185529051465709, 19947574345781438581105186436718539, 48770603600956548080699610819303235, 119261355989728246212509483676650821, 291684525929087013741255678799558755] ------------------------------------------------------------ Theorem Number, 37 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH H + HH H H HH + H HH HH H + HH H H HH 4+ H HH HH H + H HH HH H + HH HH H HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 2 (x + 4 x + 6 x + x - 4 x - x + 1) P(x) 5 4 3 2 4 3 2 + (-2 x - 5 x - 2 x + 5 x + 2 x - 2) P(x) + x + x - x - x + 1 = 0 The sequence a(n) satisfies the linear recurrence (4 n - 11) a(n - 1) (n - 5) a(n - 2) (13 n - 50) a(n - 3) a(n) = ------------------- + ---------------- - -------------------- n - 2 n - 2 n - 2 3 (n - 5) a(n - 4) 2 (7 n - 32) a(n - 5) (11 n - 52) a(n - 6) - ------------------ + --------------------- + -------------------- n - 2 n - 2 n - 2 3 (n - 5) a(n - 7) + ------------------ n - 2 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 3, a(5) = 5, a(6) = 12, a(7) = 25] Just for fun, a(1000), equals 2892929439472754275639110962905703645191735357727095508899750650229296998699\ 650250417702312500933508160542528875502594632077456926638556488398534818\ 079656524695088294746931723795733904343776817447344284277088298632959438\ 188606625737348008403331499207787392501687009279451173947644782842916335\ 590525409631737159819076331285702848545178346291410865239808433214918383\ 281914221503672809369919922485847168730055063398763474767613956751723358\ 415883171519041713477635602814169613 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 3, 5, 12, 25, 58, 133, 317, 766, 1891, 4741, 12070, 31132, 81253, 214281, 570355, 1530619, 4137681, 11258218, 30810977, 84762419, 234280495, 650288121, 1811922049, 5066241036, 14210585700, 39976144968, 112758502436, 318834665353, 903583180747, 2566163949117, 7302124082604, 20816325699097, 59442392855164, 170011454608695, 486974888157768, 1396825475257367, 4011890812853959, 11537047874589090, 33216120587777263, 95737962218271909, 276232723988433327, 797807894315299271, 2306386750043567194, 6673543245670751681, 19326479099806588116, 56014905144409127161, 162477391986594286662, 471634639910545402699, 1370022841010420490639, 3982411331837222431784, 11583726101568771675180, 33714965843652599376280, 98188034231868635978533, 286118252811544040061832, 834207468756277346585543, 2433520415189651964513444, 7102638836095197443964402, 20740544969288284445616965, 60593857605397876664140545, 177107831170009544910107609, 517893841632468570994296682, 1515064053665165872006908549, 4434067310543872167835372653, 12982211772826628997427774566, 38024593915428031841352012403, 111415240793505685251038663461, 326575523622006641554282378168, 957583807196547533996136789550, 2808791050774407741236831189929, 8241515930178006758508408779475, 24189982354208029625605722410770, 71023278166409550987269096563282, 208592503751217119003468709661551, 612809900635087260733886986413235, 1800853439393854697704563428184329, 5293623629682496252191869130693941, 15564907984243243464692606902277257, 45777883078979090173886494418209242, 134672050453738930657164966639280471, 396286196017995203972219876472995851, 1166399487230985962929210807281724423, 3433917863777186820431620943202976729, 10111931029595215538737170868048324250, 29783614937181281767108547739284676661, 87743999090625245792470311866733184860, 258554336136611861603433787894215869383, 762041063139495898839722169297774206845, 2246440452120897232386872856999881987840, 6623680132878369046356230732069581005915, 19533932545131858458849086269537812355069, 57618767371669844191146990726871805226294, 169988801410185552666885303822533699739870, 501599289695207480435044190215096219606659, 1480375913258479345517696378084308279464613, 4369823586894576366277873359867799338612566, 12901225049094700532567305828699773777530580, 38095314014096636968163517056229082815915739] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 38 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH H + HH H H HH + H HH HH H + HH H H HH 4+ H HH HH H + H HH HH H + HH HH H HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 5 4 3 2 2 (x - x - 3 x - x + 4 x + 6 x + x - 4 x - x + 1) P(x) 7 5 4 3 2 5 4 2 + (2 x - 4 x - 5 x - x + 5 x + 2 x - 2) P(x) + x + x - x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 1, 1, 2, 4, 7, 15, 29, 61, 127, 271, 586, 1284, 2851, 6401, 14528, 33290, 76959, 179339, 420963, 994662, 2364307, 5650550, 13571285, 32741773, 79316223, 192861905, 470563080, 1151736762, 2827109374, 6958049903, 17167211458, 42452181624, 105200309767, 261208021774, 649756213067, 1619035095366, 4040697134706, 10099670334604, 25279556358129, 63358802672237, 158996185196047, 399464668161477, 1004740965736694, 2529812039268552, 6376129439265816, 16085657549407848, 40617486655086834, 102650478258730707, 259636115804741083, 657217503786048531, 1664860404141452886, 4220432959077922278, 10706151265174191810, 27176559799636294517, 69028531975005217927, 175438486812291859892, 446141746173007201393, 1135175174326041517925, 2889919674420257510760, 7360933087674662702827, 18758409049146171218297, 47826371351778426624173, 121994382563029184515362, 311319455992724823415243, 794804160537153370095763, 2029996005196491065924440, 5186871274062376282446698, 13258223448198050178705836, 33902317674660164993362575, 86722642295960754001377990, 221916593448940994006321966, 568063053741139676307806450, 1454615936876682135322731839, 3725983311900217306369726158, 9547071137390468894115840462, 24469900240113377669214108747, 62736933815488987081687982903, 160894000268832831086440842640, 412741793134750931643941634128, 1059097119722585880864665634906, 2718370879626463051457619565941, 6979015727134402004113883394055, 17922115529659196187567283920954, 46035322551173580481730925186008, 118276164309522454450874267190602, 303951856047645514295978341829247, 781288429499139011734727825531415, 2008697788259892889838102290450524, 5165496879477621065689421148927643, 13286226741382655254979104080622679, 34180712388182352411790013754992185, 87952536583905315844522451217854971, 226360906527982608810828063429502302, 582691028785537848253212044104105443, 1500227810151160279886420960353428046, 3863280561577537668583192110278752568, 9950242989134400621172019920081191727, 25632314653984564319749433172509735893, 66041514186845210984568247528134518963] ------------------------------------------------------------ Theorem Number, 39 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H H + HH H H H H + H H H H H H + HH H HH H HH HH + H H H H H H 1+ H H H H + H H + H H + HH HH +H H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 (x + 3 x + 5 x + 4 x + x - 2 x - x - x + 1) P(x) 6 5 4 3 2 4 + (-2 x - 3 x - 2 x + 2 x + x + 2 x - 2) P(x) + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 2, 5, 12, 23, 49, 112, 253, 579, 1358, 3227, 7748, 18813, 46125, 114013, 283897, 711584, 1794015, 4546671, 11577180, 29604270, 75993121, 195754315, 505862986, 1311058676, 3407029364, 8875702299, 23175084811, 60639980667, 158983044883, 417578461535, 1098672159454, 2895297229743, 7641346128597, 20195697132110, 53447096195890, 141622665569225, 375711616241302, 997842401302760, 2652950875842241, 7060456732514495, 18808358609957636, 50149116458310735, 133829293008862050, 357435036603330737, 955399082257794831, 2555636540797558539, 6841096693744642690, 18325314910917937883, 49120613334009563827, 131750124641517182896, 353591823117062576312, 949526236370539620659, 2551264891918074784473, 6858651978260183087958, 18447941994455093871814, 49644919963685742485007, 133663142170818395702285, 360040301506875909353955, 970255659958764262554670, 2615834388900122750507795, 7055324647718768885562043, 19037086039122051642684945, 51387200762903904407707405, 138763435324440300691465335, 374848303703121681265255860, 1012958261247020257079200669, 2738283054836353056704524953, 7404767361195354074854837770, 20030255658327162082560716465, 54200020376631062771396758558, 146705486580589416371369182716, 397212990517280503627304325553, 1075789001720150725327259291407, 2914431574550658198089515312139, 7897695965796251173360710095773, 21407381708490941390986231724109, 58041713566580520540000612211825, 157408247812499632998921407099676, 426994713831049906072706401927609, 1158570706633902438570168165164839, 3144307401839123283932256343422602, 8535467976231382791366044856647984, 23175394141450371675485289924071740, 62939317377251538244357435249004576, 170966009465362716902406440776984218, 464502640812098078131177397502868347, 1262278429524071490537345499684630382, 3430904761435168832449731839903173677, 9327102072221139737607603519725203248, 25361063671374774615982527071315358111, 68971398680913371077401476687705762424, 187607269403742559963414016714492354623, 510396439637631516248262323508946565135, 1388805072843575940295077209058378663716, 3779627748100460288176596967204517247922, 10287960378019101772156447629346127060676] ------------------------------------------------------------ Theorem Number, 40 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH H + HH H H HH + H HH HH H + HH H H HH 4+ H HH HH H + H HH HH H + HH HH H HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 3 2 (x + 4 x + 4 x + 2 x - 3 x - x + 1) P(x) 7 6 5 4 3 2 6 4 3 + (-2 x - 4 x - x - x + 5 x - x + 2 x - 2) P(x) + x + x - 2 x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 16, 30, 56, 110, 216, 427, 860, 1746, 3569, 7364, 15307, 32009, 67334, 142416, 302663, 646031, 1384486, 2977909, 6426645, 13911978, 30200915, 65732898, 143413923, 313595163, 687144819, 1508571730, 3317922569, 7309678595, 16129313045, 35643218886, 78875659428, 174774870620, 387750760231, 861260255477, 1915124365594, 4262999134446, 9498739083878, 21184957996924, 47291188848596, 105658705445218, 236257445857091, 528694503932384, 1183990265577090, 2653392279620791, 5950474873926279, 13353228687666239, 29984296182635404, 67369519671951483, 151455416547760267, 340681282599710033, 766734008185798102, 1726495929499880140, 3889579288927147642, 8766943991330924627, 19769471877891215161, 44600139100704220528, 100661893945840143266, 227287461047749510237, 513406564122610083700, 1160158342765246864333, 2622633765230015042003, 5930856706035176271770, 13416885846141972833882, 30362383327524771174841, 68733011599111138672309, 155645297432095952323778, 352568632584310655173635, 798885376847742072720203, 1810734038572540097719042, 4105354542744134743448103, 9310415132809649662747530, 21120611967784825680218897, 47924750352934920938803331, 108774247977720255491636377, 246946153325595537553849633, 560771040004278719756311221, 1273718100463445848541909110, 2893762556614309851409015460, 6575848332156955986692361543, 14946434073696128773556355402, 33979577457896340549152610437, 77266402617848204240564334985, 175733104543382019162751390086, 399764798964070432229145750542, 909581419077380496710545668751, 2069963664743655297207114411687, 4711574821783439857678518992020, 10726298267792692107831401356403, 24423746955362712321076292407699, 55622647415401209237519460346954, 126696994280216636321751775842705, 288638769125621515697430788944262] ------------------------------------------------------------ Theorem Number, 41 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 4+ H + HH + HH HH + H H + H H 3+ H H + H H + H H + H H + H H 2+ H H H + H H HH + H H H H + HH HH H HH + H H H H 1+ H H H H H H + H H H H H H H H + H H H H H H H H + HH HH HH HH HH HH HH HH +H HH HH HH H -*-+-+-+-+*-+-+-+-+*-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 5 4 3 2 3 (x - 5 x + 17 x - 33 x + 48 x - 50 x + 38 x - 21 x + 7 x - 1) P(x) 8 7 6 5 4 3 2 2 + (2 x - 14 x + 36 x - 65 x + 81 x - 71 x + 46 x - 18 x + 3) P(x) 7 6 5 4 3 2 6 + (3 x - 11 x + 26 x - 40 x + 41 x - 32 x + 15 x - 3) P(x) + x 5 4 3 2 - 3 x + 6 x - 7 x + 7 x - 4 x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 5, 12, 29, 73, 191, 515, 1423, 4016, 11550, 33782, 100285, 301601, 917453, 2819032, 8739226, 27306172, 85917302, 272020626, 866042142, 2771054911, 8906471230, 28743122005, 93103701715, 302595131090, 986491638639, 3225155513120, 10571477201642, 34734525664670, 114380001028568, 377426487739339, 1247802486733972, 4132704433632150, 13710305670791170, 45555230759506051, 151588381485252350, 505117103962315635, 1685318390817555108, 5629939570533566457, 18829039535489653704, 63041766068983498846, 211290295092257519477, 708857262476170927125, 2380372322293683444246, 8000519212343233816197, 26912891103278271708156, 90605470625550145989495, 305270266739420930224776, 1029285358763534837968591, 3472917212610870719742343, 11725933706847799816131273, 39617095639065173521260768, 133933308282058583840095252, 453058801258483109746801548, 1533452885279882538913708039, 5193099587030088664738418302, 17596010355689674118628585974, 59651944220632387061740782490, 202325111110360981815515125967, 686565766543993889712912661484, 2330850700786944843575127811490, 7916621371417009688220182573159, 26899976341585164858850143207558, 91441712187753668305149533593626, 310964886869350257470014517247471, 1057906572289872839937979802873309, 3600368132007536340493979956839736, 12257588977235388766990065303733393, 41746195764971019221719546619210564, 142225619636382332708473747666868965, 484711608545868758871039594575802936, 1652454223827522842694891077792167211, 5635232514959349536452881788006391475, 19223247478353196836769619347624426025, 65594961158855736489917835304683184500, 223892451416477792097573483364152113873, 764416871676281018630000391579467925265, 2610595855682553759241158792056987908755, 8917938033037119811160946389492968441096, 30472049379779812370737771183808007567895, 104147387896357198112766868077020865346598, 356042478207032964420160366674586329912842, 1217472830482654367123710375363511208848469, 4164071572807548222532254086946394843957380, 14245445909023137909472207851417680078927899, 48745043939037526704472699105679124683015870, 166831912108575354641975997808571637444928802, 571110047846431480187447237404210280787983461, 1955466295225403890769384626670001428938599481, 6696820388761040155082887663506978766100582208, 22938908226837376211520173476618193978940347640, 78588803144148146411902589572617354596253534902, 269296414258721844770482354845230421821153368717, 922955199188638766026473549203556385751488028029, 3163800372694296765359516020827859298678154703554, 10847114705369298297448060139211994422412852426744, 37195847616260924175209839052026809176807308013863, 127569885555564875549553141747312074602015517696386] ------------------------------------------------------------ Theorem Number, 42 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H H + HH HH 4+ H H + H H + HH HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H H + H HH H H HHH + HH H HH HH H HH + H HH H H HH H 1+ H H H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 (x + 3 x + 5 x + 4 x + x - 2 x - x - x + 1) P(x) 6 5 4 3 2 4 + (-2 x - 3 x - 2 x + 2 x + x + 2 x - 2) P(x) + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 2, 5, 12, 23, 49, 112, 253, 579, 1358, 3227, 7748, 18813, 46125, 114013, 283897, 711584, 1794015, 4546671, 11577180, 29604270, 75993121, 195754315, 505862986, 1311058676, 3407029364, 8875702299, 23175084811, 60639980667, 158983044883, 417578461535, 1098672159454, 2895297229743, 7641346128597, 20195697132110, 53447096195890, 141622665569225, 375711616241302, 997842401302760, 2652950875842241, 7060456732514495, 18808358609957636, 50149116458310735, 133829293008862050, 357435036603330737, 955399082257794831, 2555636540797558539, 6841096693744642690, 18325314910917937883, 49120613334009563827, 131750124641517182896, 353591823117062576312, 949526236370539620659, 2551264891918074784473, 6858651978260183087958, 18447941994455093871814, 49644919963685742485007, 133663142170818395702285, 360040301506875909353955, 970255659958764262554670, 2615834388900122750507795, 7055324647718768885562043, 19037086039122051642684945, 51387200762903904407707405, 138763435324440300691465335, 374848303703121681265255860, 1012958261247020257079200669, 2738283054836353056704524953, 7404767361195354074854837770, 20030255658327162082560716465, 54200020376631062771396758558, 146705486580589416371369182716, 397212990517280503627304325553, 1075789001720150725327259291407, 2914431574550658198089515312139, 7897695965796251173360710095773, 21407381708490941390986231724109, 58041713566580520540000612211825, 157408247812499632998921407099676, 426994713831049906072706401927609, 1158570706633902438570168165164839, 3144307401839123283932256343422602, 8535467976231382791366044856647984, 23175394141450371675485289924071740, 62939317377251538244357435249004576, 170966009465362716902406440776984218, 464502640812098078131177397502868347, 1262278429524071490537345499684630382, 3430904761435168832449731839903173677, 9327102072221139737607603519725203248, 25361063671374774615982527071315358111, 68971398680913371077401476687705762424, 187607269403742559963414016714492354623, 510396439637631516248262323508946565135, 1388805072843575940295077209058378663716, 3779627748100460288176596967204517247922, 10287960378019101772156447629346127060676] ------------------------------------------------------------ Theorem Number, 43 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The first, 100, terms of the sequence are [1, 1, 1, 2, 4, 8, 16, 34, 79, 192, 476, 1205, 3117, 8202, 21862, 58903, 160237, 439627, 1215183, 3381113, 9463222, 26627171, 75282466, 213771449, 609427584, 1743659152, 5005369843, 14412151256, 41613615979, 120465670005, 349566090912, 1016620564056, 2962681380013, 8650612537558, 25304033410884, 74141917021821, 217582370248315, 639481630397627, 1882084441180307, 5546539854086081, 16366105303169195, 48348100488181193, 142987259628346710, 423325232357070412, 1254546438835307457, 3721462694511360819, 11049306709677391611, 32834642645227845540, 97653586933071494907, 290660912704862402202, 865789930135905150786, 2580784805776051101702, 7698235745374769712748, 22978324989123978841646, 68631204528064528230156, 205111495788189876255130, 613356793930911636521594, 1835191613066792276038238, 5493960253965491205770697, 16455715951324348999607550, 49313633379898889383021321, 147852403332952465816197775, 443499937196185249492023797, 1330930952563415067021010170, 3995835763837222559472030687, 12001721897852429953820979133, 36062622506766467496575761991, 108403461606339502866666276467, 325983520046328130381892475344, 980639979368827478597885018844, 2951073621637971872321729244055, 8883871358958775278108385153642, 26752952917178966661864184404800, 80590548945481353467310849227153, 242848425702065499168041757204311, 732017227618580456858104232280374, 2207183043062491082609878280634167, 6657065112030109888425946619564084, 20084044635837786789659159873446803, 60609419675266659974990802255630360, 182955907469972214616578562393858803, 552416989599590831672441085315089411, 1668395194698627559174021943303615018, 5040101658352210125184502724643516494, 15229490745377914834385666536838756861, 46029318528145265253023134766777022745, 139150331306488380420022781714896252332, 420757658780151834853174354854926328169, 1272552003558211125982655302297999307459, 3849572424165465921985880068152401141924, 11647714029664066597674746017912728596349, 35249911641944228198049502407530716646740, 106699515335448698654432522514590860518079, 323036773191508705791478975401884091541634, 978193272221972534338772148378218988040824, 2962638915057084285359994625667259114603818, 8974542762966136343978490828013505912402263, 27190911558147114826908131657907359266414155, 82396992097773926675683259813624964817788931, 249731590963131293109712594865208051813258897, 757022131523278837558490057093509799167471482] ------------------------------------------------------------ Theorem Number, 44 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 19 17 16 15 14 13 12 11 10 9 8 (x - 3 x - 3 x + x + 6 x + 5 x + 5 x - 4 x - 7 x - 12 x - x 7 6 5 4 3 3 16 14 13 + x + 10 x + 2 x + 2 x - 4 x - x + 1) P(x) + (3 x - 6 x - 6 x 12 11 10 9 8 6 5 4 3 2 - 3 x + 7 x + 7 x + 16 x - x - 18 x - 2 x - 6 x + 10 x - x 2 13 11 10 9 8 7 6 5 + 3 x - 3) P(x) + (3 x - 3 x - 3 x - 6 x + 2 x - x + 10 x - x 4 3 2 10 6 5 4 3 2 + 6 x - 8 x + 2 x - 3 x + 3) P(x) + x - 2 x + x - 2 x + 2 x - x + x - 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 5, 8, 13, 29, 58, 109, 224, 469, 962, 2004, 4252, 9040, 19301, 41563, 89980, 195525, 426833, 935706, 2058178, 4541767, 10053231, 22314028, 49652620, 110747551, 247557767, 554494496, 1244340818, 2797369080, 6299090767, 14206237723, 32085826176, 72567625371, 164336472104, 372610625607, 845820085338, 1922098715982, 4372439993387, 9956343355891, 22692518763045, 51766984724848, 118193127925529, 270074678950222, 617606728681446, 1413389735165284, 3236829392842055, 7417758864228774, 17010163891685398, 39031546594762198, 89615630050729955, 205874421628959923, 473218932824758023, 1088313251325592598, 2504201652272691910, 5765011816121712227, 13278193824888813318, 30596973864393382114, 70536151123238024372, 162679316350017450709, 375347930528973340133, 866385011098516035066, 2000587011422283874074, 4621343255190585868302, 10679186807475470377805, 24686666622984511599253, 57086890984494449675458, 132055217721375981546693, 305573531184986377077384, 707314945608209847639832, 1637732567993388063478190, 3793171352332367061541348, 8787952385570629968379636, 20365508673681019718977166, 47208682141769560467266124, 109462253636911869115750581, 253874894309395675440725721, 588959015626156096858744585, 1366650733136206654486516020, 3172009302049365401821933578, 7363990486154164923351386224, 17099812077001249363177056269, 39716082047248874128446242438, 92264807603841023580831955900, 214386848828249772877195858443, 498253569181261894314960926934, 1158219376625202957075370231454, 2692882159291120533526978907126, 6262215947461674813212714459568, 14565353079098544379221773377774, 33883983435461603108197061482155, 78839999338725373583734755178870, 183474523257584763789771107474471, 427051528522691727246479996839525, 994165194829588457736112300985513, 2314776392397718522798745297649492] ------------------------------------------------------------ Theorem Number, 45 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 7+ HHH + H HH + HH HH 6+ H H + H H + HH HH 5+ HH HH + H H + HH HH 4+ HH H H + HH HH HH HH 3+ HH HHH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 7 6 5 4 3 3 (x - 2 x - 5 x - 3 x - 2 x + 3 x + x - 1) P(x) 7 6 5 4 3 2 2 + (3 x + 8 x + 5 x + 6 x - 7 x + x - 3 x + 3) P(x) 7 6 5 4 3 2 6 4 3 + (-x - 4 x - 2 x - 6 x + 5 x - 2 x + 3 x - 3) P(x) + x + 2 x - x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 43, 88, 181, 389, 845, 1831, 4021, 8945, 19980, 44876, 101560, 231010, 527584, 1210453, 2788476, 6445177, 14945402, 34762125, 81073108, 189553176, 444229775, 1043335656, 2455327148, 5789089278, 13673287432, 32347923587, 76645650172, 181868152606, 432131947999, 1028094409010, 2448927381166, 5840049815756, 13942114199763, 33318698172213, 79702566442900, 190836157036246, 457334328621662, 1096917010956494, 2633078585793776, 6325404794268179, 15206619342001349, 36583326860847321, 88069812063175602, 212154504630779797, 511384369072275705, 1233394476770162448, 2976499939131468798, 7187032770060977636, 17362981676250949732, 41968309776585627857, 101492313014921186269, 245556679552748148628, 594388103914626682818, 1439399709664903706555, 3487218741318126496572, 8451957378207794626600, 20493205067305036075159, 49708582948760009479704, 120619210443382870051297, 292792615812251132529932, 710980136125717425588637, 1727046021860570120941582, 4196575501209940797812052, 10200621702455464613659962, 24802455440798284389096014, 60324716048583750707955995, 146765765674535755148007651, 357173774922625517418331537, 869473358467539692845081275, 2117150070857988843015817831, 5156588999541444162109803583, 12562783839142431859361805544, 30613913629937142427400197707, 74620580935239731515082935125, 181929247599493983737210197721, 443657637304923176383168028461, 1082162432233326872644453144656, 2640180374641018514672900497678, 6442717456740989437135783799985, 15725218378158726422687662180370, 38389658112166505912696121858576, 93738863949937324569728179979473, 228934372538591942169266458710404, 559224587509088449210338239175827, 1366292159965124659559109224182072, 3338728826968157284819565578655591, 8160132627279040287185529051465709, 19947574345781438581105186436718539, 48770603600956548080699610819303235, 119261355989728246212509483676650821, 291684525929087013741255678799558755] ------------------------------------------------------------ Theorem Number, 46 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 3 2 (x + 4 x + 4 x + 2 x - 3 x - x + 1) P(x) 7 6 5 4 3 2 6 4 3 + (-2 x - 4 x - x - x + 5 x - x + 2 x - 2) P(x) + x + x - 2 x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 16, 30, 56, 110, 216, 427, 860, 1746, 3569, 7364, 15307, 32009, 67334, 142416, 302663, 646031, 1384486, 2977909, 6426645, 13911978, 30200915, 65732898, 143413923, 313595163, 687144819, 1508571730, 3317922569, 7309678595, 16129313045, 35643218886, 78875659428, 174774870620, 387750760231, 861260255477, 1915124365594, 4262999134446, 9498739083878, 21184957996924, 47291188848596, 105658705445218, 236257445857091, 528694503932384, 1183990265577090, 2653392279620791, 5950474873926279, 13353228687666239, 29984296182635404, 67369519671951483, 151455416547760267, 340681282599710033, 766734008185798102, 1726495929499880140, 3889579288927147642, 8766943991330924627, 19769471877891215161, 44600139100704220528, 100661893945840143266, 227287461047749510237, 513406564122610083700, 1160158342765246864333, 2622633765230015042003, 5930856706035176271770, 13416885846141972833882, 30362383327524771174841, 68733011599111138672309, 155645297432095952323778, 352568632584310655173635, 798885376847742072720203, 1810734038572540097719042, 4105354542744134743448103, 9310415132809649662747530, 21120611967784825680218897, 47924750352934920938803331, 108774247977720255491636377, 246946153325595537553849633, 560771040004278719756311221, 1273718100463445848541909110, 2893762556614309851409015460, 6575848332156955986692361543, 14946434073696128773556355402, 33979577457896340549152610437, 77266402617848204240564334985, 175733104543382019162751390086, 399764798964070432229145750542, 909581419077380496710545668751, 2069963664743655297207114411687, 4711574821783439857678518992020, 10726298267792692107831401356403, 24423746955362712321076292407699, 55622647415401209237519460346954, 126696994280216636321751775842705, 288638769125621515697430788944262] ------------------------------------------------------------ Theorem Number, 47 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 19 17 16 15 14 13 12 11 10 9 8 (x - 3 x - 3 x + x + 6 x + 5 x + 5 x - 4 x - 7 x - 12 x - x 7 6 5 4 3 3 16 14 13 + x + 10 x + 2 x + 2 x - 4 x - x + 1) P(x) + (3 x - 6 x - 6 x 12 11 10 9 8 6 5 4 3 2 - 3 x + 7 x + 7 x + 16 x - x - 18 x - 2 x - 6 x + 10 x - x 2 13 11 10 9 8 7 6 5 + 3 x - 3) P(x) + (3 x - 3 x - 3 x - 6 x + 2 x - x + 10 x - x 4 3 2 10 6 5 4 3 2 + 6 x - 8 x + 2 x - 3 x + 3) P(x) + x - 2 x + x - 2 x + 2 x - x + x - 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 5, 8, 13, 29, 58, 109, 224, 469, 962, 2004, 4252, 9040, 19301, 41563, 89980, 195525, 426833, 935706, 2058178, 4541767, 10053231, 22314028, 49652620, 110747551, 247557767, 554494496, 1244340818, 2797369080, 6299090767, 14206237723, 32085826176, 72567625371, 164336472104, 372610625607, 845820085338, 1922098715982, 4372439993387, 9956343355891, 22692518763045, 51766984724848, 118193127925529, 270074678950222, 617606728681446, 1413389735165284, 3236829392842055, 7417758864228774, 17010163891685398, 39031546594762198, 89615630050729955, 205874421628959923, 473218932824758023, 1088313251325592598, 2504201652272691910, 5765011816121712227, 13278193824888813318, 30596973864393382114, 70536151123238024372, 162679316350017450709, 375347930528973340133, 866385011098516035066, 2000587011422283874074, 4621343255190585868302, 10679186807475470377805, 24686666622984511599253, 57086890984494449675458, 132055217721375981546693, 305573531184986377077384, 707314945608209847639832, 1637732567993388063478190, 3793171352332367061541348, 8787952385570629968379636, 20365508673681019718977166, 47208682141769560467266124, 109462253636911869115750581, 253874894309395675440725721, 588959015626156096858744585, 1366650733136206654486516020, 3172009302049365401821933578, 7363990486154164923351386224, 17099812077001249363177056269, 39716082047248874128446242438, 92264807603841023580831955900, 214386848828249772877195858443, 498253569181261894314960926934, 1158219376625202957075370231454, 2692882159291120533526978907126, 6262215947461674813212714459568, 14565353079098544379221773377774, 33883983435461603108197061482155, 78839999338725373583734755178870, 183474523257584763789771107474471, 427051528522691727246479996839525, 994165194829588457736112300985513, 2314776392397718522798745297649492] ------------------------------------------------------------ Theorem Number, 48 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 13 12 11 10 9 8 7 6 3 2 (x + x - 2 x - 4 x - 3 x + 3 x + 4 x + 6 x - 3 x - x + 1) P(x) 10 9 8 7 6 5 3 2 7 + (2 x + 2 x - 2 x - 4 x - 6 x + x + 5 x - x + 2 x - 2) P(x) + x 6 3 2 + x - 2 x + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 6, 10, 19, 33, 58, 108, 200, 369, 693, 1317, 2514, 4829, 9344, 18189, 35592, 69989, 138246, 274200, 545940, 1090783, 2186343, 4395255, 8860180, 17906170, 36272959, 73639706, 149804622, 305325432, 623406188, 1274971376, 2611597375, 5357319501, 11004919758, 22635485192, 46614898206, 96108678206, 198370269901, 409866012138, 847687251598, 1754836188328, 3636009242250, 7540193770668, 15649198235420, 32504074898770, 67562356231747, 140533256797216, 292514736909839, 609254134880606, 1269754726390174, 2647896576486016, 5524994718069158, 11534607248785519, 24093773197898278, 50353464184534819, 105285448864725132, 220249118224532352, 460956131234061995, 965156511973390642, 2021725152698343735, 4236688329900954289, 8881883438230807832, 18627397182916201898, 39080715060247020314, 82022115622422152747, 172207762570687826247, 361678877010846007041, 759867396916920746124, 1596954324667818296589, 3357245924609224890483, 7060019872578142781287, 14851047729721395149780, 31248783492263029171750, 65770414878714109406951, 138466981593815657698333, 291592875976612309859386, 614213871210552443656780, 1294110799316931990023822, 2727279498663073507672105, 5748989295111391492048961, 12121446207030768286650281, 25563244831836776909695275, 53922960715096892884159306, 113769377614144303450467979, 240087037262751143517465894, 506759120361514589789712132, 1069847544922340934122333579, 2259059523710516236809422262, 4771083098316998636523021295, 10078315424868890475447526836, 21293093745583655561384647024, 44995356141694588627315053710, 95098362701380027278219259613, 201026434433379944826868248653] ------------------------------------------------------------ Theorem Number, 49 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} To make it crystal clear here is such a path of semi-length 8 2+ H H H + HH HH HH + HH HH HH + H H H H H H + H H H H H H 1.5+ H HH H H H H + H H H H H H + H H H H H H + H H H H H H + H H H H H H 1+ H H H H H H H H + H H H H H H H H + H H H H H H H H H H + H H H H H H H H H H + H H H H H H H H H H 0.5+ HH H H H H H H H H HH + H H H H H H H H H H + H H H H H H H H H H +H HH HH HH HH H +H HH HH HH HH H -*-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+*-+-+-+-+-*+-+-+-+-++-+-+-+-*+-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 5 4 3 2 2 2 1 + (x + 2 x + 3 x - 2 x - 2 x + 1) P(x) + (3 x + 2 x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence 6 (n - 3) a(n - 1) 6 (n - 4) a(n - 2) (11 n - 34) a(n - 3) a(n) = ------------------ - ------------------ - -------------------- n - 2 n - 2 n - 2 2 (5 n - 19) a(n - 4) (7 n - 26) a(n - 5) 2 (2 n - 7) a(n - 6) + --------------------- + ------------------- + -------------------- n - 2 n - 2 n - 2 subject to the initial conditions [a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 8, a(5) = 17, a(6) = 38] Just for fun, a(1000), equals 7472004740884143994679750011505414385089449080809549227151747775583714181956\ 536837803404283279988418353929138376866003733009327141623241983843394689\ 049761892634452461232604005614720971176536316575409273226450456885738387\ 261012234298147398550894624041521122725841806595200083186863945206452239\ 380842400951251068488049992909370751069592425905243432500414876191403085\ 422392998286571046477077108918278795775455036332196840751414469909154932\ 710321709382180009144619429722667562379640581778042866289048485864956256\ 397761757402418687172541856803678959868399371154511099339433238463533730\ 9261531230970400 For the sake of the OEIS here are the first, 100, terms. [1, 1, 2, 4, 8, 17, 38, 90, 227, 610, 1740, 5231, 16430, 53451, 178765, 610952, 2123890, 7484497, 26667782, 95889889, 347447909, 1267218066, 4648105412, 17133976089, 63437990117, 235800295775, 879570828277, 3291416367487, 12352509949332, 46481454639292, 175332289038462, 662855505694621, 2511177005933507, 9531739239005258, 36244755979304360, 138052480995439209, 526650177127152839, 2012040434973057757, 7697488024923790849, 29486555056302509409, 113091416346905424722, 434246610471577928762, 1669234743713214339676, 6423144903467831561863, 24740268666835142791959, 95381968184080688365610, 368055802098615512682135, 1421439329607234789344251, 5494058530699568495063881, 21251652820815936901293000, 82264384899019041597380619, 318666863628427531898776858, 1235250227381056102563572513, 4791305320905739355458681460, 18596115753583770685482939290, 72218660300452603618940007138, 280624340282747425055997772472, 1091039511064409681927555315341, 4244102801573701993376341470144, 16517835095918365595066095764300, 64318242777699810581616493883447, 250565604269225983617823365293988, 976579938819461663675644636173453, 3807907549446985472345350767324693, 14854251801658753984917302474799470, 57968867177482568971392567897529856, 226314640386805519972549960751423804, 883891034561703142803775846416233021, 3453406910362724794068783849169665799, 13497541567092128156893682776496812146, 52773350725230020327955665836740829313, 206406391143006196508097884689083504175, 807561656690124872027268502632461290619, 3160589661866307365468951804920933356158, 12373609145312399276074282140584626026607, 48457016900985196116589509516153468081710, 189821442767550390589566330641108650947247, 743804229231814557118659953299105166002398, 2915368139773662012977827245253839175035808, 11430001697124323088243408312404535628069164, 44824369810474325129002130363708608531476586, 175830452306177589350420861233872781050367633, 689895179808416620565325289843633156027799162, 2707562361211079408186654144198490594798343636, 10628635033790706798569689912848940589309815101, 41732808991772269882722008909648164365896942416, 163899021288349899015430415806146246175897724269, 643830186596036759992299609403969951390706682073, 2529648859286937851335004647764867988985203416118, 9941246848440089013558999387357999331331774637962, 39076084838762898178966671362274770273111734065927, 153627417987827651782542935407503030555655631332009, 604104295506691716943633347251389544820725665394277, 2375957630716937708966398798345021837644239728513549, 9346460415541814696518796715621755643159440331073609, 36773550107674900865523765407780448727774691254097140, 144711188680621822662978681802688710000708165930118288, 569567422291104441001487849953608340313659015741018254, 2242141491136055465482963645514603818158677927424224947, 8827833904587517114116271159376422193553347263079557949, 34762981953234488727019402777565101895466837261569932414] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 50 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ HH HH + HH HH + H H + HH HH 4+ H H + H H + HH HH 3+ H H + HH HH + HH HH 2+ H HH HH + HHH H H + HH H HH HH + H HH H H 1+ HH HH HH HH + H H H H +HH HHHH HH -*-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 2 3 2 1 + (x + 2 x + 2 x - x - 3 x + 1) P(x) + (x + 3 x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence (3 n - 10) a(n - 1) (3 n - 8) a(n - 2) 2 (4 n - 15) a(n - 3) a(n) = ------------------- + ------------------ - --------------------- n - 2 n - 2 n - 2 (5 n - 18) a(n - 4) 2 (2 n - 9) a(n - 5) 5 (n - 4) a(n - 6) - ------------------- + -------------------- + ------------------ n - 2 n - 2 n - 2 3 (n - 4) a(n - 7) + ------------------ n - 2 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 3, a(6) = 6, a(7) = 11] Just for fun, a(1000), equals 1145247889784214898460130876069454497699198945209860899382804351131772528282\ 133731708344230812745657626667920291158213887747665601298356727927944776\ 030647389055090332321758191672696872867145930254564580810357005416798425\ 235280336717321020337663125309372117332677581781296695840911474312424353\ 109238551195056650414827517078765491770489867998868102973083917330485510\ 399043443105952336355540946876142869551090788288618966120768479952859514\ 795957207874664550283182756465317216 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 2, 3, 6, 11, 23, 49, 111, 260, 631, 1571, 3998, 10351, 27183, 72222, 193765, 524152, 1427918, 3913839, 10785131, 29860663, 83023346, 231707013, 648868066, 1822699373, 5134485859, 14501047305, 41051801922, 116470558685, 331116492750, 943112497105, 2690965323833, 7690669891937, 22013351910519, 63100511237332, 181120160013138, 520539572163698, 1497829936462364, 4314837199837584, 12443230444884596, 35920670606415061, 103794869271147607, 300197286521208002, 868996122922789762, 2517626225693246437, 7299790270596987425, 21181646928330883083, 61507036728946177581, 178727788506850517912, 519695401194125167769, 1512109642114056218557, 4402345591213388938044, 12824513841726472678399, 37380380499502693529327, 109014237042999565846414, 318090431309338867876386, 928618961502649280575439, 2712290291874573897385879, 7925725232544474142754753, 23170692019202436270094753, 67768744210089068725511744, 198291428265081823208297909, 580438829259492218057110181, 1699735035454079974628644198, 4979351238773816135366226285, 14592370164402130306544769983, 42779476843772237493471224940, 125457578082203857583718106213, 368048952058834293418581033508, 1080083138285505007796767563496, 3170644490401645406403850948413, 9310493512451363834365498930284, 27348205767536943373804271403750, 80354890729020603260436860850253, 236167229309395496479949093808441, 694300446923883916492764288370007, 2041702813338529207790328193243177, 6005536220991345043954472164788529, 17669418816060756131445534993099489, 51999730668275388400877829723757133, 153068396153069627851719888972166535, 450684823412170479752466953309262686, 1327274512101663332984053719899520102, 3909727981272968899078067938135412974, 11519348937895569170200408275308392163, 33947097692236579161442595800013425794, 100061841976125938681066186451886631053, 295000900653572234496534231947047618528, 869891621951872165451134841310889932774, 2565617645671780764466166332700650893684, 7568360110286842554479538057942198692096, 22330212873582761969352565997259163646634, 65896655484064735791290558939780113001569, 194496409912782516835188524447856832105677, 574163843122884606850063550924349788068566, 1695252941486902059890063951026991901471039, 5006174931932292551782715214924205548964976, 14785937476305642720296210359047476219203690] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 51 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ H HH HH + HH H HH HH + H HH H H + HH HHH HH 4+ H H H + H H + HH HH 3+ H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HHH HH HH + H H H H +HH HHHH HH -*-+-+-+-+*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 7 6 4 2 3 1 + (x + 2 x - x - x - 2 x + 3 x - 1) P(x) 6 5 4 2 2 3 2 + (2 x + x + 2 x + x - 6 x + 3) P(x) + (x + x + 3 x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 4, 8, 17, 38, 90, 226, 595, 1626, 4583, 13255, 39152, 117632, 358379, 1104540, 3437534, 10787176, 34092504, 108416342, 346645635, 1113679878, 3593289935, 11638329754, 37826257535, 123327887440, 403249754800, 1321986604313, 4344384539207, 14308585907725, 47223762974482, 156154993535171, 517279512721917, 1716395204199984, 5704077870995768, 18983996816831575, 63268228255622253, 211127426884278728, 705395591765359037, 2359504087156624476, 7900995290800278546, 26484466908922828087, 88864073724588088699, 298445919260326762000, 1003205605602019452638, 3375055833370038475415, 11363752620369614839002, 38290991969078239124495, 129119188192510283827134, 435702970067638336401740, 1471237997718381515655961, 4971139384784656516703965, 16807309393261681339710744, 56858974801937271214176618, 192463501961806458549367052, 651833685891728036085694672, 2208794626314526388838255955, 7488512324566195972361000419, 25400920138005481936189410149, 86200438627398423962889600077, 292663432884995520011132877727, 994076051219370632401679112136, 3377975277695539232604789875543, 11483463522638435443420165861052, 39053778764590735260214744028411, 132868310291185248291927887818515, 452212595533463639447561125683626, 1539649070809086278831859924742111, 5243893547522691259065527773303234, 17866291918724072977961447260036381, 60891830322474637750552436362104287, 207598148314178819145969881814596126, 707984497135539785608629387494969248, 2415216119078895254512880464157359266, 8241695045438919774148317893449721900, 28132079874956214994051496481913538658, 96052464350363092991383456220764683736, 328044905624843906486485910858513572753, 1120657949699764053438714184735518723226, 3829348091994830090540765936502828433868, 13088374765183661773959550315278141251762, 44745867155596722406431032729662747065064, 153011402892271561815196042063134630398688, 523354250006880412472190047285558113292304, 1790467027162243322753693430925261434443773, 6126791872075190111368519055950746871602422, 20969779058966314248851114808976871975726427, 71787080271027796432625484374156603053117008, 245803630113711348682394578531982069883458376, 841817199668759568384197048764335720527595287, 2883585288228841543815032314714643011117463365, 9879418496568057882669009954460305569698680220, 33854131408980382617618344444952182795596663927, 116030423557513326707883799432379866683283721065, 397750052241093250199289662453849289266840594823, 1363719630603529068278211367514741724664144560802, 4676433740888708159615030564012359089021337713132, 16039016859440355938804763268391989690074811992256, 55018967565298271506684475923762005973565491298128] ------------------------------------------------------------ Theorem Number, 52 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 6 5 3 2 3 1 + (x - 3 x - x + 2 x + x - 1) P(x) 6 5 3 2 2 3 2 + (3 x + x - 4 x - 2 x + 3) P(x) + (2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 24, 50, 100, 207, 447, 965, 2097, 4641, 10355, 23208, 52434, 119238, 272302, 624715, 1439586, 3329035, 7723353, 17974332, 41946862, 98136543, 230136785, 540859834, 1273645217, 3004841530, 7101517437, 16810654036, 39854525348, 94621732620, 224950428542, 535465514846, 1276128784737, 3044720948720, 7272175007841, 17386856915967, 41609801199087, 99670684111928, 238955727084254, 573360774386420, 1376837089630860, 3308764126191171, 7957256958283794, 19149694328506228, 46115679249370442, 111125023198040762, 267941826636175204, 646433889341639098, 1560460708781195478, 3768924211315976522, 9107707171651752850, 22020112055371883983, 53264882178765929245, 128903877018767159586, 312095509354848105514, 755960176581781422429, 1831866563301055461943, 4440851470520127715854, 10769869026085811825291, 26128886252691178913799, 63415069809483920442764, 153964356091782976724423, 373937747991975936017966, 908500784048788194813896, 2207973525844179375897501, 5367855788092394722780239, 13053960160887471570162186, 31755167086049496587497855, 77270472173378202155837143, 188077261542308295134223872, 457909065550704597273291014, 1115165194681535923031110438, 2716521249676194665104160483, 6619085378281378272215833397, 16132103128989793305097961362, 39326870592610533254429937069, 95893803896808211685222661778, 233879375049458391590317811366, 570546544663074707446614357597, 1392148793303975476592593878598, 3397608858830076013984401770568, 8293771449881197142382854212644, 20249746445904137513663328249672, 49450867782510794412659596235458, 120785019511666197733598404565551, 295076860334680412091808688807665, 721005064284596647207054443970788, 1762060340462307761691050403140135, 4307058934161988158319017861316542, 10529718506027909698423650877921477, 25747021917792169515323378704085982, 62966561145792275331841952840077732, 154015405894044114004769528686025159] ------------------------------------------------------------ Theorem Number, 53 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H + HH + HH HH + H H + H H 3+ H H + H H + HH H + H H + H H 2+ H H H H + HH HH H H + H H H H H H + HH H HH H HH HH + H H H H H H 1+ H H H H H H + H H H H H H + H H H H H H + HH HH HH HH HH HH +H HH HH H -*-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 2 3 2 1 + (x + 2 x + 2 x - x - 3 x + 1) P(x) + (x + 3 x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence (3 n - 10) a(n - 1) (3 n - 8) a(n - 2) 2 (4 n - 15) a(n - 3) a(n) = ------------------- + ------------------ - --------------------- n - 2 n - 2 n - 2 (5 n - 18) a(n - 4) 2 (2 n - 9) a(n - 5) 5 (n - 4) a(n - 6) - ------------------- + -------------------- + ------------------ n - 2 n - 2 n - 2 3 (n - 4) a(n - 7) + ------------------ n - 2 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 3, a(6) = 6, a(7) = 11] Just for fun, a(1000), equals 1145247889784214898460130876069454497699198945209860899382804351131772528282\ 133731708344230812745657626667920291158213887747665601298356727927944776\ 030647389055090332321758191672696872867145930254564580810357005416798425\ 235280336717321020337663125309372117332677581781296695840911474312424353\ 109238551195056650414827517078765491770489867998868102973083917330485510\ 399043443105952336355540946876142869551090788288618966120768479952859514\ 795957207874664550283182756465317216 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 2, 3, 6, 11, 23, 49, 111, 260, 631, 1571, 3998, 10351, 27183, 72222, 193765, 524152, 1427918, 3913839, 10785131, 29860663, 83023346, 231707013, 648868066, 1822699373, 5134485859, 14501047305, 41051801922, 116470558685, 331116492750, 943112497105, 2690965323833, 7690669891937, 22013351910519, 63100511237332, 181120160013138, 520539572163698, 1497829936462364, 4314837199837584, 12443230444884596, 35920670606415061, 103794869271147607, 300197286521208002, 868996122922789762, 2517626225693246437, 7299790270596987425, 21181646928330883083, 61507036728946177581, 178727788506850517912, 519695401194125167769, 1512109642114056218557, 4402345591213388938044, 12824513841726472678399, 37380380499502693529327, 109014237042999565846414, 318090431309338867876386, 928618961502649280575439, 2712290291874573897385879, 7925725232544474142754753, 23170692019202436270094753, 67768744210089068725511744, 198291428265081823208297909, 580438829259492218057110181, 1699735035454079974628644198, 4979351238773816135366226285, 14592370164402130306544769983, 42779476843772237493471224940, 125457578082203857583718106213, 368048952058834293418581033508, 1080083138285505007796767563496, 3170644490401645406403850948413, 9310493512451363834365498930284, 27348205767536943373804271403750, 80354890729020603260436860850253, 236167229309395496479949093808441, 694300446923883916492764288370007, 2041702813338529207790328193243177, 6005536220991345043954472164788529, 17669418816060756131445534993099489, 51999730668275388400877829723757133, 153068396153069627851719888972166535, 450684823412170479752466953309262686, 1327274512101663332984053719899520102, 3909727981272968899078067938135412974, 11519348937895569170200408275308392163, 33947097692236579161442595800013425794, 100061841976125938681066186451886631053, 295000900653572234496534231947047618528, 869891621951872165451134841310889932774, 2565617645671780764466166332700650893684, 7568360110286842554479538057942198692096, 22330212873582761969352565997259163646634, 65896655484064735791290558939780113001569, 194496409912782516835188524447856832105677, 574163843122884606850063550924349788068566, 1695252941486902059890063951026991901471039, 5006174931932292551782715214924205548964976, 14785937476305642720296210359047476219203690] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 54 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 7 6 5 4 3 2 2 4 3 2 1 + (x + 2 x + 2 x + x - x - 3 x + 1) P(x) + (x + x + 3 x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 1, 1, 2, 3, 5, 9, 16, 30, 58, 116, 239, 506, 1097, 2427, 5463, 12477, 28848, 67394, 158836, 377178, 901495, 2166867, 5234213, 12699051, 30930290, 75598789, 185359855, 455784844, 1123668716, 2776873715, 6877498572, 17068229197, 42439024789, 105707007672, 263725516660, 658967894157, 1648915263775, 4131583829601, 10365363804074, 26035798398589, 65470688357808, 164811011072201, 415302561925389, 1047513699571611, 2644544397931547, 6682191535862199, 16898420509896319, 42767701945366204, 108321066048548465, 274550837288738769, 696357117682632614, 1767372462308960760, 4488478712318269104, 11406021832716039226, 29001653107017572484, 73782782204311220057, 187811324578820557484, 478315299598201141936, 1218777069501669912391, 3107019537324250284192, 7924393223792736964588, 20220103680933308897535, 51616603776023631799277, 131818952594264610824282, 336777197104868241753975, 860752297305782872714291, 2200791837469090360102192, 5629111296473102966453270, 14403096346209760209248825, 36865688736322206868993176, 94391915584188999964681518, 241762486800264653818633439, 619413388059785426667077461, 1587471328993104064922016395, 4069687452247926398792681845, 10436202123998847589674392197, 26769897363547185608920104004, 68686345920426647296144500475, 176282993708252565466531022075, 452546969785698724707339960583, 1162056396731017551043333923461, 2984683088317573075660756014720, 7667856918842597699990931757657, 19703887367936868082006875399023, 50644169471185007417823062512338, 130197961586413959569843116196477, 334790997742176319121603071307000, 861065043064209575846281411734462, 2215075682539561817347226211311255, 5699404812652210175645233057118791, 14667521630026537828113104987883985, 37754463990531766806166131082876284, 97199118700228488192153500861825564, 250286233847847220249286755657934286, 644600218742256194485115463829505246, 1660431912815134961187878791724614232, 4277866004858347047255111494171913904, 11023186660234240231599180829063203219, 28409230651584433392323145563646776771] ------------------------------------------------------------ Theorem Number, 55 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 4 3 2 1 + (x + x + x + x - x - x - x + 1) P(x) + (x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 9, 20, 43, 94, 216, 505, 1198, 2888, 7046, 17359, 43137, 107986, 272046, 689198, 1754652, 4486913, 11519069, 29677863, 76709558, 198858270, 516900481, 1346928260, 3517812058, 9206982999, 24144187305, 63430675750, 166926209959, 439987494362, 1161457799001, 3070255463861, 8126765806794, 21537753472145, 57146877005927, 151798160546375, 403642671170599, 1074387128983240, 2862437766174048, 7633132414976498, 20372410447459630, 54417377540703801, 145469599771072044, 389162759164454635, 1041839048797994230, 2791046978146480666, 7482013275489230337, 20069799089461878199, 53867882389413538721, 144666793314662499979, 388731932228847260560, 1045116905170070363120, 2811280699994517254169, 7565891179843013798830, 20371584272556101353559, 54877087351154823046446, 147894461371360500600404, 398750098993789333077497, 1075552191594669068647971, 2902271444012572374471826, 7834562869658274177153008, 21157109326212823185371492, 57155440767212455161258473, 154459142393974795917404387, 417560856723240437282759617, 1129201890148987098172382917, 3054673524396563025177593345, 8265998275897394297578700464, 22374802833318826872122652538, 60583279272141959914982521230, 164086330649285086653717627671, 444543898955742904386559063447, 1204692560613252897219505954158, 3265532089955788106865668972679, 8854107516472847109120249657859, 24012966115006271427327717275224, 65140965498791000199417379317401, 176753178774204821202519803270294, 479713926425134540506383824683065, 1302257781851035311627912843109992, 3535971195076524443101782142887928, 9603184357286987364921458068688114, 26086412423311750697378497706485168, 70876753808269982532179552022657879, 192611209804903641890607349672943792, 523534801206376171916739816450584025, 1423291500980846384191200772841684210, 3870121247506880066064768564504438295, 10525333776394036585049608115707480862, 28630307203934076060493508135753907692, 77892072069382575356484169897529778150, 211951244648000788109732632639813482287, 576836257741440102366311224036503009129, 1570151092397116337614826283811806160336, 4274655842190291606027844557863134575971] ------------------------------------------------------------ Theorem Number, 56 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 1 + (x + 2 x + x + x - 2 x - x - x + 1) P(x) 4 3 2 + (2 x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 5, 9, 15, 27, 52, 99, 191, 380, 767, 1561, 3212, 6676, 13980, 29462, 62457, 133095, 284904, 612325, 1320835, 2858528, 6204794, 13504896, 29467046, 64443065, 141232079, 310127824, 682241562, 1503390100, 3318125585, 7334286838, 16234066006, 35980324422, 79843282903, 177385141221, 394524626312, 878382600262, 1957594535824, 4366863711975, 9749996633366, 21787566703548, 48726502251068, 109058113252455, 244271064473613, 547511721413636, 1228030388231890, 2756175501803298, 6189779845643445, 13909237626615638, 31273831159967158, 70355639977282178, 158360839466453771, 356631087517226215, 803535502075715446, 1811330859313674816, 4084981786161499396, 9216695405000005915, 20803989056321610682, 46978386063151976380, 106126487292393541621, 239838068767767185118, 542220278033115896654, 1226285592006812766558, 2774347696927967271630, 6278832855959591570828, 14214818277513329553301, 32191694467833655881358, 72926011196323997616532, 165254507682930495490441, 374586998426594973124882, 849331174613651352346800, 1926295937143221477067260, 4370058634352064414375042, 9916689051600915661994716, 22509108319009393216650851, 51104501330215476319608970, 116055707226039146541380078, 263619574451311192836750119, 598949259458766324199929106, 1361134842968359225485601500, 3093916712876970178692847268, 7034125719967269865497972628, 15995707321136515987498854278, 36381989171910295666778485227, 82766964402142036623410315006, 188327268860574074593683978712, 428600872006951760458663393481, 975606498851851793659981066972, 2221141899698881971264987257058, 5057736078329066998909302286110, 11518941426334046103018614433700, 26238795102693120770431186038224, 59778984596015548358887856377059, 136215011845586736829636505040690] ------------------------------------------------------------ Theorem Number, 57 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H H + HH H HH H + H HH H HH + HH HHH H 4+ H H H H H + H HH HHH HHH + HH H H HH H HH + H HH H H HH H 3+ H H H H + HH HH + H H + HH HH 2+ H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 7 6 4 2 3 1 + (x + 2 x - x - x - 2 x + 3 x - 1) P(x) 6 5 4 2 2 3 2 + (2 x + x + 2 x + x - 6 x + 3) P(x) + (x + x + 3 x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 4, 8, 17, 38, 90, 226, 595, 1626, 4583, 13255, 39152, 117632, 358379, 1104540, 3437534, 10787176, 34092504, 108416342, 346645635, 1113679878, 3593289935, 11638329754, 37826257535, 123327887440, 403249754800, 1321986604313, 4344384539207, 14308585907725, 47223762974482, 156154993535171, 517279512721917, 1716395204199984, 5704077870995768, 18983996816831575, 63268228255622253, 211127426884278728, 705395591765359037, 2359504087156624476, 7900995290800278546, 26484466908922828087, 88864073724588088699, 298445919260326762000, 1003205605602019452638, 3375055833370038475415, 11363752620369614839002, 38290991969078239124495, 129119188192510283827134, 435702970067638336401740, 1471237997718381515655961, 4971139384784656516703965, 16807309393261681339710744, 56858974801937271214176618, 192463501961806458549367052, 651833685891728036085694672, 2208794626314526388838255955, 7488512324566195972361000419, 25400920138005481936189410149, 86200438627398423962889600077, 292663432884995520011132877727, 994076051219370632401679112136, 3377975277695539232604789875543, 11483463522638435443420165861052, 39053778764590735260214744028411, 132868310291185248291927887818515, 452212595533463639447561125683626, 1539649070809086278831859924742111, 5243893547522691259065527773303234, 17866291918724072977961447260036381, 60891830322474637750552436362104287, 207598148314178819145969881814596126, 707984497135539785608629387494969248, 2415216119078895254512880464157359266, 8241695045438919774148317893449721900, 28132079874956214994051496481913538658, 96052464350363092991383456220764683736, 328044905624843906486485910858513572753, 1120657949699764053438714184735518723226, 3829348091994830090540765936502828433868, 13088374765183661773959550315278141251762, 44745867155596722406431032729662747065064, 153011402892271561815196042063134630398688, 523354250006880412472190047285558113292304, 1790467027162243322753693430925261434443773, 6126791872075190111368519055950746871602422, 20969779058966314248851114808976871975726427, 71787080271027796432625484374156603053117008, 245803630113711348682394578531982069883458376, 841817199668759568384197048764335720527595287, 2883585288228841543815032314714643011117463365, 9879418496568057882669009954460305569698680220, 33854131408980382617618344444952182795596663927, 116030423557513326707883799432379866683283721065, 397750052241093250199289662453849289266840594823, 1363719630603529068278211367514741724664144560802, 4676433740888708159615030564012359089021337713132, 16039016859440355938804763268391989690074811992256, 55018967565298271506684475923762005973565491298128] ------------------------------------------------------------ Theorem Number, 58 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 7+ HH + H HH + HH HH 6+ H HH + H HH + HH HH 5+ HH HH HHH + H HH HH H + HH HHH HH 4+ H H H + HH HH 3+ HH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 4 3 2 1 + (x + x + x + x - x - x - x + 1) P(x) + (x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 9, 20, 43, 94, 216, 505, 1198, 2888, 7046, 17359, 43137, 107986, 272046, 689198, 1754652, 4486913, 11519069, 29677863, 76709558, 198858270, 516900481, 1346928260, 3517812058, 9206982999, 24144187305, 63430675750, 166926209959, 439987494362, 1161457799001, 3070255463861, 8126765806794, 21537753472145, 57146877005927, 151798160546375, 403642671170599, 1074387128983240, 2862437766174048, 7633132414976498, 20372410447459630, 54417377540703801, 145469599771072044, 389162759164454635, 1041839048797994230, 2791046978146480666, 7482013275489230337, 20069799089461878199, 53867882389413538721, 144666793314662499979, 388731932228847260560, 1045116905170070363120, 2811280699994517254169, 7565891179843013798830, 20371584272556101353559, 54877087351154823046446, 147894461371360500600404, 398750098993789333077497, 1075552191594669068647971, 2902271444012572374471826, 7834562869658274177153008, 21157109326212823185371492, 57155440767212455161258473, 154459142393974795917404387, 417560856723240437282759617, 1129201890148987098172382917, 3054673524396563025177593345, 8265998275897394297578700464, 22374802833318826872122652538, 60583279272141959914982521230, 164086330649285086653717627671, 444543898955742904386559063447, 1204692560613252897219505954158, 3265532089955788106865668972679, 8854107516472847109120249657859, 24012966115006271427327717275224, 65140965498791000199417379317401, 176753178774204821202519803270294, 479713926425134540506383824683065, 1302257781851035311627912843109992, 3535971195076524443101782142887928, 9603184357286987364921458068688114, 26086412423311750697378497706485168, 70876753808269982532179552022657879, 192611209804903641890607349672943792, 523534801206376171916739816450584025, 1423291500980846384191200772841684210, 3870121247506880066064768564504438295, 10525333776394036585049608115707480862, 28630307203934076060493508135753907692, 77892072069382575356484169897529778150, 211951244648000788109732632639813482287, 576836257741440102366311224036503009129, 1570151092397116337614826283811806160336, 4274655842190291606027844557863134575971] ------------------------------------------------------------ Theorem Number, 59 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 4+ H + HH + HH HH + H H + H H 3+ H H H + H H H H + H HH H H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H H + H H H H H H + H H H H H H + HH HH HH HH HH HH +H HH HH H -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-*+-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 16 15 14 13 12 11 10 9 8 1 + (x - x + 6 x - 6 x + 18 x - 20 x + 34 x - 38 x + 40 x 7 6 5 4 3 2 4 13 - 40 x + 30 x - 26 x + 21 x - 14 x + 10 x - 5 x + 1) P(x) + (x 12 11 10 9 8 7 6 5 4 - 2 x + 9 x - 11 x + 29 x - 30 x + 45 x - 40 x + 36 x - 39 x 3 2 3 + 26 x - 21 x + 15 x - 4) P(x) + 9 8 7 6 5 4 3 2 (-x + 6 x - 7 x + 16 x - 11 x + 22 x - 17 x + 12 x - 15 x + 6) 2 5 4 3 2 P(x) + (x - 2 x + 5 x - x + 5 x - 4) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 4, 8, 16, 33, 72, 164, 385, 929, 2302, 5840, 15111, 39757, 106109, 286706, 782895, 2157299, 5991418, 16754254, 47133780, 133304680, 378797699, 1080937675, 3096280119, 8899537396, 25659333522, 74191700152, 215076839680, 624984670580, 1820129112192, 5311545870919, 15529666967498, 45485087616947, 133441401606583, 392085514552888, 1153713520188876, 3399415949251933, 10029177048139525, 29624392272099090, 87604798820875723, 259342755462266401, 768535092178287956, 2279681122434373908, 6768375941367910082, 20112936791682062882, 59817630236510978087, 178044647096438862509, 530345732633818958522, 1580902724946170334773, 4715780087511170459341, 14076398258518775876396, 42044309205950898883276, 125657775418684525282240, 375774621674018331570617, 1124375927719242353602914, 3366140922258788228051364, 10082793760566366735918753, 30216834851528764843156805, 90600097910177860813128100, 271776877001635112478414645, 815630397943282664266638306, 2448863906442666183560448424, 7355629884158953643930957993, 22103090309753427783590493944, 66444383239904652378075241174, 199815976489513500075595905999, 601123318248188990641792230597, 1809061723036316227985596979165, 5446216242789368898725261988498, 16401498208300928395050396115929, 49410012350418902703302101326748, 148896693672221242827909559044314, 448838211413611377469327964824974, 1353397796957554208981282171507347, 4082144142443893016917450931257201, 12316149177447690389881162822938496, 37169089927293809170079493663299174, 112203416185148453398803931080693932, 338800706501319369696382907969208572, 1023278009928211254986459641428529088, 3091371705830182285136346003095432953, 9341450589312301474826731216085100489, 28234511565375956705606901847882309569, 85358457499283402993308375153738965078, 258113493055080447379653073472541104222, 780675170598065191844926634793792438671, 2361691994037213726820461707344108161220, 7146068639249103743135894366785630040238, 21627186493192152310573023594624255353631, 65466592489038222003770498496859393128085, 198209443092575491290699990137175971768889, 600222027601206427086019256822220933481158, 1817944647598719163400906295383418984699691, 5507173198404019898922370364442446872032905, 16686084596223294364458606091843282550184879, 50565709242439808017941737571515784259619335, 153261158470844421358596187137846703234962646, 464601788900210564675892966077306594592266035] ------------------------------------------------------------ Theorem Number, 60 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 13 12 11 10 9 8 7 6 4 3 2 1 + (x + x + 2 x + x + x - 3 x - x - 4 x + x + 2 x + x - 1) 3 9 8 7 6 5 4 3 2 2 P(x) + (x + 3 x + x + 4 x - x - 2 x - 4 x - 2 x + 3) P(x) 5 4 3 2 + (x + x + 2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 6, 10, 20, 38, 71, 141, 287, 583, 1202, 2521, 5325, 11323, 24282, 52406, 113664, 247757, 542480, 1192313, 2629716, 5818748, 12912436, 28729813, 64079490, 143246101, 320883803, 720195496, 1619321024, 3647055270, 8226810087, 18584803194, 42041964361, 95229445722, 215968599608, 490356105984, 1114569328616, 2536017260869, 5775957109341, 13167432219753, 30044425752651, 68611060035416, 156810360349206, 358665774457819, 820965478557937, 1880467407759439, 4310223033012889, 9885862452845120, 22688139458714478, 52100512493277179, 119710934944822697, 275210416296780580, 633032733279400641, 1456831279717572075, 3354332676055975953, 7726969805373657763, 17807843238326490566, 41058742911104314735, 94707794486141835055, 218547278951873016885, 504520534310020995281, 1165146383227358880327, 2691813944679198933254, 6221103655060471930962, 14382781044804077557878, 33263407167250290261739, 76954607918587159600611, 178091158645065231338585, 412273954575974881103440, 954687857089931041054750, 2211389126498755280117222, 5123817785575377937814225, 11875270986534273727976053, 27530333413891900687626772, 63840220572849222278104788, 148077560084497124425798169, 343552566984827422999062338, 797266374037095566105098803, 1850620324705013667862682500, 4296672645787599932254592314, 9978052601349627898050764057, 23176908557720826553542054224, 53846703494421872456717997242, 125127973383987991144787652593, 290830021503801381930721105937, 676100921983193825969329450941, 1572060634527419283928666538120, 3656037156311472113594424613186, 8504202409975272430583173001536, 19785018164064980155520108020417, 46038106439011476416261131849967, 107145737836783124653205894881028, 249406152369906834599933074660180, 580647632830318254723094356140953, 1352040998078175397319979202051475] ------------------------------------------------------------ Theorem Number, 61 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 7+ HHH + H HH + HH HH 6+ H H + H H + HH HH 5+ HH HH + H H + HH HH 4+ HH H H + HH HH HH HH 3+ HH HHH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 6 5 3 2 3 1 + (x - 3 x - x + 2 x + x - 1) P(x) 6 5 3 2 2 3 2 + (3 x + x - 4 x - 2 x + 3) P(x) + (2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 24, 50, 100, 207, 447, 965, 2097, 4641, 10355, 23208, 52434, 119238, 272302, 624715, 1439586, 3329035, 7723353, 17974332, 41946862, 98136543, 230136785, 540859834, 1273645217, 3004841530, 7101517437, 16810654036, 39854525348, 94621732620, 224950428542, 535465514846, 1276128784737, 3044720948720, 7272175007841, 17386856915967, 41609801199087, 99670684111928, 238955727084254, 573360774386420, 1376837089630860, 3308764126191171, 7957256958283794, 19149694328506228, 46115679249370442, 111125023198040762, 267941826636175204, 646433889341639098, 1560460708781195478, 3768924211315976522, 9107707171651752850, 22020112055371883983, 53264882178765929245, 128903877018767159586, 312095509354848105514, 755960176581781422429, 1831866563301055461943, 4440851470520127715854, 10769869026085811825291, 26128886252691178913799, 63415069809483920442764, 153964356091782976724423, 373937747991975936017966, 908500784048788194813896, 2207973525844179375897501, 5367855788092394722780239, 13053960160887471570162186, 31755167086049496587497855, 77270472173378202155837143, 188077261542308295134223872, 457909065550704597273291014, 1115165194681535923031110438, 2716521249676194665104160483, 6619085378281378272215833397, 16132103128989793305097961362, 39326870592610533254429937069, 95893803896808211685222661778, 233879375049458391590317811366, 570546544663074707446614357597, 1392148793303975476592593878598, 3397608858830076013984401770568, 8293771449881197142382854212644, 20249746445904137513663328249672, 49450867782510794412659596235458, 120785019511666197733598404565551, 295076860334680412091808688807665, 721005064284596647207054443970788, 1762060340462307761691050403140135, 4307058934161988158319017861316542, 10529718506027909698423650877921477, 25747021917792169515323378704085982, 62966561145792275331841952840077732, 154015405894044114004769528686025159] ------------------------------------------------------------ Theorem Number, 62 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 1 + (x + 2 x + x + x - 2 x - x - x + 1) P(x) 4 3 2 + (2 x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 5, 9, 15, 27, 52, 99, 191, 380, 767, 1561, 3212, 6676, 13980, 29462, 62457, 133095, 284904, 612325, 1320835, 2858528, 6204794, 13504896, 29467046, 64443065, 141232079, 310127824, 682241562, 1503390100, 3318125585, 7334286838, 16234066006, 35980324422, 79843282903, 177385141221, 394524626312, 878382600262, 1957594535824, 4366863711975, 9749996633366, 21787566703548, 48726502251068, 109058113252455, 244271064473613, 547511721413636, 1228030388231890, 2756175501803298, 6189779845643445, 13909237626615638, 31273831159967158, 70355639977282178, 158360839466453771, 356631087517226215, 803535502075715446, 1811330859313674816, 4084981786161499396, 9216695405000005915, 20803989056321610682, 46978386063151976380, 106126487292393541621, 239838068767767185118, 542220278033115896654, 1226285592006812766558, 2774347696927967271630, 6278832855959591570828, 14214818277513329553301, 32191694467833655881358, 72926011196323997616532, 165254507682930495490441, 374586998426594973124882, 849331174613651352346800, 1926295937143221477067260, 4370058634352064414375042, 9916689051600915661994716, 22509108319009393216650851, 51104501330215476319608970, 116055707226039146541380078, 263619574451311192836750119, 598949259458766324199929106, 1361134842968359225485601500, 3093916712876970178692847268, 7034125719967269865497972628, 15995707321136515987498854278, 36381989171910295666778485227, 82766964402142036623410315006, 188327268860574074593683978712, 428600872006951760458663393481, 975606498851851793659981066972, 2221141899698881971264987257058, 5057736078329066998909302286110, 11518941426334046103018614433700, 26238795102693120770431186038224, 59778984596015548358887856377059, 136215011845586736829636505040690] ------------------------------------------------------------ Theorem Number, 63 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 13 12 11 10 9 8 7 6 4 3 2 1 + (x + x + 2 x + x + x - 3 x - x - 4 x + x + 2 x + x - 1) 3 9 8 7 6 5 4 3 2 2 P(x) + (x + 3 x + x + 4 x - x - 2 x - 4 x - 2 x + 3) P(x) 5 4 3 2 + (x + x + 2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 6, 10, 20, 38, 71, 141, 287, 583, 1202, 2521, 5325, 11323, 24282, 52406, 113664, 247757, 542480, 1192313, 2629716, 5818748, 12912436, 28729813, 64079490, 143246101, 320883803, 720195496, 1619321024, 3647055270, 8226810087, 18584803194, 42041964361, 95229445722, 215968599608, 490356105984, 1114569328616, 2536017260869, 5775957109341, 13167432219753, 30044425752651, 68611060035416, 156810360349206, 358665774457819, 820965478557937, 1880467407759439, 4310223033012889, 9885862452845120, 22688139458714478, 52100512493277179, 119710934944822697, 275210416296780580, 633032733279400641, 1456831279717572075, 3354332676055975953, 7726969805373657763, 17807843238326490566, 41058742911104314735, 94707794486141835055, 218547278951873016885, 504520534310020995281, 1165146383227358880327, 2691813944679198933254, 6221103655060471930962, 14382781044804077557878, 33263407167250290261739, 76954607918587159600611, 178091158645065231338585, 412273954575974881103440, 954687857089931041054750, 2211389126498755280117222, 5123817785575377937814225, 11875270986534273727976053, 27530333413891900687626772, 63840220572849222278104788, 148077560084497124425798169, 343552566984827422999062338, 797266374037095566105098803, 1850620324705013667862682500, 4296672645787599932254592314, 9978052601349627898050764057, 23176908557720826553542054224, 53846703494421872456717997242, 125127973383987991144787652593, 290830021503801381930721105937, 676100921983193825969329450941, 1572060634527419283928666538120, 3656037156311472113594424613186, 8504202409975272430583173001536, 19785018164064980155520108020417, 46038106439011476416261131849967, 107145737836783124653205894881028, 249406152369906834599933074660180, 580647632830318254723094356140953, 1352040998078175397319979202051475] ------------------------------------------------------------ Theorem Number, 64 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 4 3 2 2 1 + (x + 2 x + 2 x + x - 2 x - x - x + 1) P(x) 5 4 3 2 + (x + 2 x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 4, 7, 12, 19, 32, 57, 102, 184, 338, 631, 1193, 2278, 4386, 8512, 16640, 32727, 64698, 128497, 256288, 513091, 1030652, 2076551, 4195389, 8497633, 17251501, 35097685, 71545640, 146109780, 298889934, 612391857, 1256577012, 2581964630, 5312226040, 10942958613, 22568085609, 46593874760, 96296775298, 199214136675, 412508168198, 854927060291, 1773336983575, 3681307349391, 7647959447370, 15900359496772, 33080520927256, 68869800315583, 143471049030427, 299066191297844, 623774018719246, 1301769563196978, 2718180873152235, 5678728121019830, 11869812600023930, 24822684507644365, 51934768256973512, 108708894175429681, 227647361551092841, 476918760095371805, 999549561565926862, 2095736135278881599, 4395777411189686495, 9223512701958825628, 19360365984653297116, 40652059395072526234, 85388375869205205243, 179414611502079962724, 377099148730245402023, 792844459482558248785, 1667444254566230073271, 3507856998483416898418, 7381696118116215545446, 15537845884622259716287, 32714673959131873020032, 68898287436832289381615, 145139401140260625493636, 305823205702351606859661, 644556519941560063235675, 1358796405023640572966082, 2865154456516923094879448, 6042817833843456908071545, 12747540906403185592012971, 26897164309201694344507690, 56764595715167086292090332, 119822245028973861534159847, 252978817559414921672077464, 534214521684417321803286560, 1128314411252867525432280150, 2383557588085094296518911183, 5036170748773190613622743871, 10642723994868396804864305247, 22494743116136658770993174434, 47553615795218186780095160580, 100544600599431729835469608976] ------------------------------------------------------------ Theorem Number, 65 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H H HH H + H H H H + H HH H 2+ H H H H + HH H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 2 2 1 + (x + 2 x) P(x) + (-2 x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (7 n - 5) a(n - 1) (2 n - 1) a(n - 2) a(n) = 1/2 ------------------ + ------------------ n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 1] Just for fun, a(1000), equals 9092286347408461381190066458644902332454977875247293311387713927335022651708\ 790591454794080031581991630628884894078589863816755856423465455809978902\ 215379119222272899206264366066173821986022926458407504948455831289366140\ 238545615125470513702613444381026177032665897911379872276676494537250861\ 786577236406823049366761456188177013994590407524775514493010776149916793\ 192693983980293384686624135032669547056492248204325339662817306751885575\ 741928461674191908125700977179251990434492028091611489047246596478022834\ 937256638120591432185377578709965401290745742725035892110974195554730651\ 90163315088837952 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 2, 6, 18, 57, 186, 622, 2120, 7338, 25724, 91144, 325878, 1174281, 4260282, 15548694, 57048048, 210295326, 778483932, 2892818244, 10786724388, 40347919626, 151355847012, 569274150156, 2146336125648, 8110508473252, 30711521221376, 116518215264492, 442862000693438, 1686062250699433, 6429286894263738, 24552388991392230, 93891870710425440, 359526085719652662, 1378379704593824300, 5290709340633314596, 20330047491994213884, 78201907647506243758, 301111732041234778316, 1160507655117628665252, 4476705468260134734384, 17283862221822154612428, 66784808491631598524136, 258257655550682547281952, 999430099263282762985884, 3870449306134945313530218, 14999162225528050786449636, 58164368008320657345771132, 225693938585648240354418720, 876283859585256206634677868, 3404250905464530089298069144, 13232458020154047930422174184, 51462706717814193160065835848, 200248505655069639143771836788, 779583055476874615924354019592, 3036436936183999440812652222360, 11832257960295426231716284345152, 46128101164413438900891017233464, 179908446981581773414876213066824, 701971258807269134635948282986036, 2740078349888793257925566282115150, 10699829696578416783892946132079081, 41798010431096781502021219407150522, 163340579722359938988493911653198214, 638538996714279864941997991295333568, 2497071293923016435025958857276905766, 9768326864016750514166296878390022092, 38225298567471767606009671892655543204, 149630101971372103975387060781201540748, 585896003224455083257277558204514001926, 2294836960710825238080831317895275857212, 8991026714499287312424083625179682584244, 35236231325289045264392077003584793150128, 138130323492907137936572107257612170438556, 541632665468735836946094307375174980176232, 2124388132447284037191145950509844740424144, 8334377297181300269010107664288850226412948, 32705433157175579095340022095179239506461326, 128372324009044660179471911967155763878965452, 503993626969145805876080482518755846394324244, 1979148211509587338293580407493688668552665248, 7773718402053594748220780055210983483921784516, 30540361365363299979833635989728831829531710792, 120008544837015909024404413649719867295569784824, 471672579042628963474316122196878295713766809848, 1854203747256947885520189852987014581357492252236, 7290555543704239455774477553887935781005049071352, 28671352521029337457361079966129122840838243094824, 112776402891485729533296180406356262122854564465888, 443679098954434526198051535065092801119803262742456, 1745816567314785079609901018849128354327114208640672, 6870770627588552163759212972464607438843408459597464, 27045052953875931793851675304182520726563160223920268, 106474146282100536687532050659647101139858833632154666, 419249825847663835582536909897646247848685962301374692, 1651096689279145274990666815740800205930410367538659804, 6503416333425215081013105515458116277891272291680547648, 25619970868088071350587297546292370028064729666538031676, 100944429963070639155616121424698241963988476467501507832, 397787758563530428765776974324701195228424522535622200744] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 66 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 3+ H H + H H H H + HH HH HH HH 2.5+ H H H H + H H H H + H H H H + H H H H 2+ H H H H H + H H HH H H + H H H H H H 1.5+ HH HH HH HH HH HH + H H H H H H + H H H H H H 1+ H H H H H H + H H H H H H + H H H H H H + H H H H H H 0.5+ H H H H H H + H H H H H H +H HH HH H -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 2 2 1 + (x + x) P(x) + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence 2 (n - 1) a(n - 1) 3 (n - 1) a(n - 2) a(n) = ------------------ + ------------------ n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 1] Just for fun, a(1000), equals 1530036790419022830094068760617849229908556040379774601663141696520714432010\ 505204701121726795033893282006994307845948795123386656757478216103147793\ 598816422588288234800268711441603480859153836164038767267854510780820952\ 045840116183968093556741601444043981217670681866042373596290010701135538\ 838865417082061461269201435105172255513381628729705648108431316186711898\ 379048365060433595579837015920602409701854829405120869376654505085317762\ 1901271877999637563068542637350933457 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 3, 6, 15, 36, 91, 232, 603, 1585, 4213, 11298, 30537, 83097, 227475, 625992, 1730787, 4805595, 13393689, 37458330, 105089229, 295673994, 834086421, 2358641376, 6684761125, 18985057351, 54022715451, 154000562758, 439742222071, 1257643249140, 3602118427251, 10331450919456, 29671013856627, 85317692667643, 245613376802185, 707854577312178, 2042162142208813, 5897493615536452, 17047255430494497, 49320944483427000, 142816973618414817, 413887836110423787, 1200394300050487935, 3484084625456932134, 10119592485062548155, 29412628894558563849, 85543870540455597789, 248952602654003411640, 724947137834104063053, 2112261618875209962525, 6157879192714893166503, 17961708307164474879078, 52418979494565497284659, 153054402342387832806318, 447107296039753836151995, 1306709599137475613111808, 3820682066516442811470123, 11176109832763802046866481, 32705601410484246215745189, 95747934502509579263312730, 280418620117853741708024169, 821578511126260089293299449, 2407969409295125052827266131, 7060047856454817331912175136, 20706869494801128932088054675, 60752886013114747805209321971, 178303876727715987264460117881, 523470228309211183146132538770, 1537292873088850037153655419037, 4515968752463518800863883219540, 13270012942708831885430137279857, 39004474517326916925520791131352, 114677148186439520720469807592881, 337252779926837166106515094143507, 992081497804863208806272348440575, 2919102839611001046292805620867782, 8591299535354652688143556684853307, 25291409899803750802286391976502211, 74471367333926485356188553908612137, 219333623772940705482181740240713080, 646127582088681087104424825527569497, 1903821367984370378973123565306390657, 5610824882652789101159297569379125339, 16539320523461975633674292210615157006, 48763733724885023891037362712600616695, 143801214724243338894845383828477461126, 424143211957453170823189308473526283071, 1251252511019022217034672218022874172864, 3691969061033252211450145056818715608239, 10895571836533928224569430533625487349525, 32160232558185513877392752232595140415729, 94943198059462752589589672745512131329394, 280338312871514003720592178984834743192165, 827891507006386759684746014201909924531418, 2445317582470051292788355811433410180110685, 7223813569919277908209909876381273600472448, 21343507566294190307011455122677671120241053, 63071286725499290051879587077009179781061461, 186407292265725088628262974383000850686750119] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 67 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ HH HH + HH HH + H H + HH HH 4+ HH H H + H HH H H + HH HH HH HH 3+ H H H H + HH HH HH HH + HH H HH HH 2+ HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 3 2 2 1 + (x - 2 x ) P(x) + (2 x + 3 x) P(x) + (-3 x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (49 n - 47) (4 n - 19) a(n - 1) a(n) = -1/30 ------------------------------- n + 1 3 2 (245 n - 1539 n + 2668 n - 1344) a(n - 2) + 1/15 ------------------------------------------- (n + 1) n 3 2 (784 n - 7077 n + 20591 n - 20088) a(n - 3) + 1/30 --------------------------------------------- (n + 1) n 3 2 (833 n - 8208 n + 25975 n - 25890) a(n - 4) - 1/15 --------------------------------------------- (n + 1) n 2 (n - 4) (5096 n - 36799 n + 57573) a(n - 5) + 1/30 -------------------------------------------- (n + 1) n 23 (49 n - 114) (n - 4) (n - 5) a(n - 6) - -- ------------------------------------- 15 (n + 1) n subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 3, a(5) = 7, a(6) = 18] Just for fun, a(1000), equals 8285753824575282171237736810876391700652864188682665250700311816643909088081\ 740167319151314749343292526671229912995080190975960695105004845384117359\ 550223217834239575419266549192928461360692000053432638309996050919186095\ 907052022209642829343933619226021629018440257816968234872985459965942995\ 902211404162554818217928608387947212444875612898283374173056638264779688\ 558540496134691108473988828496805652904561206773771195742292987269898405\ 768813354175322802310734911441016369620881305330270963221714925690427024\ 31672286840180788726922817377 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 3, 7, 18, 52, 155, 463, 1395, 4262, 13180, 41125, 129255, 408894, 1301135, 4161931, 13374303, 43156096, 139776848, 454256933, 1480848446, 4841145512, 15867701546, 52133878697, 171667424383, 566432002597, 1872566404785, 6201547675370, 20572436572000, 68351546658109, 227429256825375, 757777682321235, 2528134584062483, 8444808323816588, 28241096657374228, 94547210450136505, 316859413682657578, 1062949033845573918, 3569152374102987282, 11995130282384546723, 40347244007088692849, 135823479793795385863, 457585329831936192647, 1542733194189268925160, 5204958005513667511025, 17572694709496417495461, 59366586865641058742194, 200685950728220645425265, 678816357319717323791505, 2297407099038975283078237, 7779728097598900819568075, 26358620755902129775681950, 89351993326136419432517550, 303040774664725700342200534, 1028266131733988247934831715, 3490683062361256894764333668, 11855199860920676386148000966, 40280459060860905722891766161, 136918089824987549705086383863, 465588516672932180756227372547, 1583846537426598620500186460386, 5389990695167719718442210729452, 18349402299875637900069393860543, 62489919823137506828230790400195, 212886203188640436511561018301475, 725487845904165320012186316558188, 2473167934513409887379213139577172, 8433616774470195331910961305808241, 28767825025385974308053954552503390, 98158833354892024463301671951105402, 335025125255997309876779809550820710, 1143793387808407014013535669093152535, 3906040285527171586266398906189803285, 13342635998795059588815839368274340279, 45588921216192755951650329289301081335, 155806955325311934434109596609356226676, 532624839844004651866828284710041081393, 1821211357623494697508602049368632540739, 6228752135735518975258955765244344704338, 21307918012139839483720484206369257471691, 72908447064947418645153917682330154817475, 249522218013902109146777586967035845275043, 854147556195567692746084595250893037359701, 2924467036377210577059714903815120963324398, 10014946228395443582363282499939992196606846, 34303352349295313135250224774177655807340518, 117519142362780866628671736242098360158814745, 402682624215279576645406080446967297301044788, 1380058758848888117622239121195687802525836778, 4730542176440464452679333673679344912645040499, 16218146280843529518476588769126425094301110529, 55611782436172524201000593348812398940585695579, 190724344424514906348575694415488052247372949567, 654210660751968900715411465414395762619254649274, 2244397509032304014376895973090623005302536121570, 7701071658905632652978679705499375079366268192481, 26428369560464143631836075301292862589934528565073, 90710181259990414936315690306452532930705261786641, 311391557080827122419031902870170886925786584873523] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 68 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 2 1 + P(x) x + P(x) x + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence 2 (518 n - 1031 n + 780) a(n - 1) a(n) = -1/195 -------------------------------- n + 2 3 2 (1036 n - 4335 n + 4367 n - 1170) a(n - 2) - 1/195 -------------------------------------------- (n + 2) (n + 1) 3 2 (2590 n - 11115 n + 11939 n - 1812) a(n - 3) + 1/195 ---------------------------------------------- (n + 2) (n + 1) 2 (n - 3) (1036 n - 1865 n + 1074) a(n - 4) + 1/65 ------------------------------------------ (n + 2) (n + 1) 161 (74 n - 55) (n - 3) (n - 4) a(n - 5) + --- ------------------------------------ 195 (n + 2) (n + 1) subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 1456286415952438964652364581682787386720721652723427223868652268704217615734\ 013661511014559584083968554107289545538396109682528052843351355027899510\ 336875839145124807012715538480932028100745003225030603205050905923754789\ 120913436417646081847745149534148691270052401108733311860953962982834165\ 142352139116820596104418788220221329115699403613078628786955744338071295\ 785210326056171164845618097 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 4, 8, 13, 31, 71, 144, 318, 729, 1611, 3604, 8249, 18803, 42907, 98858, 228474, 528735, 1228800, 2865180, 6693712, 15676941, 36807239, 86584783, 204060509, 481823778, 1139565120, 2699329341, 6403500057, 15211830451, 36183117255, 86171536894, 205459894230, 490417795075, 1171809537516, 2802705573178, 6709741472212, 16077633792535, 38557470978465, 92543928713413, 222292252187223, 534346494991678, 1285375448785399, 3094086155360823, 7452774657572624, 17962853486458385, 43320517724198861, 104535253974750259, 252390609561764423, 609699521485999914, 1473608895007574938, 3563397472304118410, 8620933920694737411, 20866285270840567272, 50527739882573522706, 122405798503806040965, 296657476936886314309, 719256672330233630085, 1744545018773625965860, 4232966162296687137816, 10274640808649742568633, 24948409910226791411907, 60599469176019760272535, 147244830551438483529790, 357892285312932720768838, 870164978202711215226483, 2116332778149031726542808, 5148677470688468794716894, 12529484860432553111063488, 30499528076281485551927755, 74262935424150986850220725, 180870207357836188170341069, 440630810522439231975963839, 1073724026979876073732588658, 2617084008072054452326466527, 6380388743696091681799439061, 15558886191823487315141504508, 37949779254079097680017924283, 92584221613943667202286169191, 225922433438019608173638548805, 551409374869368431174631180753, 1346105758407373798214652723522, 3286793196606678831626181940722, 8026971507652658919500628768874, 19607185640982275440523729181033, 47902823795394726062462438796736, 117054324948067108888357885549374, 286083328086281443692788683942327, 699317934466646636669713105276839, 1709748244857899463255482766619759, 4180839106559551650627420507840427, 10225086357877339275440446600065760, 25011587559313816213376886296261184, 61190615947546730766871192612064833, 149725673741988363612845353038008205, 366415846156516055604065564079552535, 896845046441850691487935128438196159] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 69 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 3+ H H + H H H H + HH HH HH HH 2.5+ H H H H + H H H H + H H H H + H H H H 2+ H H H H H H + HH H H HH H H + H H H H H H H H 1.5+ HH HH HH HH HH HH HH HH + H H H H H H H H + H H H H H H H H 1+ H H H H H H + H H H H + H H H H + H H H H 0.5+ H H H H + H H H H +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 2 2 1 + (x + x) P(x) + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence 2 (n - 1) a(n - 1) 3 (n - 1) a(n - 2) a(n) = ------------------ + ------------------ n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 1] Just for fun, a(1000), equals 1530036790419022830094068760617849229908556040379774601663141696520714432010\ 505204701121726795033893282006994307845948795123386656757478216103147793\ 598816422588288234800268711441603480859153836164038767267854510780820952\ 045840116183968093556741601444043981217670681866042373596290010701135538\ 838865417082061461269201435105172255513381628729705648108431316186711898\ 379048365060433595579837015920602409701854829405120869376654505085317762\ 1901271877999637563068542637350933457 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 3, 6, 15, 36, 91, 232, 603, 1585, 4213, 11298, 30537, 83097, 227475, 625992, 1730787, 4805595, 13393689, 37458330, 105089229, 295673994, 834086421, 2358641376, 6684761125, 18985057351, 54022715451, 154000562758, 439742222071, 1257643249140, 3602118427251, 10331450919456, 29671013856627, 85317692667643, 245613376802185, 707854577312178, 2042162142208813, 5897493615536452, 17047255430494497, 49320944483427000, 142816973618414817, 413887836110423787, 1200394300050487935, 3484084625456932134, 10119592485062548155, 29412628894558563849, 85543870540455597789, 248952602654003411640, 724947137834104063053, 2112261618875209962525, 6157879192714893166503, 17961708307164474879078, 52418979494565497284659, 153054402342387832806318, 447107296039753836151995, 1306709599137475613111808, 3820682066516442811470123, 11176109832763802046866481, 32705601410484246215745189, 95747934502509579263312730, 280418620117853741708024169, 821578511126260089293299449, 2407969409295125052827266131, 7060047856454817331912175136, 20706869494801128932088054675, 60752886013114747805209321971, 178303876727715987264460117881, 523470228309211183146132538770, 1537292873088850037153655419037, 4515968752463518800863883219540, 13270012942708831885430137279857, 39004474517326916925520791131352, 114677148186439520720469807592881, 337252779926837166106515094143507, 992081497804863208806272348440575, 2919102839611001046292805620867782, 8591299535354652688143556684853307, 25291409899803750802286391976502211, 74471367333926485356188553908612137, 219333623772940705482181740240713080, 646127582088681087104424825527569497, 1903821367984370378973123565306390657, 5610824882652789101159297569379125339, 16539320523461975633674292210615157006, 48763733724885023891037362712600616695, 143801214724243338894845383828477461126, 424143211957453170823189308473526283071, 1251252511019022217034672218022874172864, 3691969061033252211450145056818715608239, 10895571836533928224569430533625487349525, 32160232558185513877392752232595140415729, 94943198059462752589589672745512131329394, 280338312871514003720592178984834743192165, 827891507006386759684746014201909924531418, 2445317582470051292788355811433410180110685, 7223813569919277908209909876381273600472448, 21343507566294190307011455122677671120241053, 63071286725499290051879587077009179781061461, 186407292265725088628262974383000850686750119] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 70 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 2+ H H H H + HH HH HH HH + HH HH HH HH + H H H H H H H H + H H H H H H H H 1.5+ H HH H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H 1+ H H H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H 0.5+ HH H H H H H H HH + H H H H H H H H + H H H H H H H H +H HH HH HH H +H HH HH HH H -*-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-*+-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 2 2 1 + P(x) x + (x - x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 1) a(n - 1) (n - 2) a(n - 2) (2 n - 7) a(n - 3) a(n) = ------------------ + ---------------- + ------------------ n + 1 n + 1 n + 1 (n - 5) a(n - 4) - ---------------- n + 1 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2] Just for fun, a(1000), equals 1258976834224262773199644936372839877301103060006870706162324567937209163404\ 664302285780840930405187497908781943435725250693212008816722364247013884\ 666083868760309087497764707542210251550367470907064283210333730955232698\ 389440175341142704919913139331694533974031860259334845579975521445364654\ 700207290194856330574815120730458306294903984090244990972375467298717489\ 01445379751297218627473874798563437753301835209165 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 2, 4, 8, 17, 37, 82, 185, 423, 978, 2283, 5373, 12735, 30372, 72832, 175502, 424748, 1032004, 2516347, 6155441, 15101701, 37150472, 91618049, 226460893, 560954047, 1392251012, 3461824644, 8622571758, 21511212261, 53745962199, 134474581374, 336908488839, 845139060165, 2122553644686, 5336735929371, 13432403613621, 33843022209066, 85349327734485, 215440028338359, 544288586926914, 1376230297675914, 3482537223611046, 8819175375714063, 22349794473772659, 56678600914995057, 143830921235537742, 365225623668676437, 927972354829010775, 2359192024476568203, 6001174121892988758, 15273713134056377698, 38893747432145085266, 99090832134641995427, 252579381177903040849, 644118340220292169786, 1643348924746923013481, 4194532932723720267271, 10710773165730370402070, 27361217667381195152609, 69923263927774760117419, 178761583832906815958299, 457180542019634361749654, 1169653910683020997823700, 2993493968182857335738916, 7663836950023084292126586, 19627124209913879819201256, 50281185027971273570344779, 128851301008215990676245297, 330295607482296149639113771, 846922848867278127081934118, 2172243398314031502060434813, 5573055540747246795936497203, 14301951559375317288722742625, 36712267090479571354186761752, 94262318866766131085885820862, 242087967735412291153757221292, 621890217530867044998372625244, 1597927417599990976164331285618, 4106772441264045401019924649921, 10557037252659735639822884541089, 27144318295978988020876731613899, 69808615378820015816460193046634, 179568484819409906464233459965565, 461998012612770916903282585499931, 1188877068859680412470053314034196, 3059980617900905437254279674385261, 7877408686568953921404087246339411, 20282861001228149602530202549462410, 52234134723235412099021791134645474, 134541797507827311283829108795938674, 346605946314513254492433135097630809, 893077485129793636878895057273178901, 2301521609537728551186835553085928773, 5932151623905973421624468114595077244, 15292526112023196667544094397358322057, 39428894200253097844359818441341768857, 101675651191856047918093722573856681533, 262231577763896699071580185616362344474] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 71 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 3 2 2 2 1 + (x + x + x + x) P(x) + (-x - x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (n - 2) a(n - 1) 2 (n - 2) a(n - 2) (4 n - 11) a(n - 3) a(n) = ---------------- + ------------------ + ------------------- n + 1 n + 1 n + 1 (8 n - 25) a(n - 4) 6 (n - 4) a(n - 5) (5 n - 22) a(n - 6) + ------------------- + ------------------ + ------------------- n + 1 n + 1 n + 1 3 (n - 5) a(n - 7) + ------------------ n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 2, a(5) = 3, a(6) = 7, a(7) = 17] Just for fun, a(1000), equals 5032496365637955067683347870950409710915701764522282276774675157243603802582\ 866298211231210653006394070833138976334822500759759891785781976829184637\ 643038378788330330417356109675556672422365107611262498459448932887962130\ 643216278920681957917488069930040634733745054315448220108085618003032954\ 450492248812276239145688825121717358769928635223051275485462691699729085\ 004111315387526723342034239842070602836639078569636160428330200510115437\ 8 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 2, 3, 7, 17, 39, 91, 219, 533, 1307, 3234, 8067, 20255, 51150, 129839, 331109, 847876, 2179289, 5620427, 14540036, 37721539, 98116305, 255818920, 668475637, 1750371384, 4592023450, 12068440795, 31770262976, 83765896945, 221181546661, 584827643203, 1548351907603, 4104332864117, 10892248356011, 28937953375916, 76960653386603, 204879046734386, 545925151025906, 1455981438846053, 3886400469418666, 10382245378672359, 27756918405827983, 74263029726249001, 198829910977439625, 532703401910546531, 1428143266813608445, 3831150646223428804, 10283645508579576861, 27619465812286909488, 74220560789749220334, 199556083749746908195, 536820382086347568072, 1444801763827645841740, 3890408061198204485087, 10480511016119495728047, 28246414132024260752912, 76160755522727900890180, 205437651850071834885351, 554375257812010095795798, 1496570004169021588945828, 4041609047370900210653163, 10918690306434774952578648, 29508078135961103546885935, 79773895103635321195662633, 215737548410098076340647126, 583621812616151827778503173, 1579334324213728450307002233, 4275132085798309736678255810, 11575884594778904645167342803, 31353387126118713173448037627, 84944829539009789690078850478, 230201612728140081149731098573, 624015753012404624623636602384, 1691981062322825431740014538160, 4588865052096950827456946241043, 12448646109079949902773787405256, 33778729419959874702068588413132, 91678238541123561975081610471607, 248879128104875104185719943760541, 675783451463072308953369784249936, 1835359604551371265619822938995004, 4985710857337519919668459945842445, 13546376824511567847839151376237576, 36813514224468113012736613431442804, 100063905775460344278497620851740509, 272039349875194325299625521607283254, 739721482329877908140333154428499903, 2011801570464323989823815200231900135, 5472436183516459408824867590415304632, 14888577978323084494016869894352271445, 40513618989439805975549944975329369435, 110261163534686841700263792854738579832, 300134742948778688458516646666561272035, 817110224328981701161007905911583958297, 2224919190647836747971327271445212007300, 6059205160189587412826547467523630270925, 16503783917191051549681811499901147080856, 44958988651672462155755561208723586036494] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 72 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 1 + (x + x) P(x) + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 1) a(n - 1) (2 n - 1) a(n - 2) 3 (2 n - 5) a(n - 3) a(n) = ------------------ - ------------------ + -------------------- n + 1 n + 1 n + 1 (n - 2) a(n - 4) 2 (2 n - 7) a(n - 5) - ---------------- + -------------------- n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 3134627918704846396589045582068293464668313386749891925759492579301366817419\ 266879466277019895299966556024437351904982229786354769359885950645415854\ 487762568754124509622513897418551096745213564746196714896494879980931898\ 708502918937008296833152741264657083164783324054454272242916859440693717\ 64365001116217342900388877265443604205502181219138765202835544779530 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 42, 84, 170, 354, 740, 1549, 3269, 6945, 14811, 31711, 68177, 147091, 318313, 690837, 1503351, 3279445, 7169907, 15708485, 34482475, 75830981, 167042763, 368548926, 814341362, 1801867812, 3992172298, 8855912464, 19668236110, 43729885814, 97330067640, 216844435346, 483571453768, 1079354856828, 2411240775366, 5391038526802, 12062679137620, 27010935099196, 60526554883902, 135721815227704, 304535849636366, 683755276072670, 1536121565708096, 3453050076522008, 7766456476976834, 17477390474177854, 39350957376284872, 88644320911742304, 199781711595784890, 450466438264400446, 1016164744759318568, 2293264805607373804, 5177575503207130558, 11694317639185843274, 26423715558659122260, 59728010995453379421, 135058602016778295285, 305507289981338348257, 691307640696528081979, 1564829833276593856647, 3543272754560789200681, 8025639916478510155339, 18183959683820504446497, 41212333247384488368873, 93431281455612601484243, 211875294974960472423745, 480604617462452614426815, 1090465639820281676643073, 2474853003558648387670983, 5618201029263006344110809, 12757123617698163352573711, 28974313940315478259062979, 65822724585319538752564981, 149567876677718197286698671, 339936780414026450535924453, 772774892529317395366017603, 1757116730487566683312107729, 3996122690354078859729209647, 9090033908542877936842438557, 20681339112467629598326301035, 47062642985297219558085765929, 107116551464339266855499450015, 243847122183568901454753070765, 555210537090801909689698475727, 1264372572489361483075032316117, 2879838126594606020044275873923, 6560472630207720552639437125041, 14947710059210582396858604974621, 34063179534729044501719950750975, 77636370512116064921945474111629, 176975591352231844422902101233579, 403485801251068480867972873270661, 920043891167408188870554973636539] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 73 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH + HH H + H H + HH HH 4+ HH H H + H HH H H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH H + H H HHH + HH HH H HH + H H HH H 1+ HH H HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 3 2 2 1 + (x - 2 x ) P(x) + (2 x + 3 x) P(x) + (-3 x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (49 n - 47) (4 n - 19) a(n - 1) a(n) = -1/30 ------------------------------- n + 1 3 2 (245 n - 1539 n + 2668 n - 1344) a(n - 2) + 1/15 ------------------------------------------- (n + 1) n 3 2 (784 n - 7077 n + 20591 n - 20088) a(n - 3) + 1/30 --------------------------------------------- (n + 1) n 3 2 (833 n - 8208 n + 25975 n - 25890) a(n - 4) - 1/15 --------------------------------------------- (n + 1) n 2 (n - 4) (5096 n - 36799 n + 57573) a(n - 5) + 1/30 -------------------------------------------- (n + 1) n 23 (49 n - 114) (n - 4) (n - 5) a(n - 6) - -- ------------------------------------- 15 (n + 1) n subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 3, a(5) = 7, a(6) = 18] Just for fun, a(1000), equals 8285753824575282171237736810876391700652864188682665250700311816643909088081\ 740167319151314749343292526671229912995080190975960695105004845384117359\ 550223217834239575419266549192928461360692000053432638309996050919186095\ 907052022209642829343933619226021629018440257816968234872985459965942995\ 902211404162554818217928608387947212444875612898283374173056638264779688\ 558540496134691108473988828496805652904561206773771195742292987269898405\ 768813354175322802310734911441016369620881305330270963221714925690427024\ 31672286840180788726922817377 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 3, 7, 18, 52, 155, 463, 1395, 4262, 13180, 41125, 129255, 408894, 1301135, 4161931, 13374303, 43156096, 139776848, 454256933, 1480848446, 4841145512, 15867701546, 52133878697, 171667424383, 566432002597, 1872566404785, 6201547675370, 20572436572000, 68351546658109, 227429256825375, 757777682321235, 2528134584062483, 8444808323816588, 28241096657374228, 94547210450136505, 316859413682657578, 1062949033845573918, 3569152374102987282, 11995130282384546723, 40347244007088692849, 135823479793795385863, 457585329831936192647, 1542733194189268925160, 5204958005513667511025, 17572694709496417495461, 59366586865641058742194, 200685950728220645425265, 678816357319717323791505, 2297407099038975283078237, 7779728097598900819568075, 26358620755902129775681950, 89351993326136419432517550, 303040774664725700342200534, 1028266131733988247934831715, 3490683062361256894764333668, 11855199860920676386148000966, 40280459060860905722891766161, 136918089824987549705086383863, 465588516672932180756227372547, 1583846537426598620500186460386, 5389990695167719718442210729452, 18349402299875637900069393860543, 62489919823137506828230790400195, 212886203188640436511561018301475, 725487845904165320012186316558188, 2473167934513409887379213139577172, 8433616774470195331910961305808241, 28767825025385974308053954552503390, 98158833354892024463301671951105402, 335025125255997309876779809550820710, 1143793387808407014013535669093152535, 3906040285527171586266398906189803285, 13342635998795059588815839368274340279, 45588921216192755951650329289301081335, 155806955325311934434109596609356226676, 532624839844004651866828284710041081393, 1821211357623494697508602049368632540739, 6228752135735518975258955765244344704338, 21307918012139839483720484206369257471691, 72908447064947418645153917682330154817475, 249522218013902109146777586967035845275043, 854147556195567692746084595250893037359701, 2924467036377210577059714903815120963324398, 10014946228395443582363282499939992196606846, 34303352349295313135250224774177655807340518, 117519142362780866628671736242098360158814745, 402682624215279576645406080446967297301044788, 1380058758848888117622239121195687802525836778, 4730542176440464452679333673679344912645040499, 16218146280843529518476588769126425094301110529, 55611782436172524201000593348812398940585695579, 190724344424514906348575694415488052247372949567, 654210660751968900715411465414395762619254649274, 2244397509032304014376895973090623005302536121570, 7701071658905632652978679705499375079366268192481, 26428369560464143631836075301292862589934528565073, 90710181259990414936315690306452532930705261786641, 311391557080827122419031902870170886925786584873523] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 74 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH H + HH H HH H + H HH H H + HH HHH HH 4+ H H H + H H + HH HH 3+ H H H + HH HH HH HH + HH HH H HH 2+ HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 3 2 2 2 1 + (x + x + x + x) P(x) + (-x - x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (n - 2) a(n - 1) 2 (n - 2) a(n - 2) (4 n - 11) a(n - 3) a(n) = ---------------- + ------------------ + ------------------- n + 1 n + 1 n + 1 (8 n - 25) a(n - 4) 6 (n - 4) a(n - 5) (5 n - 22) a(n - 6) + ------------------- + ------------------ + ------------------- n + 1 n + 1 n + 1 3 (n - 5) a(n - 7) + ------------------ n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 2, a(5) = 3, a(6) = 7, a(7) = 17] Just for fun, a(1000), equals 5032496365637955067683347870950409710915701764522282276774675157243603802582\ 866298211231210653006394070833138976334822500759759891785781976829184637\ 643038378788330330417356109675556672422365107611262498459448932887962130\ 643216278920681957917488069930040634733745054315448220108085618003032954\ 450492248812276239145688825121717358769928635223051275485462691699729085\ 004111315387526723342034239842070602836639078569636160428330200510115437\ 8 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 2, 3, 7, 17, 39, 91, 219, 533, 1307, 3234, 8067, 20255, 51150, 129839, 331109, 847876, 2179289, 5620427, 14540036, 37721539, 98116305, 255818920, 668475637, 1750371384, 4592023450, 12068440795, 31770262976, 83765896945, 221181546661, 584827643203, 1548351907603, 4104332864117, 10892248356011, 28937953375916, 76960653386603, 204879046734386, 545925151025906, 1455981438846053, 3886400469418666, 10382245378672359, 27756918405827983, 74263029726249001, 198829910977439625, 532703401910546531, 1428143266813608445, 3831150646223428804, 10283645508579576861, 27619465812286909488, 74220560789749220334, 199556083749746908195, 536820382086347568072, 1444801763827645841740, 3890408061198204485087, 10480511016119495728047, 28246414132024260752912, 76160755522727900890180, 205437651850071834885351, 554375257812010095795798, 1496570004169021588945828, 4041609047370900210653163, 10918690306434774952578648, 29508078135961103546885935, 79773895103635321195662633, 215737548410098076340647126, 583621812616151827778503173, 1579334324213728450307002233, 4275132085798309736678255810, 11575884594778904645167342803, 31353387126118713173448037627, 84944829539009789690078850478, 230201612728140081149731098573, 624015753012404624623636602384, 1691981062322825431740014538160, 4588865052096950827456946241043, 12448646109079949902773787405256, 33778729419959874702068588413132, 91678238541123561975081610471607, 248879128104875104185719943760541, 675783451463072308953369784249936, 1835359604551371265619822938995004, 4985710857337519919668459945842445, 13546376824511567847839151376237576, 36813514224468113012736613431442804, 100063905775460344278497620851740509, 272039349875194325299625521607283254, 739721482329877908140333154428499903, 2011801570464323989823815200231900135, 5472436183516459408824867590415304632, 14888577978323084494016869894352271445, 40513618989439805975549944975329369435, 110261163534686841700263792854738579832, 300134742948778688458516646666561272035, 817110224328981701161007905911583958297, 2224919190647836747971327271445212007300, 6059205160189587412826547467523630270925, 16503783917191051549681811499901147080856, 44958988651672462155755561208723586036494] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 75 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H H + HH H HH H + H HH H HH + HH HHH H 4+ H H H + H HH + HH H + H HH 3+ H H H H + HH H H HH H HH + H H HH H HH H + HH HHH HHH HH 2+ H H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 4 7 6 5 4 3 2 3 1 + (x + x + x + 2 x ) P(x) + (x + x - x - 2 x - 4 x - 5 x ) P(x) 5 4 3 2 2 3 2 + (-x - x + 2 x + 6 x + 4 x) P(x) + (x - x - 4 x - 1) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 3, 7, 19, 49, 126, 334, 904, 2471, 6807, 18900, 52849, 148660, 420325, 1193916, 3405356, 9749315, 28006238, 80699997, 233193793, 675593665, 1961964795, 5710259469, 16653669719, 48662152261, 142443957281, 417654984426, 1226497273459, 3607026596578, 10622535716536, 31323384649059, 92478170657610, 273344828523410, 808829368871051, 2395812288512228, 7103555194631171, 21081658560415003, 62621043999998018, 186167637984100791, 553910404651518108, 1649341313799809201, 4914762103943287884, 14655533417492755428, 43731564162964959917, 130578154947305593767, 390136691342430677837, 1166334464962171793565, 3488826563147025513752, 10441817538848120708927, 31268301499079297668355, 93681852256006666871818, 280815844275551968876990, 842160965545315530460268, 2526787756956710916630648, 7584653426446653692938131, 22776634231078054999318237, 68426450785595158659311052, 205652158240197961696980021, 618317776112017941406111945, 1859747879074395360795305719, 5595710199921850604197082341, 16842645774363143311927501180, 50712468088516494428061586910, 152743976017712644884609899610, 460207952849370521066932409154, 1387013814730304512849197034930, 4181578172580671471565780770141, 12610395543848302021153298783051, 38040191652886821720625138124523, 114783316697048201969133416495004, 346444486545798342601434483383134, 1045933817623064809106511702581739, 3158547413412678138066329153481835, 9540702106575664362609547298074556, 28825723549378934132892601532367494, 87113277977566854392116909251958404, 263323830453547890259076927410689777, 796150455079863126820455796886402815, 2407669685362533408970236516652788690, 7282711709782702678205141028294830587, 22033401838824602010088844667529320699, 66674544664906621110815143730131240508, 201802502558318981858404023479256934913, 610912501809690455024179644165273239949, 1849760545524744739752616042442045606861, 5601884736702958489513062189576218002735, 16968098173952219574284267878678695040233, 51405638256107368924192489833379955866113, 155763333479290433036255358055713301252260, 472057558650639852594918245244759095978183, 1430863921943530828625065503453420087356285, 4337842940941595294802923858063552473499040, 13152851981613022996733864146678412516374559, 39887352667484077301756190330700374370119210, 120981303885417192791508518925368069466662430, 367001394690732499730651433919754966661846934, 1113479571760043105992054849433299269641213503] ------------------------------------------------------------ Theorem Number, 76 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H H + H HH HHH + HH H H HH + H HH HH H 3+ H H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH H H 1+ H H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 3 2 3 1 + P(x) x + (-x + x) P(x) + (x - x - 1) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 45, 90, 187, 401, 851, 1815, 3925, 8517, 18528, 40542, 89084, 196294, 433960, 962393, 2139725, 4768821, 10653090, 23847852, 53488512, 120189110, 270525714, 609872556, 1376944183, 3113165351, 7047902436, 15975594178, 36254691983, 82367121385, 187327128656, 426463015182, 971794420430, 2216459060360, 5059629805741, 11559371030889, 26429562274813, 60474319746292, 138472393707020, 317287830492994, 727493771194568, 1669087821473347, 3831706461817534, 8801529240735440, 20228664934673007, 46516855554565603, 107023436481333016, 246356874740823900, 567361687949009896, 1307246629320351536, 3013355124126816199, 6949151353923786831, 16032292599544117909, 37002915021440054034, 85437203538695002412, 197343801364604072373, 455995108480664878088, 1054028117510157798185, 2437220586009809344922, 5637462008726047404420, 13044104958896714457266, 30191362314092522717088, 69901256387316040143189, 161889000866928364911673, 375038792667449550811129, 869076638050116790409663, 2014464785302280486552292, 4670655362261462738855867, 10832019860018976585731355, 25127635081038417757157497, 58304423258510204427627370, 135318255823699614530347478, 314133072780421992235087867, 729408461607772465334189902, 1694046655330387389384834123, 3935274500031711929656356229, 9143606614506172522351127413, 21249596815155851886618287769, 49393797017310482802102109124, 114836688005465996914314854101, 267038248650994935249780214935, 621082017773923851102098581060, 1444791837130910209523805410400, 3361558483284669762056309772047, 7822641317082942993733966553444, 18207145325605714380902195077507, 42384238894815170678689367097517, 98682343013046975787558512672747, 229797628412303496855469928034966, 535206276701024419566301617271889, 1246708738543803536060775647603284, 2904528122457933390105864083112466, 6767863837948236105079296085122986] ------------------------------------------------------------ Theorem Number, 77 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ HH HH + HH HH + H H + HH HH 4+ HH H H + H HH H H + HH HH HH HH 3+ H H H H + HH HH HH HH + HH H HH HH 2+ HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 2 1 + P(x) x + P(x) x + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (1103 n - 944) a(n - 1) a(n) = -3/2 ----------------------- (n + 2) (259 n - 354) 3 2 3 (777 n - 2165 n + 1588 n - 708) a(n - 2) + -------------------------------------------- (259 n - 354) (n + 2) (n + 1) 3 2 (2072 n - 22485 n + 55975 n - 38424) a(n - 3) - 1/2 ----------------------------------------------- (259 n - 354) (n + 2) (n + 1) 2 (n - 3) (2849 n + 6033 n - 23246) a(n - 4) + ------------------------------------------- (259 n - 354) (n + 2) (n + 1) (148 n - 305) (n - 3) (n - 4) a(n - 5) - 161/2 -------------------------------------- (259 n - 354) (n + 2) (n + 1) subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 1456286415952438964652364581682787386720721652723427223868652268704217615734\ 013661511014559584083968554107289545538396109682528052843351355027899510\ 336875839145124807012715538480932028100745003225030603205050905923754789\ 120913436417646081847745149534148691270052401108733311860953962982834165\ 142352139116820596104418788220221329115699403613078628786955744338071295\ 785210326056171164845618097 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 4, 8, 13, 31, 71, 144, 318, 729, 1611, 3604, 8249, 18803, 42907, 98858, 228474, 528735, 1228800, 2865180, 6693712, 15676941, 36807239, 86584783, 204060509, 481823778, 1139565120, 2699329341, 6403500057, 15211830451, 36183117255, 86171536894, 205459894230, 490417795075, 1171809537516, 2802705573178, 6709741472212, 16077633792535, 38557470978465, 92543928713413, 222292252187223, 534346494991678, 1285375448785399, 3094086155360823, 7452774657572624, 17962853486458385, 43320517724198861, 104535253974750259, 252390609561764423, 609699521485999914, 1473608895007574938, 3563397472304118410, 8620933920694737411, 20866285270840567272, 50527739882573522706, 122405798503806040965, 296657476936886314309, 719256672330233630085, 1744545018773625965860, 4232966162296687137816, 10274640808649742568633, 24948409910226791411907, 60599469176019760272535, 147244830551438483529790, 357892285312932720768838, 870164978202711215226483, 2116332778149031726542808, 5148677470688468794716894, 12529484860432553111063488, 30499528076281485551927755, 74262935424150986850220725, 180870207357836188170341069, 440630810522439231975963839, 1073724026979876073732588658, 2617084008072054452326466527, 6380388743696091681799439061, 15558886191823487315141504508, 37949779254079097680017924283, 92584221613943667202286169191, 225922433438019608173638548805, 551409374869368431174631180753, 1346105758407373798214652723522, 3286793196606678831626181940722, 8026971507652658919500628768874, 19607185640982275440523729181033, 47902823795394726062462438796736, 117054324948067108888357885549374, 286083328086281443692788683942327, 699317934466646636669713105276839, 1709748244857899463255482766619759, 4180839106559551650627420507840427, 10225086357877339275440446600065760, 25011587559313816213376886296261184, 61190615947546730766871192612064833, 149725673741988363612845353038008205, 366415846156516055604065564079552535, 896845046441850691487935128438196159] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 78 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H H + HH H H HH + H HH HH H + HH H H HH 4+ H H H H + H HH HH H + HH H H HH + H HH H H 3+ H H H H + HH H HH HH + H HH H H + HH HHH HH 2+ H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 1 + (x + x) P(x) + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 1) a(n - 1) (2 n - 1) a(n - 2) 3 (2 n - 5) a(n - 3) a(n) = ------------------ - ------------------ + -------------------- n + 1 n + 1 n + 1 (n - 2) a(n - 4) 2 (2 n - 7) a(n - 5) - ---------------- + -------------------- n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 3134627918704846396589045582068293464668313386749891925759492579301366817419\ 266879466277019895299966556024437351904982229786354769359885950645415854\ 487762568754124509622513897418551096745213564746196714896494879980931898\ 708502918937008296833152741264657083164783324054454272242916859440693717\ 64365001116217342900388877265443604205502181219138765202835544779530 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 42, 84, 170, 354, 740, 1549, 3269, 6945, 14811, 31711, 68177, 147091, 318313, 690837, 1503351, 3279445, 7169907, 15708485, 34482475, 75830981, 167042763, 368548926, 814341362, 1801867812, 3992172298, 8855912464, 19668236110, 43729885814, 97330067640, 216844435346, 483571453768, 1079354856828, 2411240775366, 5391038526802, 12062679137620, 27010935099196, 60526554883902, 135721815227704, 304535849636366, 683755276072670, 1536121565708096, 3453050076522008, 7766456476976834, 17477390474177854, 39350957376284872, 88644320911742304, 199781711595784890, 450466438264400446, 1016164744759318568, 2293264805607373804, 5177575503207130558, 11694317639185843274, 26423715558659122260, 59728010995453379421, 135058602016778295285, 305507289981338348257, 691307640696528081979, 1564829833276593856647, 3543272754560789200681, 8025639916478510155339, 18183959683820504446497, 41212333247384488368873, 93431281455612601484243, 211875294974960472423745, 480604617462452614426815, 1090465639820281676643073, 2474853003558648387670983, 5618201029263006344110809, 12757123617698163352573711, 28974313940315478259062979, 65822724585319538752564981, 149567876677718197286698671, 339936780414026450535924453, 772774892529317395366017603, 1757116730487566683312107729, 3996122690354078859729209647, 9090033908542877936842438557, 20681339112467629598326301035, 47062642985297219558085765929, 107116551464339266855499450015, 243847122183568901454753070765, 555210537090801909689698475727, 1264372572489361483075032316117, 2879838126594606020044275873923, 6560472630207720552639437125041, 14947710059210582396858604974621, 34063179534729044501719950750975, 77636370512116064921945474111629, 176975591352231844422902101233579, 403485801251068480867972873270661, 920043891167408188870554973636539] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 79 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 3 2 3 1 + P(x) x + (-x + x) P(x) + (x - x - 1) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 45, 90, 187, 401, 851, 1815, 3925, 8517, 18528, 40542, 89084, 196294, 433960, 962393, 2139725, 4768821, 10653090, 23847852, 53488512, 120189110, 270525714, 609872556, 1376944183, 3113165351, 7047902436, 15975594178, 36254691983, 82367121385, 187327128656, 426463015182, 971794420430, 2216459060360, 5059629805741, 11559371030889, 26429562274813, 60474319746292, 138472393707020, 317287830492994, 727493771194568, 1669087821473347, 3831706461817534, 8801529240735440, 20228664934673007, 46516855554565603, 107023436481333016, 246356874740823900, 567361687949009896, 1307246629320351536, 3013355124126816199, 6949151353923786831, 16032292599544117909, 37002915021440054034, 85437203538695002412, 197343801364604072373, 455995108480664878088, 1054028117510157798185, 2437220586009809344922, 5637462008726047404420, 13044104958896714457266, 30191362314092522717088, 69901256387316040143189, 161889000866928364911673, 375038792667449550811129, 869076638050116790409663, 2014464785302280486552292, 4670655362261462738855867, 10832019860018976585731355, 25127635081038417757157497, 58304423258510204427627370, 135318255823699614530347478, 314133072780421992235087867, 729408461607772465334189902, 1694046655330387389384834123, 3935274500031711929656356229, 9143606614506172522351127413, 21249596815155851886618287769, 49393797017310482802102109124, 114836688005465996914314854101, 267038248650994935249780214935, 621082017773923851102098581060, 1444791837130910209523805410400, 3361558483284669762056309772047, 7822641317082942993733966553444, 18207145325605714380902195077507, 42384238894815170678689367097517, 98682343013046975787558512672747, 229797628412303496855469928034966, 535206276701024419566301617271889, 1246708738543803536060775647603284, 2904528122457933390105864083112466, 6767863837948236105079296085122986] ------------------------------------------------------------ Theorem Number, 80 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 2 3 1 + P(x) x + (x - x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 1) a(n - 1) (n - 2) a(n - 2) (2 n - 7) a(n - 3) a(n) = ------------------ - ---------------- + ------------------ n + 1 n + 1 n + 1 2 (n - 5) a(n - 4) (n - 8) a(n - 6) + ------------------ - ---------------- n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 2] Just for fun, a(1000), equals 1176950714989472039879867097164787467731168548390670636801992136416332972086\ 898277348698087993919571204091110820742448542481636855988358314899690938\ 672843649138648682133884841687298042373689270639308856986357452480963423\ 383401026848945091071625435678738959251923453512607097256262920947479103\ 356615910327226686348752585420237437 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 22, 42, 80, 152, 292, 568, 1112, 2185, 4313, 8557, 17050, 34089, 68370, 137542, 277475, 561185, 1137595, 2311014, 4704235, 9593662, 19598920, 40103635, 82185653, 168666493, 346613232, 713200114, 1469254621, 3030218948, 6256281188, 12930039374, 26748697772, 55386529370, 114785051382, 238083048103, 494216315763, 1026681547651, 2134372036796, 4440242721757, 9243424565624, 19254704030249, 40133535117994, 83701671288887, 174665494666782, 364684302692317, 761824952311410, 1592257031239222, 3329531677118927, 6965586177249102, 14579064797995464, 30527584089316653, 63949861857983311, 134018617814709631, 280972131660117384, 589289169477022354, 1236390172104441711, 2595012857019532078, 5448483097227962922, 11443510685976418890, 24042863051171641274, 50530333059247995344, 106231480858589059892, 223401249061635751536, 469943442677589917028, 988848424941723500999, 2081299761445939780379, 4381845845937552096915, 9227711560622784969822, 19437589516267668693817, 40954291792943394506568, 86310235142767979004036, 181940086112454163969993, 383614626529901470057969, 809021619679751516526841, 1706557543555582414238756, 3600603520716009828550165, 7598380174796313007709308, 16038212465609702505254728, 33859308082950214419570804, 71496599403133792320146486, 150999659234389445825596123, 318968628398739129595019281, 673906439126292155812743214, 1424062795029526232414783635, 3009781004697296226116225432, 6362316258145714947705909656, 13451435620129537321656044592, 28444187117982772743078140610, 60157315325649798906283809598, 127248293230730230066048509796, 269204737445574334829247116243, 569612221084905567519367331211] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 81 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ H HH HH + HH H HH HH + H HH H H + HH HHH HH 4+ HH H H H + H HH H H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 1 + (x - x + 1) P(x) + (x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence (5 n - 11) a(n - 1) 2 (2 n - 5) a(n - 2) a(n) = ------------------- - -------------------- - a(n - 3) n - 1 n - 1 2 (2 n - 5) a(n - 4) + -------------------- n - 1 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 3] Just for fun, a(1000), equals 1675978308135788135555412396405303632928466859340501442364100837069866375756\ 143830236932569333704797901522480611549720329590326987171892098531904771\ 466453107768098182871283916076007024057272453507131637619369572309468876\ 130987772391909302723877633064871393981999037109914838987914425332911519\ 447650360295622958355268349302968204433902311876460210104211735640135872\ 892108675519457654191767687303809099792690765881222743803945419489542083\ 349449221136431928180499606284456123753574096873116913480230599918574454\ 325396127287685991259538575058218293841337429219756301763767131838531848\ 55225615857545571 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 3, 7, 20, 59, 184, 593, 1964, 6642, 22845, 79667, 281037, 1001092, 3595865, 13009673, 47366251, 173415176, 638044203, 2357941142, 8748646386, 32576869203, 121701491701, 456012458965, 1713339737086, 6453584646837, 24364925260024, 92185136438942, 349479503542465, 1327356794933809, 5050158160587171, 19245314696271615, 73452022778386004, 280737995030041943, 1074436722483501092, 4117270194612416350, 15796330585442827899, 60672698187581069171, 233289290779975354221, 897918331924508326712, 3459372676013419366361, 13340001976871342146160, 51486513556851277778688, 198880620085923210073027, 768840737702048561014907, 2974472078222445356489939, 11515920169669695536463232, 44615853189158693861877815, 172969279353105613983379776, 671005604363053056501525116, 2604651408930055016263421757, 10116467800091265787510858137, 39314629141500838681151750533, 152867849188353217915442174656, 594711099416215423575540825943, 2314811086883533455886868951382, 9014400165540053735521085241038, 35120641922898690216190765327759, 136894291125137460793802215188457, 533824886087174392788924573314531, 2082555208760395745259506587025668, 8127788876220113934952785219053547, 31733701733178566367875319192012328, 123946997083190150410551197551366785, 484298378082786695954607455045860188, 1892983249500527798459222030097713210, 7401717836977650383042586538395491525, 28951144480851381150446531697406581287, 113277085230311139078172950331009976577, 443360869903549464251914157247673109752, 1735831833842980353591409802740496073065, 6798124673258774773932177046404551812838, 26631654193064625172609261457411519728786, 104359311724299022660226089287020292540221, 409058065362143569023830203759424874754623, 1603822065599457755679981664309975485816857, 6289871684238462644348141783417819654301156, 24673956345686203650535672858745939972796573, 96815377891619204354545843049083293726315316, 379975587382645641164481547295156241336406390, 1491661208984295709445578751441077758031223757, 5857135881080996987554274205898127647804563181, 23003745309315128181124933176518627042864391071, 90366525233111027179567406459746196427818373414, 355066840391425148220655595543016635200685090673, 1395416798004383856012567320409974466880924104122, 5485130346327797563347695742121135728881857045028, 21565378340601799212196185107341005237048762349295, 84803222128360329726680255273077212232849373506173, 333542250086033328687286000309797723590514888487745, 1312111472545786311404489065738998582715927216089250, 5162620484001430667095662383229270880257109522606877, 20316436056407286745536703346697816388680525761954732, 79965201119201916185594777861788466697934327453303482, 314795926153277490687414892102035918823958045418926205, 1239453288074298411794484059210277699272508277891875523, 4880931291361050881172759822727735892284080647817266341, 19224064721337348827502222777282732735173077553298095236, 75727969502864408197895439326115311369921300900162830713] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 82 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 7+ HH + H HH + HH HH 6+ H HH + H HH + HH HH 5+ HH HH + H HH + HH HH 4+ H H HH + HH HH HH HH 3+ HH HHH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 3 2 2 2 1 + (x + x - x - x + 1) P(x) + (x + x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence 3 (n - 2) a(n - 1) 2 (n - 4) a(n - 2) 6 (n - 3) a(n - 3) a(n) = ------------------ + ------------------ - ------------------ n - 1 n - 1 n - 1 2 (n - 4) a(n - 4) (5 n - 17) a(n - 5) 3 (n - 4) a(n - 6) - ------------------ + ------------------- + ------------------ n - 1 n - 1 n - 1 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 4, a(6) = 8] Just for fun, a(1000), equals 3763397943187744689557058298717036661701274707108865401916940111974607872077\ 658288045952433378143909073181283205238953536359626707954324027248931203\ 293142036206158332931880545284748241890280047855652032379329135254532027\ 418033699746548607596676475393378221638914474356962386199724649823596587\ 328530039058456194196361496424523167984070089076512580890702546756287815\ 032033874024553898640073004897040106444624152147473336171196216009277955\ 755465619716781799721328921010420649 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 2, 4, 8, 18, 41, 98, 240, 602, 1538, 3990, 10484, 27845, 74636, 201641, 548520, 1501147, 4130169, 11417537, 31697323, 88335563, 247032739, 693016665, 1949776933, 5500153093, 15553283123, 44080461094, 125191586993, 356241890080, 1015542517685, 2899897830889, 8293768547892, 23755451317723, 68136244293124, 195686735756336, 562704829653673, 1619967823913525, 4668866393038729, 13470098409287510, 38901041194789990, 112450505301046743, 325350499795352311, 942134675221160158, 2730415764681587458, 7919228360313768631, 22985836060498323909, 66764735360530456928, 194057426731047586386, 564413570865224960203, 1642620166663351240445, 4783420332146312182912, 13937710312803494005796, 40633684407300598033641, 118526042023024400779817, 345911977622328522715727, 1010028323307390500414420, 2950597469676623473580492, 8623579438992579475081299, 25215028507019728783868248, 73759693396276342547566305, 215854080907620693861830681, 631941720978248049121784338, 1850818211226686502653504084, 5422694078322422657486457016, 15893733753413182801895787348, 46600585250786617171607185753, 136680562222566567550660388647, 401021482382448314332554769888, 1176981830637742252320304842085, 3455485266944898602230379995380, 10148026678529994810685008317507, 29811490327627053532556549200311, 87601537368610156299641801091562, 257491138454528754065273560703219, 757063086930258548848658706013351, 2226475475420250467994521359008779, 6549630223943234361577342672535706, 19271953848944351261094294070549004, 56720754945695480096258405339418098, 166979380329006582686255781263536965, 491683542308681667998212844009195570, 1448131419704670211913722491631048010, 4266063776811701283206969539871248671, 12570178524515842724568644540915080142, 37046572745428956862697268525140367578, 109205610321775427616856875957768924740, 321980889215005203257661276943201041326, 949514174948532341005286932136918262543, 2800638604072808547641255591093610467486, 8262187851536834686256841645171148545066, 24378872289013286125654724617628842666447, 71946711756248222025227206505790090246738, 212366188207300131383966258865261445645756, 626953350753974988907463750838598419202540, 1851223774793914375545886271767185599658694, 5467073315060779063517258382870052587610843, 16148108703282747519769980332748062103004374, 47704326029009105434055253692071275392467484] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 83 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H H + HHHH HHHH + H H H H 5+ HH HHH HH + HH H + H H + HH HH 4+ H H HH + H H HH H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 3 -1 + (x + 3 x - x + 2 x - x + 1) P(x) 4 3 2 2 2 + (-3 x + x - 4 x + 2 x - 3) P(x) + (2 x - x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 4, 10, 27, 75, 213, 618, 1829, 5502, 16766, 51638, 160510, 502936, 1586915, 5037898, 16080235, 51573656, 166126983, 537210234, 1743335620, 5675610217, 18531855864, 60672908113, 199134937139, 655080466555, 2159549717786, 7133254708699, 23605300357009, 78248448348415, 259799644349161, 863882438059948, 2876633874448522, 9591630118557203, 32021763420976533, 107031813937668680, 358152189955471182, 1199730957398468158, 4022889016490524078, 13502318085820052452, 45360149060387967073, 152516792901902446460, 513239903623380228634, 1728488199311870556341, 5825597234987627520020, 19648449862710960974468, 66315737830868245888713, 223971572659045980013396, 756912089375355151840065, 2559547558143509099472648, 8660357142068863862107857, 29319290281835060734836478, 99313188455823088233137906, 336579373400955588030931881, 1141266121031792548722158167, 3871661712452036848055847470, 13140489346376262760361654857, 44619255232323372515516410045, 151573431350978167631715794500, 515118930493289031968941039048, 1751336452045707226974825075965, 5956670663491505612092714479448, 20267681246079204003255012865920, 68986771314773128486278704491480, 234900475364654332600465402316920, 800117117051503591337045132089775, 2726278992841291666370074042805552, 9292440348616198047813541633462860, 31683115635946347629623785374232487, 108058924915649277676057259022053450, 368658517648600298195076388866196657, 1258099938148112126731753336113447860, 4294669672171115651292913118800043505, 14664417073874898414157875861063587347, 50086080693227785224603418797689014647, 171113162673527886251575588257860242944, 584737505909401426725277499566850600057, 1998696255538134167700268645424394567403, 6833421694220982386351699617760291091198, 23368594017576346412900144749396392537494, 79933226555544808570154485203703170303082, 273476527083818725459119768102976825183174, 935854562232197114417776416047740913475791, 3203243746003208985602790946785892576204084, 10966366023767169445457040146336010139424470, 37551258376630702470631501349315514985767247, 128609549273471674748920099576880547705705392, 440561888663121764943313503687497097424833621, 1509467316392012568455815312423254958860615388, 5172752952822098618520373830943462076045468471, 17729611172602185872624136898905502208011067108, 60779120265889375477319148720241010268357965932, 208394216696886394027227843877823994272813709965, 714646633388317059244644903816395674073736347859, 2451150134945316687182000995138448683956077170923, 8408525152767898640307450568476492573560279508421, 28849588699589082878771616007427495302335252329135, 98998333177530072769467116809527730538555485758775] ------------------------------------------------------------ Theorem Number, 84 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH + HH H + H HH + HH H 4+ H HH HH + H HH HH H + HH HH HH HH 3+ H H H H + HH H HH HH + HH HH H HH 2+ HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 3 4 3 2 2 -1 + (x + 2 x - x + x - x + 1) P(x) + (-2 x + x - 2 x + 2 x - 3) P(x) 2 + (x - x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 46, 94, 199, 439, 961, 2113, 4728, 10638, 24004, 54567, 124738, 286100, 658902, 1523561, 3533694, 8219807, 19175241, 44844787, 105117745, 246938893, 581271855, 1370801044, 3238379640, 7662884761, 18160172211, 43099390904, 102426035307, 243726755578, 580655580178, 1384932709103, 3306794777897, 7903693118041, 18909278995761, 45281554377074, 108530027712894, 260341203346223, 625005317744160, 1501608816996828, 3610336668014877, 8686460115399371, 20913632597618146, 50384354111881549, 121459160713524172, 292969666722436726, 707070780696631056, 1707424831884850449, 4125246497394886642, 9971935177734506502, 24116945055530422878, 58354041165562962555, 141259692927959905224, 342103485342872695337, 828862025018976189145, 2009029841925232601585, 4871515952686337473281, 11817073539321134031521, 28675998351756531656835, 69612131191902560435320, 169045763869085582882160, 410650213529812223368770, 997892674377742909288876, 2425690847473912236264926, 5898246126299524653958956, 14346379729918995608354591, 34905194678328946279956726, 84949845555910352980918162, 206802853409472003358290619, 503580166605875837883509918, 1226579626343965688411938881, 2988373791468938272294751313, 7282546274134082391128998885, 17751619375203540121720680747, 43280914162805330098090017988, 105549463086185917149368841094, 257462696364279602325677747884, 628158000160191822477935767084, 1532912888447769419915804392627, 3741604135787909331331389618256, 9134564337311934498324607449420, 22305163745068650394954160327031, 54476421551688686284866768832516, 133074664291962641673573493873036, 325135188262952797925515855558143, 794534125103762472134490936108882, 1941955965184663382273487480183164, 4747257141764020796262302049059161, 11607028692460250300264994515896655, 28383941376456748393597046710009226, 69421845442964953819449619451462120, 169820428896576356561050206615538112, 415482369434004891765163304919548259] ------------------------------------------------------------ Theorem Number, 85 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H HH H + H HH H H + HH HHH HH 2+ H H H H + H H HHH + HH HH H HH + H H HH H 1+ H H H H + HH HH H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-*+-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 3 2 2 2 1 + (x + x - x - x + 1) P(x) + (x + x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence 3 (n - 2) a(n - 1) 2 (n - 4) a(n - 2) 6 (n - 3) a(n - 3) a(n) = ------------------ + ------------------ - ------------------ n - 1 n - 1 n - 1 2 (n - 4) a(n - 4) (5 n - 17) a(n - 5) 3 (n - 4) a(n - 6) - ------------------ + ------------------- + ------------------ n - 1 n - 1 n - 1 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 4, a(6) = 8] Just for fun, a(1000), equals 3763397943187744689557058298717036661701274707108865401916940111974607872077\ 658288045952433378143909073181283205238953536359626707954324027248931203\ 293142036206158332931880545284748241890280047855652032379329135254532027\ 418033699746548607596676475393378221638914474356962386199724649823596587\ 328530039058456194196361496424523167984070089076512580890702546756287815\ 032033874024553898640073004897040106444624152147473336171196216009277955\ 755465619716781799721328921010420649 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 2, 4, 8, 18, 41, 98, 240, 602, 1538, 3990, 10484, 27845, 74636, 201641, 548520, 1501147, 4130169, 11417537, 31697323, 88335563, 247032739, 693016665, 1949776933, 5500153093, 15553283123, 44080461094, 125191586993, 356241890080, 1015542517685, 2899897830889, 8293768547892, 23755451317723, 68136244293124, 195686735756336, 562704829653673, 1619967823913525, 4668866393038729, 13470098409287510, 38901041194789990, 112450505301046743, 325350499795352311, 942134675221160158, 2730415764681587458, 7919228360313768631, 22985836060498323909, 66764735360530456928, 194057426731047586386, 564413570865224960203, 1642620166663351240445, 4783420332146312182912, 13937710312803494005796, 40633684407300598033641, 118526042023024400779817, 345911977622328522715727, 1010028323307390500414420, 2950597469676623473580492, 8623579438992579475081299, 25215028507019728783868248, 73759693396276342547566305, 215854080907620693861830681, 631941720978248049121784338, 1850818211226686502653504084, 5422694078322422657486457016, 15893733753413182801895787348, 46600585250786617171607185753, 136680562222566567550660388647, 401021482382448314332554769888, 1176981830637742252320304842085, 3455485266944898602230379995380, 10148026678529994810685008317507, 29811490327627053532556549200311, 87601537368610156299641801091562, 257491138454528754065273560703219, 757063086930258548848658706013351, 2226475475420250467994521359008779, 6549630223943234361577342672535706, 19271953848944351261094294070549004, 56720754945695480096258405339418098, 166979380329006582686255781263536965, 491683542308681667998212844009195570, 1448131419704670211913722491631048010, 4266063776811701283206969539871248671, 12570178524515842724568644540915080142, 37046572745428956862697268525140367578, 109205610321775427616856875957768924740, 321980889215005203257661276943201041326, 949514174948532341005286932136918262543, 2800638604072808547641255591093610467486, 8262187851536834686256841645171148545066, 24378872289013286125654724617628842666447, 71946711756248222025227206505790090246738, 212366188207300131383966258865261445645756, 626953350753974988907463750838598419202540, 1851223774793914375545886271767185599658694, 5467073315060779063517258382870052587610843, 16148108703282747519769980332748062103004374, 47704326029009105434055253692071275392467484] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 86 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 5 4 2 2 3 2 1 + (x + x - x - x + 1) P(x) + (x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 1, 1, 2, 3, 6, 11, 22, 45, 95, 205, 452, 1013, 2303, 5299, 12318, 28887, 68261, 162378, 388524, 934445, 2257825, 5477978, 13340342, 32597052, 79895596, 196374894, 483913063, 1195304610, 2958976037, 7339834801, 18241094639, 45412924028, 113245700219, 282834394783, 707407654401, 1771727090682, 4443029765522, 11155412690573, 28040604020532, 70559904175088, 177735079073901, 448136569131253, 1130964218420963, 2856730577032510, 7221937035815530, 18271971340356949, 46264602062392922, 117227479522291455, 297244335207374079, 754203530022169492, 1914883646655804475, 4864787119569857793, 12366373073388985492, 31453425461785677412, 80044458790641774203, 203809045620843784774, 519201725396429756541, 1323309312702391048197, 3374362078593316366604, 8608350819607942123095, 21970475292914355350454, 56097532741804671821420, 143293600571198079874492, 366170004247708309164512, 936065320726018453315109, 2393821227622028733131447, 6123989383217901770489560, 15672183956044108406708589, 40121062224202163233919784, 102744544659869498024526596, 263199097281447721431070670, 674443076084351358917512888, 1728771776052830935789889303, 4432592643567142135509390624, 11368476318425902647780179864, 29165378348002562991663729555, 74842906904770854698923268980, 192109239699919575705173539470, 493239045378109676774564459254, 1266704647942467831838666714549, 3253862825667795067086975460886, 8360387768269617869541470016906, 21485939561519119323971592020157, 55230701932447147705776562477159, 142004689399176019947282705368515, 365189625557095661180664282333447, 939346000239253860837716511164830, 2416696053162063667458685953035157, 6218787279970599797897622870739574, 16005697704745345705632449082412180, 41202810360147745584041893709115887, 106086600734727185860399731461973554, 273195725269569579262914425143253384, 703663787334094074112217219749705993, 1812728489668564198964352359374610211, 4670624475445511892818963507039155536, 12036222263015752748284469824403616053, 31022513337880797655272396083908134269, 79971237524146989974840114770529202975] ------------------------------------------------------------ Theorem Number, 87 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H H H + HHH H HH HHH + HH H HH H H HH + H HH H HH HH H 3+ H H H H H + HH H H HH + H H HH H + HH HHH HH 2+ H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - x + x - x + 1) P(x) + (x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 4, 8, 18, 40, 91, 213, 506, 1218, 2966, 7294, 18090, 45194, 113632, 287327, 730186, 1863977, 4777509, 12289898, 31720176, 82117774, 213178108, 554823435, 1447397817, 3784127064, 9913343789, 26018984599, 68410290554, 180162323411, 475196423227, 1255186628492, 3319955888745, 8792478337054, 23313863133954, 61888878237389, 164467529111164, 437514750895099, 1165005085521436, 3105022797156006, 8282937176309562, 22114068273505488, 59088226972484393, 158003152307405371, 422813155834784436, 1132232400449265104, 3033990842200935686, 8135281602930567564, 21827264601898788794, 58598143522316010039, 157404179282151204174, 423045542218716765139, 1137594743452912452613, 3060618745844589852386, 8238424391588288975038, 22186297926687272423033, 59775628331378116451380, 161122179240285688242086, 434482162221497135042578, 1172108773956891793300694, 3163278407606545361590941, 8540335279906181472947846, 23066152471490359501137078, 62320812139929346207300967, 168439506971919225588258595, 455410615205746441651620508, 1231703823760505986269633728, 3332339148450456918147838167, 9018366051490575879544837616, 24413972116862043819738815721, 66111540013999155652084562422, 179077417530677201597463612360, 485205493746336496557728723796, 1315008887370019158511357609032, 3564894162691250059222924187153, 9666662859580349116822421353996, 26218968527010913350495426276500, 71131335168680718796826444485856, 193023408967072905414753719183441, 523914112753491222458732852061593, 1422357723794427722247327153795571, 3862369444640190578023693432554655, 10490416497562008857897222598208662, 28498590066657605402946101068227090, 77436123391842858541274538722111900, 210451173947131125254485079572740558, 572064013100223785974710999433555658, 1555326255238638414011723574401111725, 4229413184640997453434597390339399864, 11503199764674649726828795101413864073, 31292149913413969843695821586747884491, 85139020625938221899250912882028413646, 231684389965423872121630053321756963050, 630577096684214752553581697741642950844, 1716529914048561172329410428733040296684, 4673420308399441399562494693207807848635, 12725860269271441688238739378854893163811] ------------------------------------------------------------ Theorem Number, 88 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - 2 x + x - x + 1) P(x) + (2 x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 23, 46, 90, 178, 362, 744, 1535, 3194, 6704, 14152, 30019, 63993, 137027, 294528, 635254, 1374563, 2982983, 6490689, 14157809, 30951925, 67809639, 148847546, 327326203, 721035228, 1590820794, 3515055239, 7777682719, 17232166570, 38226855740, 84899762419, 188767040056, 420148088468, 936077556199, 2087530911028, 4659567865103, 10409537599216, 23274106872900, 52077999630392, 116616556556144, 261322452488208, 585990624975113, 1314890839669959, 2952311871586680, 6632807182343025, 14910247851249936, 33536291670621966, 75470769540757993, 169928986936543073, 382800909986262158, 862755122454412830, 1945377625304332347, 4388490288087053044, 9904091476130975332, 22361254774054848157, 50507256400673637226, 114125267802978496554, 257973369424416665508, 583348170337435269883, 1319580231355477362189, 2986029400547774283515, 6759245279299394441018, 15305372653077265127430, 34667865012474818872923, 78549578510974131598343, 178028874433432077107813, 403611252116127504636989, 915290323330934693322979, 2076222803712862095242661, 4710915866189376517310722, 10691779547594570193232569, 24271966712323362836221071, 55114699810857040680904084, 125179934207220209125653142, 284383356087736137374477620, 646209443923063259545590287, 1468722246098351522200998572, 3338881124332810099290521494, 7591977293065388173220549042, 17266301921829442269815327549, 39276443078065295093655855985, 89361682067950853306373719225, 203355033871187599510752721774, 462850820082071205035423702170, 1053677818530795805415739939561, 2399128776874249888355418021047, 5463569785243751528140091257717, 12444429345978153227988825135555, 28349632447886106488535958107141, 64594015240375900277795861639819, 147200074451998200230485801692760, 335500536871502998738039665095175] ------------------------------------------------------------ Theorem Number, 89 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH + HH H + H HH + HH H 4+ H HH HH + H HH HH H + HH HH H HH 3+ H H H H + HH HH HH HH + HH HH H HH 2+ HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 3 -1 + (x + 3 x - x + 2 x - x + 1) P(x) 4 3 2 2 2 + (-3 x + x - 4 x + 2 x - 3) P(x) + (2 x - x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 4, 10, 27, 75, 213, 618, 1829, 5502, 16766, 51638, 160510, 502936, 1586915, 5037898, 16080235, 51573656, 166126983, 537210234, 1743335620, 5675610217, 18531855864, 60672908113, 199134937139, 655080466555, 2159549717786, 7133254708699, 23605300357009, 78248448348415, 259799644349161, 863882438059948, 2876633874448522, 9591630118557203, 32021763420976533, 107031813937668680, 358152189955471182, 1199730957398468158, 4022889016490524078, 13502318085820052452, 45360149060387967073, 152516792901902446460, 513239903623380228634, 1728488199311870556341, 5825597234987627520020, 19648449862710960974468, 66315737830868245888713, 223971572659045980013396, 756912089375355151840065, 2559547558143509099472648, 8660357142068863862107857, 29319290281835060734836478, 99313188455823088233137906, 336579373400955588030931881, 1141266121031792548722158167, 3871661712452036848055847470, 13140489346376262760361654857, 44619255232323372515516410045, 151573431350978167631715794500, 515118930493289031968941039048, 1751336452045707226974825075965, 5956670663491505612092714479448, 20267681246079204003255012865920, 68986771314773128486278704491480, 234900475364654332600465402316920, 800117117051503591337045132089775, 2726278992841291666370074042805552, 9292440348616198047813541633462860, 31683115635946347629623785374232487, 108058924915649277676057259022053450, 368658517648600298195076388866196657, 1258099938148112126731753336113447860, 4294669672171115651292913118800043505, 14664417073874898414157875861063587347, 50086080693227785224603418797689014647, 171113162673527886251575588257860242944, 584737505909401426725277499566850600057, 1998696255538134167700268645424394567403, 6833421694220982386351699617760291091198, 23368594017576346412900144749396392537494, 79933226555544808570154485203703170303082, 273476527083818725459119768102976825183174, 935854562232197114417776416047740913475791, 3203243746003208985602790946785892576204084, 10966366023767169445457040146336010139424470, 37551258376630702470631501349315514985767247, 128609549273471674748920099576880547705705392, 440561888663121764943313503687497097424833621, 1509467316392012568455815312423254958860615388, 5172752952822098618520373830943462076045468471, 17729611172602185872624136898905502208011067108, 60779120265889375477319148720241010268357965932, 208394216696886394027227843877823994272813709965, 714646633388317059244644903816395674073736347859, 2451150134945316687182000995138448683956077170923, 8408525152767898640307450568476492573560279508421, 28849588699589082878771616007427495302335252329135, 98998333177530072769467116809527730538555485758775] ------------------------------------------------------------ Theorem Number, 90 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ HH HH + HH HH + H H + HH HH 4+ HH H H + H HH H H + HH HH HH HH 3+ H H H H + HH HH HH HH + HH H HH HH 2+ HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - x + x - x + 1) P(x) + (x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 4, 8, 18, 40, 91, 213, 506, 1218, 2966, 7294, 18090, 45194, 113632, 287327, 730186, 1863977, 4777509, 12289898, 31720176, 82117774, 213178108, 554823435, 1447397817, 3784127064, 9913343789, 26018984599, 68410290554, 180162323411, 475196423227, 1255186628492, 3319955888745, 8792478337054, 23313863133954, 61888878237389, 164467529111164, 437514750895099, 1165005085521436, 3105022797156006, 8282937176309562, 22114068273505488, 59088226972484393, 158003152307405371, 422813155834784436, 1132232400449265104, 3033990842200935686, 8135281602930567564, 21827264601898788794, 58598143522316010039, 157404179282151204174, 423045542218716765139, 1137594743452912452613, 3060618745844589852386, 8238424391588288975038, 22186297926687272423033, 59775628331378116451380, 161122179240285688242086, 434482162221497135042578, 1172108773956891793300694, 3163278407606545361590941, 8540335279906181472947846, 23066152471490359501137078, 62320812139929346207300967, 168439506971919225588258595, 455410615205746441651620508, 1231703823760505986269633728, 3332339148450456918147838167, 9018366051490575879544837616, 24413972116862043819738815721, 66111540013999155652084562422, 179077417530677201597463612360, 485205493746336496557728723796, 1315008887370019158511357609032, 3564894162691250059222924187153, 9666662859580349116822421353996, 26218968527010913350495426276500, 71131335168680718796826444485856, 193023408967072905414753719183441, 523914112753491222458732852061593, 1422357723794427722247327153795571, 3862369444640190578023693432554655, 10490416497562008857897222598208662, 28498590066657605402946101068227090, 77436123391842858541274538722111900, 210451173947131125254485079572740558, 572064013100223785974710999433555658, 1555326255238638414011723574401111725, 4229413184640997453434597390339399864, 11503199764674649726828795101413864073, 31292149913413969843695821586747884491, 85139020625938221899250912882028413646, 231684389965423872121630053321756963050, 630577096684214752553581697741642950844, 1716529914048561172329410428733040296684, 4673420308399441399562494693207807848635, 12725860269271441688238739378854893163811] ------------------------------------------------------------ Theorem Number, 91 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H H + HHHH HHHH + H H H H 5+ HH HHH HH + HH H + H H + HH HH 4+ H H + H H + HH HH 3+ H H H + HH HH HH HH + HH HH H HH 2+ HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 5 3 2 4 1 + (x + x + 2 x + 3 x + 2 x + 3 x - 3 x + 2 x) P(x) 10 7 6 5 4 3 2 3 + (x + x - 6 x - x - 2 x - 8 x + 7 x - 5 x - 1) P(x) 7 6 5 4 3 2 2 + (-x + 3 x - x + x + 7 x - 5 x + 4 x + 3) P(x) 4 3 2 + (x - 2 x + x - x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 3, 7, 18, 44, 109, 280, 735, 1954, 5255, 14288, 39217, 108505, 302302, 847429, 2388592, 6765497, 19246740, 54970312, 157562416, 453097016, 1306839319, 3779538739, 10958392378, 31846583680, 92749692974, 270663243009, 791322671236, 2317564004952, 6798531375987, 19973753306397, 58765902899481, 173131371863628, 510713506115097, 1508340867919083, 4459786618285185, 13200644997642960, 39112787528848850, 116001279632237698, 344355206748910149, 1023131191333303333, 3042419221532643555, 9054267496265684000, 26966134344132354244, 80371256036254553851, 239709920294859563755, 715421183374306007562, 2136565638960719056486, 6384665872837409917126, 19090490663261938862990, 57114067020897547781467, 170964846855150763189694, 512034794172729323663479, 1534308541126455320397434, 4599796479480732910229762, 13796525827540580085304946, 41399870101856911693167035, 124285255191656772092061739, 373271847488557793060517405, 1121527133521793199398529651, 3371067683449113643208929271, 10136611244295805178335506076, 30491619569690255288608488922, 91754119429669930918439578524, 276199730126408645572551619700, 831704332336391375603521245725, 2505291996504647791163520786378, 7548962005207718512325143407531, 22753679484774428262141551847782, 68603728073198831689741859679122, 206905395935030304046916480848346, 624195196971857633889350855456976, 1883606423186986155277619004361039, 5685619623395209730334405327840435, 17166439619614140305204033759142919, 51843517807962177918812419431253021, 156609328191448317164163442286830882, 473202422733253589702545651921133223, 1430144112363376670826848484744426602, 4323282306043807828111513035709202120, 13072115360399069790669916059912438838, 39534318291653934703315545185353834326, 119590436552338302559220773081934909782, 361834731326808756208297833835888676769, 1094998489140762391359812725618698478528, 3314394486204804522148317021872597821536, 10034143201570208161313294880666515660127, 30383646042639021286863897593477035894106, 92019764881724708060825041166437152084740, 278741858403344620559751350060783939727561, 844503328063769443224654633785899215174158, 2559039323627586379562839652619725161977135, 7755813282661770390894741126403860743539833, 23509908834096397085071851401290457807032194, 71276477994521894144434592480478231691440190, 216128343340502595627075199672200914399781898, 655459690737824635537416520631391620380551798] ------------------------------------------------------------ Theorem Number, 92 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 10 6 5 4 3 2 3 1 + (x - 2 x + x - 2 x + 2 x - x + x - 1) P(x) 7 6 5 4 3 2 2 + (x + x - x + x - 3 x + 2 x - 2 x + 3) P(x) 4 3 2 + (x + x - x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 17, 34, 65, 133, 277, 572, 1199, 2548, 5435, 11661, 25202, 54731, 119348, 261396, 574697, 1267637, 2804766, 6223660, 13845748, 30876213, 69007994, 154549822, 346790404, 779545026, 1755254743, 3958377577, 8939866583, 20218263933, 45784748248, 103807846272, 235636648909, 535466068728, 1218071478804, 2773591163064, 6321494978078, 14420661400511, 32924586831606, 75232979162492, 172041452385538, 393713260872074, 901644278692778, 2066263403285064, 4738263632158243, 10872374445702404, 24962637101521442, 57346546860289679, 131815132773209257, 303148512760931479, 697540229550486854, 1605826655979897539, 3698588507211274869, 8522638804153427168, 19647441284817832815, 45313253580871653227, 104550383069514369365, 241324409472237417738, 557245419733284748247, 1287229693480919378873, 2974574026651753256696, 6876189561850918905880, 15900858902075293511482, 36782273871125357477166, 85113327142737810030492, 197012375003682813852835, 456165570455991041730121, 1056527571106193560850298, 2447738008468144928854371, 5672459058686943865573806, 13149125315278295774252093, 30488654546469742502130612, 70711902848157227103981151, 164042637933125322263068179, 380652001553129860482798995, 883494697499474927314001201, 2051075452619209350256289939, 4762759874731327435883409954, 11061975317641962863310400262, 25698117521092741572233415672, 59712096957248664358561521889, 138775742402184313315715377316, 322591509897645444074929806223, 750029645171417394902589537216, 1744167142214391071760524674061, 4056767007670417319587124446474, 9437403083691333062035763415892, 21958549988526521939432355828298, 51101285390525603579799232888122, 118942032240129949401856916912786, 276893431455538839876673911173260, 644706688082075915267897498545600, 1501351643055827208280357431912040, 3496809282282403489853994898965247] ------------------------------------------------------------ Theorem Number, 93 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ H HH HH + HH H HH HH + H HH H H + HH H HH HH 4+ H HH H H + H HH H H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 3 4 3 2 2 -1 + (x + 2 x - x + x - x + 1) P(x) + (-2 x + x - 2 x + 2 x - 3) P(x) 2 + (x - x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 46, 94, 199, 439, 961, 2113, 4728, 10638, 24004, 54567, 124738, 286100, 658902, 1523561, 3533694, 8219807, 19175241, 44844787, 105117745, 246938893, 581271855, 1370801044, 3238379640, 7662884761, 18160172211, 43099390904, 102426035307, 243726755578, 580655580178, 1384932709103, 3306794777897, 7903693118041, 18909278995761, 45281554377074, 108530027712894, 260341203346223, 625005317744160, 1501608816996828, 3610336668014877, 8686460115399371, 20913632597618146, 50384354111881549, 121459160713524172, 292969666722436726, 707070780696631056, 1707424831884850449, 4125246497394886642, 9971935177734506502, 24116945055530422878, 58354041165562962555, 141259692927959905224, 342103485342872695337, 828862025018976189145, 2009029841925232601585, 4871515952686337473281, 11817073539321134031521, 28675998351756531656835, 69612131191902560435320, 169045763869085582882160, 410650213529812223368770, 997892674377742909288876, 2425690847473912236264926, 5898246126299524653958956, 14346379729918995608354591, 34905194678328946279956726, 84949845555910352980918162, 206802853409472003358290619, 503580166605875837883509918, 1226579626343965688411938881, 2988373791468938272294751313, 7282546274134082391128998885, 17751619375203540121720680747, 43280914162805330098090017988, 105549463086185917149368841094, 257462696364279602325677747884, 628158000160191822477935767084, 1532912888447769419915804392627, 3741604135787909331331389618256, 9134564337311934498324607449420, 22305163745068650394954160327031, 54476421551688686284866768832516, 133074664291962641673573493873036, 325135188262952797925515855558143, 794534125103762472134490936108882, 1941955965184663382273487480183164, 4747257141764020796262302049059161, 11607028692460250300264994515896655, 28383941376456748393597046710009226, 69421845442964953819449619451462120, 169820428896576356561050206615538112, 415482369434004891765163304919548259] ------------------------------------------------------------ Theorem Number, 94 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - 2 x + x - x + 1) P(x) + (2 x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 23, 46, 90, 178, 362, 744, 1535, 3194, 6704, 14152, 30019, 63993, 137027, 294528, 635254, 1374563, 2982983, 6490689, 14157809, 30951925, 67809639, 148847546, 327326203, 721035228, 1590820794, 3515055239, 7777682719, 17232166570, 38226855740, 84899762419, 188767040056, 420148088468, 936077556199, 2087530911028, 4659567865103, 10409537599216, 23274106872900, 52077999630392, 116616556556144, 261322452488208, 585990624975113, 1314890839669959, 2952311871586680, 6632807182343025, 14910247851249936, 33536291670621966, 75470769540757993, 169928986936543073, 382800909986262158, 862755122454412830, 1945377625304332347, 4388490288087053044, 9904091476130975332, 22361254774054848157, 50507256400673637226, 114125267802978496554, 257973369424416665508, 583348170337435269883, 1319580231355477362189, 2986029400547774283515, 6759245279299394441018, 15305372653077265127430, 34667865012474818872923, 78549578510974131598343, 178028874433432077107813, 403611252116127504636989, 915290323330934693322979, 2076222803712862095242661, 4710915866189376517310722, 10691779547594570193232569, 24271966712323362836221071, 55114699810857040680904084, 125179934207220209125653142, 284383356087736137374477620, 646209443923063259545590287, 1468722246098351522200998572, 3338881124332810099290521494, 7591977293065388173220549042, 17266301921829442269815327549, 39276443078065295093655855985, 89361682067950853306373719225, 203355033871187599510752721774, 462850820082071205035423702170, 1053677818530795805415739939561, 2399128776874249888355418021047, 5463569785243751528140091257717, 12444429345978153227988825135555, 28349632447886106488535958107141, 64594015240375900277795861639819, 147200074451998200230485801692760, 335500536871502998738039665095175] ------------------------------------------------------------ Theorem Number, 95 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 10 6 5 4 3 2 3 1 + (x - 2 x + x - 2 x + 2 x - x + x - 1) P(x) 7 6 5 4 3 2 2 + (x + x - x + x - 3 x + 2 x - 2 x + 3) P(x) 4 3 2 + (x + x - x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 17, 34, 65, 133, 277, 572, 1199, 2548, 5435, 11661, 25202, 54731, 119348, 261396, 574697, 1267637, 2804766, 6223660, 13845748, 30876213, 69007994, 154549822, 346790404, 779545026, 1755254743, 3958377577, 8939866583, 20218263933, 45784748248, 103807846272, 235636648909, 535466068728, 1218071478804, 2773591163064, 6321494978078, 14420661400511, 32924586831606, 75232979162492, 172041452385538, 393713260872074, 901644278692778, 2066263403285064, 4738263632158243, 10872374445702404, 24962637101521442, 57346546860289679, 131815132773209257, 303148512760931479, 697540229550486854, 1605826655979897539, 3698588507211274869, 8522638804153427168, 19647441284817832815, 45313253580871653227, 104550383069514369365, 241324409472237417738, 557245419733284748247, 1287229693480919378873, 2974574026651753256696, 6876189561850918905880, 15900858902075293511482, 36782273871125357477166, 85113327142737810030492, 197012375003682813852835, 456165570455991041730121, 1056527571106193560850298, 2447738008468144928854371, 5672459058686943865573806, 13149125315278295774252093, 30488654546469742502130612, 70711902848157227103981151, 164042637933125322263068179, 380652001553129860482798995, 883494697499474927314001201, 2051075452619209350256289939, 4762759874731327435883409954, 11061975317641962863310400262, 25698117521092741572233415672, 59712096957248664358561521889, 138775742402184313315715377316, 322591509897645444074929806223, 750029645171417394902589537216, 1744167142214391071760524674061, 4056767007670417319587124446474, 9437403083691333062035763415892, 21958549988526521939432355828298, 51101285390525603579799232888122, 118942032240129949401856916912786, 276893431455538839876673911173260, 644706688082075915267897498545600, 1501351643055827208280357431912040, 3496809282282403489853994898965247] ------------------------------------------------------------ Theorem Number, 96 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 7 6 3 2 2 4 3 2 1 + (x + x - 2 x + x - x + 1) P(x) + (x + 2 x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 5, 9, 15, 26, 48, 89, 165, 310, 591, 1138, 2205, 4297, 8427, 16622, 32938, 65530, 130863, 262249, 527197, 1062810, 2148146, 4352276, 8837592, 17982158, 36658592, 74865246, 153145667, 313763676, 643771865, 1322679887, 2721049245, 5604608198, 11557189002, 23857809325, 49300998950, 101977645247, 211133587197, 437515330510, 907391552370, 1883409945578, 3912262282749, 8132599519471, 16917470461058, 35215450619627, 73351807432625, 152882167526053, 318831083771137, 665291972000431, 1389000105612281, 2901494411312661, 6064042668767917, 12679900670970242, 26526198371525921, 55517844071636598, 116247016822421760, 243509692791621693, 510304958692509935, 1069834525683015957, 2243733025635694956, 4707477231914068428, 9880131493978309950, 20743874289698006871, 43567745659235343423, 91534327932366425904, 192372244159797445202, 404423446349987694296, 850475692006620639552, 1789021153134275019838, 3764380495324636198963, 7923054498549552785817, 16680521298024087725246, 35127021615881690891674, 73992001969543494910782, 155896761247337062814246, 328545486973552202438004, 692559921201329410518511, 1460226000605629209724777, 3079507053993080839704110, 6495885392920695093764064, 13705321751718305123493788, 28922220684172584949140422, 61046872393071002819789051, 128879108693661861817539403, 272136603848726764672198432, 574744475893032262155788238, 1214071321413282376714153974, 2565035305811338986244912025, 5420264847804173772189630486, 11455762461878381622441607611, 24215988133284034517908107932, 51198061405255014156095915233, 108262097724071227150484889909, 228965199188889890733294634337] ------------------------------------------------------------ Theorem Number, 97 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H + H H + H H + H H + H H 2+ H H H H + HH HH H H + H H H H H H + HH H HH H HH HH + H H H H H H 1+ H H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 5 4 3 2 2 4 3 2 (x + 4 x + 5 x - x - 3 x + 1) P(x) + (-2 x - 6 x - x + 6 x - 2) P(x) 3 2 + x + 2 x - 3 x + 1 = 0 The sequence a(n) satisfies the linear recurrence (7 n - 20) a(n - 1) (11 n - 40) a(n - 2) 3 (3 n - 8) a(n - 3) a(n) = ------------------- - -------------------- - -------------------- n - 2 n - 2 n - 2 2 (8 n - 31) a(n - 4) 3 (5 n - 18) a(n - 5) 2 (2 n - 7) a(n - 6) + --------------------- + --------------------- + -------------------- n - 2 n - 2 n - 2 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 2, a(4) = 5, a(5) = 13, a(6) = 35] Just for fun, a(1000), equals 1141397536277704322501846165018574658093008746969633152530782442531801600158\ 366777839508577435630292069501964459409786998537490128158109597112051799\ 747871063037682003129410566116237878980613629725146865556993470115272642\ 415851903979451129144885147729011668864419662943666594000705358359171728\ 770903461342085988965538781117252578486118358457572112629629878708284803\ 566699602946787971579694327562084814956581543448574849851625281436565335\ 059303042932609363921404526100583493508090930085108912637066115811618562\ 866977262619596023882801081946332376920335459788381392885759470815172669\ 05083210887168687 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 2, 5, 13, 35, 98, 284, 850, 2620, 8293, 26877, 88936, 299676, 1025799, 3559492, 12497714, 44330844, 158647584, 572166049, 2077579863, 7589109818, 27869278312, 102828000679, 381007809398, 1417130631828, 5289097625745, 19802149582497, 74350387047977, 279891843176997, 1056189599282035, 3994453226478852, 15137872867151505, 57477842110994136, 218628547704482714, 832976248795730420, 3178566180991349186, 12146768306843537621, 46481796210474838449, 178099596815045645008, 683233366935639659604, 2624050448891200732295, 10088939818661762848852, 38829741478365531696370, 149591308856649193952303, 576833012997589208380935, 2226261950327047601573595, 8599372268040610226959593, 33243347424089143323088437, 128609955185016843047310376, 497922369066800207193538185, 1929088819422099397557679351, 7478827947199777654734485473, 29013063318037591195511931060, 112621441318522462531802836294, 437426841346633326970377602868, 1699951966187966748896489877779, 6610059068707389989556530720994, 25715953064829716138941129214497, 100096782482326500795729376640488, 389807230751296072289288587229087, 1518743516298625595992410322860621, 5919930960503634144429324404068424, 23085525119643767598999162810360345, 90063200775499924997223336041562830, 351506541906451553200199003982415269, 1372435903414139243908123868711419545, 5360654789983210624270517848507215300, 20946210820170182618124705593193522535, 81874743386978273584664548546616411224, 320144840874586227448118708175379694758, 1252248659356755036654190743299106226778, 4899793712359578298880541032308564558132, 19178030332340581078517377593563307976303, 75087077695421623030713579283509463188595, 294074644743583330143398766233526339034058, 1152066315066776960973682926662547709977188, 4514621940855463007126690320405022487350323, 17696441503164526169763774570377246777151704, 69385371945155286343156652033093954642844708, 272122351340874837449252300449397610329327003, 1067509745548733438306898082350608681350080947, 4188783918255121424899614375738507328237003105, 16440301864314258336680330622122522732565579451, 64540851021762351442001090494438003359619418283, 253431215137570091655396737631560810977494374084, 995367792733727233615526757982371751055647830353, 3910234049776701716493387280285733907810944857339, 15364390233981142995228716046252802341858718593687, 60383614911144878613107458548874487074206854966956, 237362422918222705596463289008032727672906008773364, 933236786808067542850718110241424998087879233085504, 3669921514601354449315071326059180207652724703913809, 14434602599726461390783810665900400741816868842522400, 56785069895648254613542279121750595276912035303828677, 223430771678854375859298519251712201706234657200762006, 879284794606442595066323556756383773864113712132991555, 3460925272083861895830713350377389360576326175241493927, 13624775074687391308980888625856334724921960639174520498, 53646247722632332526407902934692908025632743818892976045] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 98 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH H + HH H HH H + H HH H H + HH HHH HH 4+ H H H + H H + HH HH 3+ H H + HH HH + HH HH 2+ HH HH H + H H HHH + HH HH H HH + H H HH H 1+ HH H HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 2 (x + 4 x + 6 x + x - 4 x - x + 1) P(x) 5 4 3 2 4 3 2 + (-2 x - 5 x - 2 x + 5 x + 2 x - 2) P(x) + x + x - x - x + 1 = 0 The sequence a(n) satisfies the linear recurrence (4 n - 11) a(n - 1) (n - 5) a(n - 2) (13 n - 50) a(n - 3) a(n) = ------------------- + ---------------- - -------------------- n - 2 n - 2 n - 2 3 (n - 5) a(n - 4) 2 (7 n - 32) a(n - 5) (11 n - 52) a(n - 6) - ------------------ + --------------------- + -------------------- n - 2 n - 2 n - 2 3 (n - 5) a(n - 7) + ------------------ n - 2 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 3, a(5) = 5, a(6) = 12, a(7) = 25] Just for fun, a(1000), equals 2892929439472754275639110962905703645191735357727095508899750650229296998699\ 650250417702312500933508160542528875502594632077456926638556488398534818\ 079656524695088294746931723795733904343776817447344284277088298632959438\ 188606625737348008403331499207787392501687009279451173947644782842916335\ 590525409631737159819076331285702848545178346291410865239808433214918383\ 281914221503672809369919922485847168730055063398763474767613956751723358\ 415883171519041713477635602814169613 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 3, 5, 12, 25, 58, 133, 317, 766, 1891, 4741, 12070, 31132, 81253, 214281, 570355, 1530619, 4137681, 11258218, 30810977, 84762419, 234280495, 650288121, 1811922049, 5066241036, 14210585700, 39976144968, 112758502436, 318834665353, 903583180747, 2566163949117, 7302124082604, 20816325699097, 59442392855164, 170011454608695, 486974888157768, 1396825475257367, 4011890812853959, 11537047874589090, 33216120587777263, 95737962218271909, 276232723988433327, 797807894315299271, 2306386750043567194, 6673543245670751681, 19326479099806588116, 56014905144409127161, 162477391986594286662, 471634639910545402699, 1370022841010420490639, 3982411331837222431784, 11583726101568771675180, 33714965843652599376280, 98188034231868635978533, 286118252811544040061832, 834207468756277346585543, 2433520415189651964513444, 7102638836095197443964402, 20740544969288284445616965, 60593857605397876664140545, 177107831170009544910107609, 517893841632468570994296682, 1515064053665165872006908549, 4434067310543872167835372653, 12982211772826628997427774566, 38024593915428031841352012403, 111415240793505685251038663461, 326575523622006641554282378168, 957583807196547533996136789550, 2808791050774407741236831189929, 8241515930178006758508408779475, 24189982354208029625605722410770, 71023278166409550987269096563282, 208592503751217119003468709661551, 612809900635087260733886986413235, 1800853439393854697704563428184329, 5293623629682496252191869130693941, 15564907984243243464692606902277257, 45777883078979090173886494418209242, 134672050453738930657164966639280471, 396286196017995203972219876472995851, 1166399487230985962929210807281724423, 3433917863777186820431620943202976729, 10111931029595215538737170868048324250, 29783614937181281767108547739284676661, 87743999090625245792470311866733184860, 258554336136611861603433787894215869383, 762041063139495898839722169297774206845, 2246440452120897232386872856999881987840, 6623680132878369046356230732069581005915, 19533932545131858458849086269537812355069, 57618767371669844191146990726871805226294, 169988801410185552666885303822533699739870, 501599289695207480435044190215096219606659, 1480375913258479345517696378084308279464613, 4369823586894576366277873359867799338612566, 12901225049094700532567305828699773777530580, 38095314014096636968163517056229082815915739] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 99 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ H H HH HH + HH H HH H HH HH + H HH H HH H H + HH HHH HHH HH 4+ H H H H + H H + HH HH 3+ H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 5 4 3 2 3 (x + x + x - x + x - 7 x + 6 x - 6 x + 4 x - 1) P(x) 8 7 6 5 4 3 2 2 + (-x - x + 2 x - 6 x + 20 x - 19 x + 19 x - 12 x + 3) P(x) 6 5 4 3 2 6 5 + (-2 x + 8 x - 19 x + 20 x - 20 x + 12 x - 3) P(x) + x - 3 x 4 3 2 + 6 x - 7 x + 7 x - 4 x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 4, 10, 26, 69, 189, 531, 1523, 4450, 13214, 39780, 121172, 372891, 1157841, 3623489, 11418526, 36203892, 115417568, 369752948, 1189764048, 3843571893, 12461608602, 40535790235, 132253277271, 432683157926, 1419169488072, 4665679809568, 15372228705037, 50749615741351, 167858114645913, 556174679248023, 1845823755059302, 6135286363018579, 20422293738167404, 68071030678607611, 227182295436852998, 759119291286332542, 2539454961242279477, 8504323028655683235, 28509103072952810991, 95664229539551293317, 321304721230615703668, 1080107330420398455253, 3633968154476867921190, 12236101253073327771110, 41232266481503696468400, 139042969030779048041655, 469207500295043194438913, 1584426908012077559082580, 5353759647242302582711820, 18101463737508612665231862, 61238721345847747586354528, 207293732805554267115798277, 702076567647642127991807288, 2379097096190062476833679245, 8066049030852425998164233444, 27360411277453582873751872343, 92851718962255312374894771844, 315250452506273969045910379547, 1070811869452775673491187100550, 3638780889056815692040459024008, 12370230860068586634945510386030, 42070054248670052431362659905985, 143131815899553811758894827182731, 487149057270703125062234646569936, 1658613157297942721468241860376256, 5649124002599740384858589655957265, 19247099641481377559718403853727263, 65598406843458652891180758967331743, 223645887529048389088765981495230867, 762718297432695244570175224605077280, 2601951540566875064455988310331463664, 8878965317999131504081627961387394721, 30307499767371689951538029767632018749, 103480633340087847478989350770067364496, 353415847282997180329248471744679259016, 1207335082161837405643710869388932905386, 4125546513140157081286007702955841621874, 14100812735651950796521349139393431620179, 48207323979457308426987111029314384408133, 164848675435334280936699227763763239552423, 563843909687862405166076860861542144433452, 1928993878579247815302182418095671374399968, 6600836082939154735478247652070099358279401, 22592323306175497701179024342565073986525120, 77341835360696272354829013053901365635981487, 264824127411455282514667678545959960907654228, 906959845730606181063193387959722649685189370, 3106733574915940100214493014745981883423735082, 10643965997427390464140238370244468735194690270, 36474101462178486234855005429729702429117790729, 125010237968610953366324125500550622027722926100, 428533401674535257688426246651092314785182656607, 1469265249889580091631256537411827251420367838696, 5038375576963596942207298259524014860830560826605, 17280412006485239330048902712602133465055318778619, 59277426731427766870737721414470831784123695522883] ------------------------------------------------------------ Theorem Number, 100 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH H + HH H H HH + H HH HH H + HH H H HH 4+ H HH HH H + H HH HH H + HH HH H HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 7 6 5 4 3 3 (x - 2 x - 5 x - 3 x - 2 x + 3 x + x - 1) P(x) 7 6 5 4 3 2 2 + (3 x + 8 x + 5 x + 6 x - 7 x + x - 3 x + 3) P(x) 7 6 5 4 3 2 6 4 3 + (-x - 4 x - 2 x - 6 x + 5 x - 2 x + 3 x - 3) P(x) + x + 2 x - x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 43, 88, 181, 389, 845, 1831, 4021, 8945, 19980, 44876, 101560, 231010, 527584, 1210453, 2788476, 6445177, 14945402, 34762125, 81073108, 189553176, 444229775, 1043335656, 2455327148, 5789089278, 13673287432, 32347923587, 76645650172, 181868152606, 432131947999, 1028094409010, 2448927381166, 5840049815756, 13942114199763, 33318698172213, 79702566442900, 190836157036246, 457334328621662, 1096917010956494, 2633078585793776, 6325404794268179, 15206619342001349, 36583326860847321, 88069812063175602, 212154504630779797, 511384369072275705, 1233394476770162448, 2976499939131468798, 7187032770060977636, 17362981676250949732, 41968309776585627857, 101492313014921186269, 245556679552748148628, 594388103914626682818, 1439399709664903706555, 3487218741318126496572, 8451957378207794626600, 20493205067305036075159, 49708582948760009479704, 120619210443382870051297, 292792615812251132529932, 710980136125717425588637, 1727046021860570120941582, 4196575501209940797812052, 10200621702455464613659962, 24802455440798284389096014, 60324716048583750707955995, 146765765674535755148007651, 357173774922625517418331537, 869473358467539692845081275, 2117150070857988843015817831, 5156588999541444162109803583, 12562783839142431859361805544, 30613913629937142427400197707, 74620580935239731515082935125, 181929247599493983737210197721, 443657637304923176383168028461, 1082162432233326872644453144656, 2640180374641018514672900497678, 6442717456740989437135783799985, 15725218378158726422687662180370, 38389658112166505912696121858576, 93738863949937324569728179979473, 228934372538591942169266458710404, 559224587509088449210338239175827, 1366292159965124659559109224182072, 3338728826968157284819565578655591, 8160132627279040287185529051465709, 19947574345781438581105186436718539, 48770603600956548080699610819303235, 119261355989728246212509483676650821, 291684525929087013741255678799558755] ------------------------------------------------------------ Theorem Number, 101 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H + HH HH + H H + HH HH 2+ H H H H + HHH HHH H H + HH H HH H HH HH + H HH H HH H H 1+ H H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 2 (x + 4 x + 6 x + x - 4 x - x + 1) P(x) 5 4 3 2 4 3 2 + (-2 x - 5 x - 2 x + 5 x + 2 x - 2) P(x) + x + x - x - x + 1 = 0 The sequence a(n) satisfies the linear recurrence (4 n - 11) a(n - 1) (n - 5) a(n - 2) (13 n - 50) a(n - 3) a(n) = ------------------- + ---------------- - -------------------- n - 2 n - 2 n - 2 3 (n - 5) a(n - 4) 2 (7 n - 32) a(n - 5) (11 n - 52) a(n - 6) - ------------------ + --------------------- + -------------------- n - 2 n - 2 n - 2 3 (n - 5) a(n - 7) + ------------------ n - 2 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 3, a(5) = 5, a(6) = 12, a(7) = 25] Just for fun, a(1000), equals 2892929439472754275639110962905703645191735357727095508899750650229296998699\ 650250417702312500933508160542528875502594632077456926638556488398534818\ 079656524695088294746931723795733904343776817447344284277088298632959438\ 188606625737348008403331499207787392501687009279451173947644782842916335\ 590525409631737159819076331285702848545178346291410865239808433214918383\ 281914221503672809369919922485847168730055063398763474767613956751723358\ 415883171519041713477635602814169613 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 3, 5, 12, 25, 58, 133, 317, 766, 1891, 4741, 12070, 31132, 81253, 214281, 570355, 1530619, 4137681, 11258218, 30810977, 84762419, 234280495, 650288121, 1811922049, 5066241036, 14210585700, 39976144968, 112758502436, 318834665353, 903583180747, 2566163949117, 7302124082604, 20816325699097, 59442392855164, 170011454608695, 486974888157768, 1396825475257367, 4011890812853959, 11537047874589090, 33216120587777263, 95737962218271909, 276232723988433327, 797807894315299271, 2306386750043567194, 6673543245670751681, 19326479099806588116, 56014905144409127161, 162477391986594286662, 471634639910545402699, 1370022841010420490639, 3982411331837222431784, 11583726101568771675180, 33714965843652599376280, 98188034231868635978533, 286118252811544040061832, 834207468756277346585543, 2433520415189651964513444, 7102638836095197443964402, 20740544969288284445616965, 60593857605397876664140545, 177107831170009544910107609, 517893841632468570994296682, 1515064053665165872006908549, 4434067310543872167835372653, 12982211772826628997427774566, 38024593915428031841352012403, 111415240793505685251038663461, 326575523622006641554282378168, 957583807196547533996136789550, 2808791050774407741236831189929, 8241515930178006758508408779475, 24189982354208029625605722410770, 71023278166409550987269096563282, 208592503751217119003468709661551, 612809900635087260733886986413235, 1800853439393854697704563428184329, 5293623629682496252191869130693941, 15564907984243243464692606902277257, 45777883078979090173886494418209242, 134672050453738930657164966639280471, 396286196017995203972219876472995851, 1166399487230985962929210807281724423, 3433917863777186820431620943202976729, 10111931029595215538737170868048324250, 29783614937181281767108547739284676661, 87743999090625245792470311866733184860, 258554336136611861603433787894215869383, 762041063139495898839722169297774206845, 2246440452120897232386872856999881987840, 6623680132878369046356230732069581005915, 19533932545131858458849086269537812355069, 57618767371669844191146990726871805226294, 169988801410185552666885303822533699739870, 501599289695207480435044190215096219606659, 1480375913258479345517696378084308279464613, 4369823586894576366277873359867799338612566, 12901225049094700532567305828699773777530580, 38095314014096636968163517056229082815915739] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 102 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H + HH + HH HH + H H + H H 3+ H H + H H + HH H + H H + H H 2+ H H H H + HH HH H H + H H H H H H + HH H HH H HH HH + H H H H H H 1+ H H H H H H + H H H H H H + H H H H H H + HH HH HH HH HH HH +H HH HH H -*-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 5 4 3 2 2 (x - x - 3 x - x + 4 x + 6 x + x - 4 x - x + 1) P(x) 7 5 4 3 2 5 4 2 + (2 x - 4 x - 5 x - x + 5 x + 2 x - 2) P(x) + x + x - x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 1, 1, 2, 4, 7, 15, 29, 61, 127, 271, 586, 1284, 2851, 6401, 14528, 33290, 76959, 179339, 420963, 994662, 2364307, 5650550, 13571285, 32741773, 79316223, 192861905, 470563080, 1151736762, 2827109374, 6958049903, 17167211458, 42452181624, 105200309767, 261208021774, 649756213067, 1619035095366, 4040697134706, 10099670334604, 25279556358129, 63358802672237, 158996185196047, 399464668161477, 1004740965736694, 2529812039268552, 6376129439265816, 16085657549407848, 40617486655086834, 102650478258730707, 259636115804741083, 657217503786048531, 1664860404141452886, 4220432959077922278, 10706151265174191810, 27176559799636294517, 69028531975005217927, 175438486812291859892, 446141746173007201393, 1135175174326041517925, 2889919674420257510760, 7360933087674662702827, 18758409049146171218297, 47826371351778426624173, 121994382563029184515362, 311319455992724823415243, 794804160537153370095763, 2029996005196491065924440, 5186871274062376282446698, 13258223448198050178705836, 33902317674660164993362575, 86722642295960754001377990, 221916593448940994006321966, 568063053741139676307806450, 1454615936876682135322731839, 3725983311900217306369726158, 9547071137390468894115840462, 24469900240113377669214108747, 62736933815488987081687982903, 160894000268832831086440842640, 412741793134750931643941634128, 1059097119722585880864665634906, 2718370879626463051457619565941, 6979015727134402004113883394055, 17922115529659196187567283920954, 46035322551173580481730925186008, 118276164309522454450874267190602, 303951856047645514295978341829247, 781288429499139011734727825531415, 2008697788259892889838102290450524, 5165496879477621065689421148927643, 13286226741382655254979104080622679, 34180712388182352411790013754992185, 87952536583905315844522451217854971, 226360906527982608810828063429502302, 582691028785537848253212044104105443, 1500227810151160279886420960353428046, 3863280561577537668583192110278752568, 9950242989134400621172019920081191727, 25632314653984564319749433172509735893, 66041514186845210984568247528134518963] ------------------------------------------------------------ Theorem Number, 103 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H H + HHH HH H + HH H H HH + H HH H H 3+ H H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 (x + 3 x + 5 x + 4 x + x - 2 x - x - x + 1) P(x) 6 5 4 3 2 4 + (-2 x - 3 x - 2 x + 2 x + x + 2 x - 2) P(x) + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 2, 5, 12, 23, 49, 112, 253, 579, 1358, 3227, 7748, 18813, 46125, 114013, 283897, 711584, 1794015, 4546671, 11577180, 29604270, 75993121, 195754315, 505862986, 1311058676, 3407029364, 8875702299, 23175084811, 60639980667, 158983044883, 417578461535, 1098672159454, 2895297229743, 7641346128597, 20195697132110, 53447096195890, 141622665569225, 375711616241302, 997842401302760, 2652950875842241, 7060456732514495, 18808358609957636, 50149116458310735, 133829293008862050, 357435036603330737, 955399082257794831, 2555636540797558539, 6841096693744642690, 18325314910917937883, 49120613334009563827, 131750124641517182896, 353591823117062576312, 949526236370539620659, 2551264891918074784473, 6858651978260183087958, 18447941994455093871814, 49644919963685742485007, 133663142170818395702285, 360040301506875909353955, 970255659958764262554670, 2615834388900122750507795, 7055324647718768885562043, 19037086039122051642684945, 51387200762903904407707405, 138763435324440300691465335, 374848303703121681265255860, 1012958261247020257079200669, 2738283054836353056704524953, 7404767361195354074854837770, 20030255658327162082560716465, 54200020376631062771396758558, 146705486580589416371369182716, 397212990517280503627304325553, 1075789001720150725327259291407, 2914431574550658198089515312139, 7897695965796251173360710095773, 21407381708490941390986231724109, 58041713566580520540000612211825, 157408247812499632998921407099676, 426994713831049906072706401927609, 1158570706633902438570168165164839, 3144307401839123283932256343422602, 8535467976231382791366044856647984, 23175394141450371675485289924071740, 62939317377251538244357435249004576, 170966009465362716902406440776984218, 464502640812098078131177397502868347, 1262278429524071490537345499684630382, 3430904761435168832449731839903173677, 9327102072221139737607603519725203248, 25361063671374774615982527071315358111, 68971398680913371077401476687705762424, 187607269403742559963414016714492354623, 510396439637631516248262323508946565135, 1388805072843575940295077209058378663716, 3779627748100460288176596967204517247922, 10287960378019101772156447629346127060676] ------------------------------------------------------------ Theorem Number, 104 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H H + H HH HHH + HH H H HH + H HH HH H 3+ H H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH H H 1+ H H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 3 2 (x + 4 x + 4 x + 2 x - 3 x - x + 1) P(x) 7 6 5 4 3 2 6 4 3 + (-2 x - 4 x - x - x + 5 x - x + 2 x - 2) P(x) + x + x - 2 x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 16, 30, 56, 110, 216, 427, 860, 1746, 3569, 7364, 15307, 32009, 67334, 142416, 302663, 646031, 1384486, 2977909, 6426645, 13911978, 30200915, 65732898, 143413923, 313595163, 687144819, 1508571730, 3317922569, 7309678595, 16129313045, 35643218886, 78875659428, 174774870620, 387750760231, 861260255477, 1915124365594, 4262999134446, 9498739083878, 21184957996924, 47291188848596, 105658705445218, 236257445857091, 528694503932384, 1183990265577090, 2653392279620791, 5950474873926279, 13353228687666239, 29984296182635404, 67369519671951483, 151455416547760267, 340681282599710033, 766734008185798102, 1726495929499880140, 3889579288927147642, 8766943991330924627, 19769471877891215161, 44600139100704220528, 100661893945840143266, 227287461047749510237, 513406564122610083700, 1160158342765246864333, 2622633765230015042003, 5930856706035176271770, 13416885846141972833882, 30362383327524771174841, 68733011599111138672309, 155645297432095952323778, 352568632584310655173635, 798885376847742072720203, 1810734038572540097719042, 4105354542744134743448103, 9310415132809649662747530, 21120611967784825680218897, 47924750352934920938803331, 108774247977720255491636377, 246946153325595537553849633, 560771040004278719756311221, 1273718100463445848541909110, 2893762556614309851409015460, 6575848332156955986692361543, 14946434073696128773556355402, 33979577457896340549152610437, 77266402617848204240564334985, 175733104543382019162751390086, 399764798964070432229145750542, 909581419077380496710545668751, 2069963664743655297207114411687, 4711574821783439857678518992020, 10726298267792692107831401356403, 24423746955362712321076292407699, 55622647415401209237519460346954, 126696994280216636321751775842705, 288638769125621515697430788944262] ------------------------------------------------------------ Theorem Number, 105 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 4+ H + HH + HH HH + H H + H H 3+ H H + H H + H HH + H H + H H 2+ H H H H H H + H H HH HH HH HH + H H H H H H H H H H + HH H HH H HH HH H HH H HH + H H H H H H H H H H 1+ H H H H H H + H H + H H + HH HH +H H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 5 4 3 2 3 (x + x + x - x + x - 7 x + 6 x - 6 x + 4 x - 1) P(x) 8 7 6 5 4 3 2 2 + (-x - x + 2 x - 6 x + 20 x - 19 x + 19 x - 12 x + 3) P(x) 6 5 4 3 2 6 5 + (-2 x + 8 x - 19 x + 20 x - 20 x + 12 x - 3) P(x) + x - 3 x 4 3 2 + 6 x - 7 x + 7 x - 4 x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 4, 10, 26, 69, 189, 531, 1523, 4450, 13214, 39780, 121172, 372891, 1157841, 3623489, 11418526, 36203892, 115417568, 369752948, 1189764048, 3843571893, 12461608602, 40535790235, 132253277271, 432683157926, 1419169488072, 4665679809568, 15372228705037, 50749615741351, 167858114645913, 556174679248023, 1845823755059302, 6135286363018579, 20422293738167404, 68071030678607611, 227182295436852998, 759119291286332542, 2539454961242279477, 8504323028655683235, 28509103072952810991, 95664229539551293317, 321304721230615703668, 1080107330420398455253, 3633968154476867921190, 12236101253073327771110, 41232266481503696468400, 139042969030779048041655, 469207500295043194438913, 1584426908012077559082580, 5353759647242302582711820, 18101463737508612665231862, 61238721345847747586354528, 207293732805554267115798277, 702076567647642127991807288, 2379097096190062476833679245, 8066049030852425998164233444, 27360411277453582873751872343, 92851718962255312374894771844, 315250452506273969045910379547, 1070811869452775673491187100550, 3638780889056815692040459024008, 12370230860068586634945510386030, 42070054248670052431362659905985, 143131815899553811758894827182731, 487149057270703125062234646569936, 1658613157297942721468241860376256, 5649124002599740384858589655957265, 19247099641481377559718403853727263, 65598406843458652891180758967331743, 223645887529048389088765981495230867, 762718297432695244570175224605077280, 2601951540566875064455988310331463664, 8878965317999131504081627961387394721, 30307499767371689951538029767632018749, 103480633340087847478989350770067364496, 353415847282997180329248471744679259016, 1207335082161837405643710869388932905386, 4125546513140157081286007702955841621874, 14100812735651950796521349139393431620179, 48207323979457308426987111029314384408133, 164848675435334280936699227763763239552423, 563843909687862405166076860861542144433452, 1928993878579247815302182418095671374399968, 6600836082939154735478247652070099358279401, 22592323306175497701179024342565073986525120, 77341835360696272354829013053901365635981487, 264824127411455282514667678545959960907654228, 906959845730606181063193387959722649685189370, 3106733574915940100214493014745981883423735082, 10643965997427390464140238370244468735194690270, 36474101462178486234855005429729702429117790729, 125010237968610953366324125500550622027722926100, 428533401674535257688426246651092314785182656607, 1469265249889580091631256537411827251420367838696, 5038375576963596942207298259524014860830560826605, 17280412006485239330048902712602133465055318778619, 59277426731427766870737721414470831784123695522883] ------------------------------------------------------------ Theorem Number, 106 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 3+ H H + H H H H + HH HH HH HH 2.5+ H H H H + H H H H + H H H H + H H H H 2+ H H H H H H + H H HH H H HH + H H H H H H H H 1.5+ HH HH HH HH HH HH HH HH + H H H H H H H H + H H H H H H H H 1+ H H H H H H + H H H H + H H H H + H H H H 0.5+ H H H H + H H H H +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 (x + 3 x + 5 x + 4 x + x - 2 x - x - x + 1) P(x) 6 5 4 3 2 4 + (-2 x - 3 x - 2 x + 2 x + x + 2 x - 2) P(x) + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 2, 5, 12, 23, 49, 112, 253, 579, 1358, 3227, 7748, 18813, 46125, 114013, 283897, 711584, 1794015, 4546671, 11577180, 29604270, 75993121, 195754315, 505862986, 1311058676, 3407029364, 8875702299, 23175084811, 60639980667, 158983044883, 417578461535, 1098672159454, 2895297229743, 7641346128597, 20195697132110, 53447096195890, 141622665569225, 375711616241302, 997842401302760, 2652950875842241, 7060456732514495, 18808358609957636, 50149116458310735, 133829293008862050, 357435036603330737, 955399082257794831, 2555636540797558539, 6841096693744642690, 18325314910917937883, 49120613334009563827, 131750124641517182896, 353591823117062576312, 949526236370539620659, 2551264891918074784473, 6858651978260183087958, 18447941994455093871814, 49644919963685742485007, 133663142170818395702285, 360040301506875909353955, 970255659958764262554670, 2615834388900122750507795, 7055324647718768885562043, 19037086039122051642684945, 51387200762903904407707405, 138763435324440300691465335, 374848303703121681265255860, 1012958261247020257079200669, 2738283054836353056704524953, 7404767361195354074854837770, 20030255658327162082560716465, 54200020376631062771396758558, 146705486580589416371369182716, 397212990517280503627304325553, 1075789001720150725327259291407, 2914431574550658198089515312139, 7897695965796251173360710095773, 21407381708490941390986231724109, 58041713566580520540000612211825, 157408247812499632998921407099676, 426994713831049906072706401927609, 1158570706633902438570168165164839, 3144307401839123283932256343422602, 8535467976231382791366044856647984, 23175394141450371675485289924071740, 62939317377251538244357435249004576, 170966009465362716902406440776984218, 464502640812098078131177397502868347, 1262278429524071490537345499684630382, 3430904761435168832449731839903173677, 9327102072221139737607603519725203248, 25361063671374774615982527071315358111, 68971398680913371077401476687705762424, 187607269403742559963414016714492354623, 510396439637631516248262323508946565135, 1388805072843575940295077209058378663716, 3779627748100460288176596967204517247922, 10287960378019101772156447629346127060676] ------------------------------------------------------------ Theorem Number, 107 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H H + H HH HHH + HH H H HH + H HH HH H 3+ H H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH H H 1+ H H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The first, 100, terms of the sequence are [1, 0, 0, 1, 1, 2, 4, 9, 24, 60, 151, 394, 1046, 2803, 7577, 20672, 56877, 157609, 439476, 1232405, 3473826, 9837404, 27975596, 79862166, 228782463, 657502101, 1895183557, 5477529504, 15871135303, 46093637019, 134156502474, 391250989983, 1143177397439, 3346056525698, 9809937898529, 28805058978828, 84703394610356, 249416863284610, 735377289590846, 2170816867268179, 6415586185392840, 18981192074546899, 56216057812952382, 166657406130716492, 494533093405906067, 1468768198533248994, 4365965582607571259, 12988521040216695146, 38670141064288649907, 115216266585458677624, 343526375962277334985, 1024947778159343815242, 3060036802514688944303, 9141625911156210002359, 27326336774148420033478, 81731688390002974875938, 244591345676178877325905, 732359418513529967909911, 2193974232033164195454515, 6575893847276181131579434, 19719071539490310156528111, 59158792466863516862302320, 177560536464210335759141900, 533164702558364726323850161, 1601614270672007464769307317, 4813158616931792659000545158, 14470132556772652791282275637, 43519068147883870133172246338, 130932152089321592489342222412, 394064940051121145733955662859, 1186422600019618975005753200114, 3573194878425425140057361677005, 10765034428515308637417988091151, 32442290187763309082576099564241, 97800494973297889796395708906284, 294917358881496473621090950841384, 889581569431415898246553894998166, 2684070646417611108499950392028342, 8100681182244151677950353870259278, 24454872647551389402214368663540425, 73845238829040271913678377111379204, 223043668549023423559510471795932256, 673852424733175119559196857125969552, 2036312757623963527947824831952907597, 6154975884371544098079022062354492798, 18608351784692367111133478791658387137, 56271270503598943879763209311365271859, 170200356246312441013631303948441692517, 514904742147357260804476040090922298262, 1558058759403899816858569433556438493152, 4715515676629230868373078656530365719032, 14274501533652824293041521976538413712666, 43219242868508588953738142350692497195028, 130880791720766245941394083533047962753890, 396419859199165433041764191799973037073792, 1200919390421552440386535026613869338940760, 3638727789189328394413420135715504259639126, 11027089014341616933249130344827052173431310, 33423058909319883140543026017112624577638239, 101322047817008507894739342394380433880472519, 307208105810704761235181655296787607838141197] ------------------------------------------------------------ Theorem Number, 108 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 19 17 16 15 14 13 12 11 10 9 8 (x - 3 x - 3 x + x + 6 x + 5 x + 5 x - 4 x - 7 x - 12 x - x 7 6 5 4 3 3 16 14 13 + x + 10 x + 2 x + 2 x - 4 x - x + 1) P(x) + (3 x - 6 x - 6 x 12 11 10 9 8 6 5 4 3 2 - 3 x + 7 x + 7 x + 16 x - x - 18 x - 2 x - 6 x + 10 x - x 2 13 11 10 9 8 7 6 5 + 3 x - 3) P(x) + (3 x - 3 x - 3 x - 6 x + 2 x - x + 10 x - x 4 3 2 10 6 5 4 3 2 + 6 x - 8 x + 2 x - 3 x + 3) P(x) + x - 2 x + x - 2 x + 2 x - x + x - 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 5, 8, 13, 29, 58, 109, 224, 469, 962, 2004, 4252, 9040, 19301, 41563, 89980, 195525, 426833, 935706, 2058178, 4541767, 10053231, 22314028, 49652620, 110747551, 247557767, 554494496, 1244340818, 2797369080, 6299090767, 14206237723, 32085826176, 72567625371, 164336472104, 372610625607, 845820085338, 1922098715982, 4372439993387, 9956343355891, 22692518763045, 51766984724848, 118193127925529, 270074678950222, 617606728681446, 1413389735165284, 3236829392842055, 7417758864228774, 17010163891685398, 39031546594762198, 89615630050729955, 205874421628959923, 473218932824758023, 1088313251325592598, 2504201652272691910, 5765011816121712227, 13278193824888813318, 30596973864393382114, 70536151123238024372, 162679316350017450709, 375347930528973340133, 866385011098516035066, 2000587011422283874074, 4621343255190585868302, 10679186807475470377805, 24686666622984511599253, 57086890984494449675458, 132055217721375981546693, 305573531184986377077384, 707314945608209847639832, 1637732567993388063478190, 3793171352332367061541348, 8787952385570629968379636, 20365508673681019718977166, 47208682141769560467266124, 109462253636911869115750581, 253874894309395675440725721, 588959015626156096858744585, 1366650733136206654486516020, 3172009302049365401821933578, 7363990486154164923351386224, 17099812077001249363177056269, 39716082047248874128446242438, 92264807603841023580831955900, 214386848828249772877195858443, 498253569181261894314960926934, 1158219376625202957075370231454, 2692882159291120533526978907126, 6262215947461674813212714459568, 14565353079098544379221773377774, 33883983435461603108197061482155, 78839999338725373583734755178870, 183474523257584763789771107474471, 427051528522691727246479996839525, 994165194829588457736112300985513, 2314776392397718522798745297649492] ------------------------------------------------------------ Theorem Number, 109 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 7 6 5 4 3 3 (x - 2 x - 5 x - 3 x - 2 x + 3 x + x - 1) P(x) 7 6 5 4 3 2 2 + (3 x + 8 x + 5 x + 6 x - 7 x + x - 3 x + 3) P(x) 7 6 5 4 3 2 6 4 3 + (-x - 4 x - 2 x - 6 x + 5 x - 2 x + 3 x - 3) P(x) + x + 2 x - x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 43, 88, 181, 389, 845, 1831, 4021, 8945, 19980, 44876, 101560, 231010, 527584, 1210453, 2788476, 6445177, 14945402, 34762125, 81073108, 189553176, 444229775, 1043335656, 2455327148, 5789089278, 13673287432, 32347923587, 76645650172, 181868152606, 432131947999, 1028094409010, 2448927381166, 5840049815756, 13942114199763, 33318698172213, 79702566442900, 190836157036246, 457334328621662, 1096917010956494, 2633078585793776, 6325404794268179, 15206619342001349, 36583326860847321, 88069812063175602, 212154504630779797, 511384369072275705, 1233394476770162448, 2976499939131468798, 7187032770060977636, 17362981676250949732, 41968309776585627857, 101492313014921186269, 245556679552748148628, 594388103914626682818, 1439399709664903706555, 3487218741318126496572, 8451957378207794626600, 20493205067305036075159, 49708582948760009479704, 120619210443382870051297, 292792615812251132529932, 710980136125717425588637, 1727046021860570120941582, 4196575501209940797812052, 10200621702455464613659962, 24802455440798284389096014, 60324716048583750707955995, 146765765674535755148007651, 357173774922625517418331537, 869473358467539692845081275, 2117150070857988843015817831, 5156588999541444162109803583, 12562783839142431859361805544, 30613913629937142427400197707, 74620580935239731515082935125, 181929247599493983737210197721, 443657637304923176383168028461, 1082162432233326872644453144656, 2640180374641018514672900497678, 6442717456740989437135783799985, 15725218378158726422687662180370, 38389658112166505912696121858576, 93738863949937324569728179979473, 228934372538591942169266458710404, 559224587509088449210338239175827, 1366292159965124659559109224182072, 3338728826968157284819565578655591, 8160132627279040287185529051465709, 19947574345781438581105186436718539, 48770603600956548080699610819303235, 119261355989728246212509483676650821, 291684525929087013741255678799558755] ------------------------------------------------------------ Theorem Number, 110 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ H HH HH + HH H HH HH + H HH H H + HH H HH HH 4+ H HH H H + H HH H H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 3 2 (x + 4 x + 4 x + 2 x - 3 x - x + 1) P(x) 7 6 5 4 3 2 6 4 3 + (-2 x - 4 x - x - x + 5 x - x + 2 x - 2) P(x) + x + x - 2 x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 16, 30, 56, 110, 216, 427, 860, 1746, 3569, 7364, 15307, 32009, 67334, 142416, 302663, 646031, 1384486, 2977909, 6426645, 13911978, 30200915, 65732898, 143413923, 313595163, 687144819, 1508571730, 3317922569, 7309678595, 16129313045, 35643218886, 78875659428, 174774870620, 387750760231, 861260255477, 1915124365594, 4262999134446, 9498739083878, 21184957996924, 47291188848596, 105658705445218, 236257445857091, 528694503932384, 1183990265577090, 2653392279620791, 5950474873926279, 13353228687666239, 29984296182635404, 67369519671951483, 151455416547760267, 340681282599710033, 766734008185798102, 1726495929499880140, 3889579288927147642, 8766943991330924627, 19769471877891215161, 44600139100704220528, 100661893945840143266, 227287461047749510237, 513406564122610083700, 1160158342765246864333, 2622633765230015042003, 5930856706035176271770, 13416885846141972833882, 30362383327524771174841, 68733011599111138672309, 155645297432095952323778, 352568632584310655173635, 798885376847742072720203, 1810734038572540097719042, 4105354542744134743448103, 9310415132809649662747530, 21120611967784825680218897, 47924750352934920938803331, 108774247977720255491636377, 246946153325595537553849633, 560771040004278719756311221, 1273718100463445848541909110, 2893762556614309851409015460, 6575848332156955986692361543, 14946434073696128773556355402, 33979577457896340549152610437, 77266402617848204240564334985, 175733104543382019162751390086, 399764798964070432229145750542, 909581419077380496710545668751, 2069963664743655297207114411687, 4711574821783439857678518992020, 10726298267792692107831401356403, 24423746955362712321076292407699, 55622647415401209237519460346954, 126696994280216636321751775842705, 288638769125621515697430788944262] ------------------------------------------------------------ Theorem Number, 111 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 7+ HHH + H HH + HH HH 6+ H H + H H + HH HH 5+ HHH HH HH + H HH H H + HH HHH HH 4+ H H H + HH HH 3+ HH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 19 17 16 15 14 13 12 11 10 9 8 (x - 3 x - 3 x + x + 6 x + 5 x + 5 x - 4 x - 7 x - 12 x - x 7 6 5 4 3 3 16 14 13 + x + 10 x + 2 x + 2 x - 4 x - x + 1) P(x) + (3 x - 6 x - 6 x 12 11 10 9 8 6 5 4 3 2 - 3 x + 7 x + 7 x + 16 x - x - 18 x - 2 x - 6 x + 10 x - x 2 13 11 10 9 8 7 6 5 + 3 x - 3) P(x) + (3 x - 3 x - 3 x - 6 x + 2 x - x + 10 x - x 4 3 2 10 6 5 4 3 2 + 6 x - 8 x + 2 x - 3 x + 3) P(x) + x - 2 x + x - 2 x + 2 x - x + x - 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 5, 8, 13, 29, 58, 109, 224, 469, 962, 2004, 4252, 9040, 19301, 41563, 89980, 195525, 426833, 935706, 2058178, 4541767, 10053231, 22314028, 49652620, 110747551, 247557767, 554494496, 1244340818, 2797369080, 6299090767, 14206237723, 32085826176, 72567625371, 164336472104, 372610625607, 845820085338, 1922098715982, 4372439993387, 9956343355891, 22692518763045, 51766984724848, 118193127925529, 270074678950222, 617606728681446, 1413389735165284, 3236829392842055, 7417758864228774, 17010163891685398, 39031546594762198, 89615630050729955, 205874421628959923, 473218932824758023, 1088313251325592598, 2504201652272691910, 5765011816121712227, 13278193824888813318, 30596973864393382114, 70536151123238024372, 162679316350017450709, 375347930528973340133, 866385011098516035066, 2000587011422283874074, 4621343255190585868302, 10679186807475470377805, 24686666622984511599253, 57086890984494449675458, 132055217721375981546693, 305573531184986377077384, 707314945608209847639832, 1637732567993388063478190, 3793171352332367061541348, 8787952385570629968379636, 20365508673681019718977166, 47208682141769560467266124, 109462253636911869115750581, 253874894309395675440725721, 588959015626156096858744585, 1366650733136206654486516020, 3172009302049365401821933578, 7363990486154164923351386224, 17099812077001249363177056269, 39716082047248874128446242438, 92264807603841023580831955900, 214386848828249772877195858443, 498253569181261894314960926934, 1158219376625202957075370231454, 2692882159291120533526978907126, 6262215947461674813212714459568, 14565353079098544379221773377774, 33883983435461603108197061482155, 78839999338725373583734755178870, 183474523257584763789771107474471, 427051528522691727246479996839525, 994165194829588457736112300985513, 2314776392397718522798745297649492] ------------------------------------------------------------ Theorem Number, 112 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 13 12 11 10 9 8 7 6 3 2 (x + x - 2 x - 4 x - 3 x + 3 x + 4 x + 6 x - 3 x - x + 1) P(x) 10 9 8 7 6 5 3 2 7 + (2 x + 2 x - 2 x - 4 x - 6 x + x + 5 x - x + 2 x - 2) P(x) + x 6 3 2 + x - 2 x + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 6, 10, 19, 33, 58, 108, 200, 369, 693, 1317, 2514, 4829, 9344, 18189, 35592, 69989, 138246, 274200, 545940, 1090783, 2186343, 4395255, 8860180, 17906170, 36272959, 73639706, 149804622, 305325432, 623406188, 1274971376, 2611597375, 5357319501, 11004919758, 22635485192, 46614898206, 96108678206, 198370269901, 409866012138, 847687251598, 1754836188328, 3636009242250, 7540193770668, 15649198235420, 32504074898770, 67562356231747, 140533256797216, 292514736909839, 609254134880606, 1269754726390174, 2647896576486016, 5524994718069158, 11534607248785519, 24093773197898278, 50353464184534819, 105285448864725132, 220249118224532352, 460956131234061995, 965156511973390642, 2021725152698343735, 4236688329900954289, 8881883438230807832, 18627397182916201898, 39080715060247020314, 82022115622422152747, 172207762570687826247, 361678877010846007041, 759867396916920746124, 1596954324667818296589, 3357245924609224890483, 7060019872578142781287, 14851047729721395149780, 31248783492263029171750, 65770414878714109406951, 138466981593815657698333, 291592875976612309859386, 614213871210552443656780, 1294110799316931990023822, 2727279498663073507672105, 5748989295111391492048961, 12121446207030768286650281, 25563244831836776909695275, 53922960715096892884159306, 113769377614144303450467979, 240087037262751143517465894, 506759120361514589789712132, 1069847544922340934122333579, 2259059523710516236809422262, 4771083098316998636523021295, 10078315424868890475447526836, 21293093745583655561384647024, 44995356141694588627315053710, 95098362701380027278219259613, 201026434433379944826868248653] ------------------------------------------------------------ Theorem Number, 113 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H H H + HH H HH H HH H + H HH H HH H H + HH HHH HHH HH 4+ H H H H H + H H HHH + HH HH H HH + H H HH H 3+ H H H + HH HH + H H + HH HH 2+ H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 5 4 2 2 2 1 + (x + 2 x - 3 x + 1) P(x) + (3 x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence 2 (2 n - 7) a(n - 1) 6 (2 n - 7) a(n - 3) a(n) = -------------------- + 3 a(n - 2) - -------------------- - 2 a(n - 4) n - 2 n - 2 (7 n - 26) a(n - 5) 2 (2 n - 7) a(n - 6) + ------------------- + -------------------- n - 2 n - 2 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 4, a(6) = 9] Just for fun, a(1000), equals 3912713036158617318509638377138375533316224831546861582313093049117204732275\ 161433935146016149091473334219917287963346644573872724477108108490662812\ 418334736856586843709683084008320340869441608374880662673099589147552205\ 830294518867174015950410987205757465351935365656500404973518910520743623\ 785521779786139878853622027801281653903223035708705335807691685705680149\ 923534758306240094876196172049724395557982812195469651963074016638111982\ 825962397780150983709295031802928437449326507419430548794098456215393473\ 160670422813528453629843358361364696749022906559143549640475433181881751\ 1555823344431927 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 2, 4, 9, 23, 64, 191, 599, 1948, 6508, 22194, 76921, 270099, 958699, 3433841, 12394906, 45042074, 164642011, 604938541, 2232965570, 8276456989, 30790837634, 114937415514, 430361256871, 1615930514318, 6083172039808, 22954478946444, 86807407106172, 328949166304185, 1248882163375950, 4749841871072163, 18094713236195971, 69038884750550145, 263793556627774938, 1009312130495860923, 3866726896447161971, 14831562294610740000, 56954359015957128256, 218946131853219424616, 842545682854457000293, 3245435586350481740697, 12512815359874744138387, 48285890048233493203843, 186487887787147450650659, 720824372882141752217882, 2788310451132590455188226, 10793717234723256916153893, 41812443444966289857489007, 162080913312787705271367156, 628692130239368684964481259, 2440128652773944665150379912, 9476443457107766603493128026, 36823578634023918245193333929, 143167736653314208985832900284, 556921504494237968167924270128, 2167524811001772020077841940860, 8440097840602431798959551478812, 32880368237839515628261887600747, 128151743065287809285055291807976, 499694128714597750607105498017505, 1949258473299776834923941709564340, 7607009510412367439252914014588545, 29698329308576542196284911203749633, 115989339089514598795194860702302774, 453175946726232134647408787407649664, 1771223312925338953544644706612032242, 6925205756611443687104319331650006281, 27085717953244928584205812101284691495, 105972200036252637919408586223133525241, 414747033797373851267664531558280985037, 1623717073702891957920414652489718075519, 6358714384375125812366417845135689282190, 24909011492792325532267219541176243510280, 97604166250571754937852753518416641235755, 382561731608277139967229456305800147422005, 1499867007339640933059665732220450957858386, 5881917387818175554353173720384776726987785, 23072626404834871538982385749170888540843408, 90528268145519859087877361349671642484219190, 355285664643481955065144442356256730823783677, 1394680680600919857153498947880692379708437554, 5476119873721278429026356679356176822162331170, 21506497894723661816480900253109218602012265154, 84481701156511546225748955826674992505340413066, 331932297697720781808713021710378580736467417117, 1304451885963764997935077313216908390215091543824, 5127387850986518477878846736116468586611966704193, 20158200592348975036606398088507152110400171935126, 79267077842170516694290002695248155068539160126279, 311757924011422470856093767396218908056581237327213, 1226376606738929530058576487836836484004639545377427, 4825142216582587629435043232038875461136363081430994, 18987795286431987988878870023809849221780371271780197, 74733520259223163181361215739458589304197116516950277, 294192220239883687604579139951836378795556080975488124, 1158297697715193592607760854721400777343544742077515466, 4561219132336158884770415177955097222191477834680167284, 17964367613470277900488559109318508065913414572826620747] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 114 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 2+ H H H H + HH HH HH HH + HH HH HH HH + H H H H H H H H + H H H H H H H H 1.5+ H HH H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H 1+ H H H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H + H H H H H H H H 0.5+ HH H H H H H H HH + H H H H H H H H + H H H H H H H H +H HH HH HH H +H HH HH HH H -*-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-*+-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 2 3 2 1 + (x + 2 x + 2 x - x - 3 x + 1) P(x) + (x + 3 x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence (3 n - 10) a(n - 1) (3 n - 8) a(n - 2) 2 (4 n - 15) a(n - 3) a(n) = ------------------- + ------------------ - --------------------- n - 2 n - 2 n - 2 (5 n - 18) a(n - 4) 2 (2 n - 9) a(n - 5) 5 (n - 4) a(n - 6) - ------------------- + -------------------- + ------------------ n - 2 n - 2 n - 2 3 (n - 4) a(n - 7) + ------------------ n - 2 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 3, a(6) = 6, a(7) = 11] Just for fun, a(1000), equals 1145247889784214898460130876069454497699198945209860899382804351131772528282\ 133731708344230812745657626667920291158213887747665601298356727927944776\ 030647389055090332321758191672696872867145930254564580810357005416798425\ 235280336717321020337663125309372117332677581781296695840911474312424353\ 109238551195056650414827517078765491770489867998868102973083917330485510\ 399043443105952336355540946876142869551090788288618966120768479952859514\ 795957207874664550283182756465317216 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 2, 3, 6, 11, 23, 49, 111, 260, 631, 1571, 3998, 10351, 27183, 72222, 193765, 524152, 1427918, 3913839, 10785131, 29860663, 83023346, 231707013, 648868066, 1822699373, 5134485859, 14501047305, 41051801922, 116470558685, 331116492750, 943112497105, 2690965323833, 7690669891937, 22013351910519, 63100511237332, 181120160013138, 520539572163698, 1497829936462364, 4314837199837584, 12443230444884596, 35920670606415061, 103794869271147607, 300197286521208002, 868996122922789762, 2517626225693246437, 7299790270596987425, 21181646928330883083, 61507036728946177581, 178727788506850517912, 519695401194125167769, 1512109642114056218557, 4402345591213388938044, 12824513841726472678399, 37380380499502693529327, 109014237042999565846414, 318090431309338867876386, 928618961502649280575439, 2712290291874573897385879, 7925725232544474142754753, 23170692019202436270094753, 67768744210089068725511744, 198291428265081823208297909, 580438829259492218057110181, 1699735035454079974628644198, 4979351238773816135366226285, 14592370164402130306544769983, 42779476843772237493471224940, 125457578082203857583718106213, 368048952058834293418581033508, 1080083138285505007796767563496, 3170644490401645406403850948413, 9310493512451363834365498930284, 27348205767536943373804271403750, 80354890729020603260436860850253, 236167229309395496479949093808441, 694300446923883916492764288370007, 2041702813338529207790328193243177, 6005536220991345043954472164788529, 17669418816060756131445534993099489, 51999730668275388400877829723757133, 153068396153069627851719888972166535, 450684823412170479752466953309262686, 1327274512101663332984053719899520102, 3909727981272968899078067938135412974, 11519348937895569170200408275308392163, 33947097692236579161442595800013425794, 100061841976125938681066186451886631053, 295000900653572234496534231947047618528, 869891621951872165451134841310889932774, 2565617645671780764466166332700650893684, 7568360110286842554479538057942198692096, 22330212873582761969352565997259163646634, 65896655484064735791290558939780113001569, 194496409912782516835188524447856832105677, 574163843122884606850063550924349788068566, 1695252941486902059890063951026991901471039, 5006174931932292551782715214924205548964976, 14785937476305642720296210359047476219203690] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 115 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH H H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 6 5 3 2 3 6 5 3 2 2 1 + (x - 2 x - x + x + x - 1) P(x) + (2 x + x - 2 x - 2 x + 3) P(x) 3 2 + (x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 5, 11, 28, 76, 211, 603, 1765, 5264, 15937, 48839, 151202, 472225, 1485993, 4706844, 14994745, 48013004, 154436696, 498783089, 1616850496, 5258677510, 17155495909, 56122636054, 184069825882, 605132075407, 1993718666131, 6581959257798, 21770219454604, 72132609144399, 239393313735202, 795717620574009, 2648693476182953, 8828624078450717, 29465173088811459, 98457437552634323, 329369272090148464, 1103027614595877015, 3697721671613957100, 12408068651321649605, 41674994654944877763, 140097075013073845341, 471353613687260606177, 1587128931879860902479, 5348224467495188034491, 18035349882379680910115, 60861612575446205158642, 205519739753068972678883, 694453035266610905992136, 2348011630987864114638090, 7943557937841427302485261, 26889170203032263300230997, 91070528550580309538274714, 308608282745862412163668173, 1046304728325015826334785857, 3549128824657861954759780787, 12044556132054484371730517178, 40893867252310281572689219400, 138904770482112069033827176260, 472021002594407564000257953774, 1604665713022310408351788221724, 5457341266348644083436776116764, 18567157426246872236210550366105, 63193474271403094358924753152457, 215157522081650694011677394062788, 732813623056840252859387665367371, 2496770705583635702188829698635748, 8509568535300028664477282084567852, 29011888168716459915175295573805305, 98941849113584962386331971475792390, 337532653082144472910275106796152219, 1151806792788709943749157992804603915, 3931588267430959334859999826417811290, 13423865794966053710448317354049404630, 45846386205274160315105968060094314811, 156620078987872854793233932094448257435, 535182127030444009092860857528364660887, 1829214861034953381096946003245244760703, 6253656635830852640527004901838412684886, 21384882714487230010667560715941555639300, 73144332311382758073193270954887001912118, 250237841171995850710686988931368133141850, 856291055935496524320477080966931768867563, 2930782940632295758122099872732927105983101, 10033154805261385069977353492775158958494793, 34354279676122797433331027757543454548418875, 117655326756676047434274855936575922349640998, 403021081540695451691200672776973956592503536, 1380789361838233623157088485298286083558419854, 4731607803480864352462590028891014127013768721, 16216976478813348552294470551221507198158908641, 55591595066663998204221299846137505201070752220, 190600836132803636720816502467271281512353886715, 653604834893454714182973008142627835759411284024, 2241707379109917718335468588579714695453663057368, 7689786265286437722107208751585796001746302957504, 26382735148028890878132386512999786745614107129861] ------------------------------------------------------------ Theorem Number, 116 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 6 5 3 2 3 1 + (x - 3 x - x + 2 x + x - 1) P(x) 6 5 3 2 2 3 2 + (3 x + x - 4 x - 2 x + 3) P(x) + (2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 24, 50, 100, 207, 447, 965, 2097, 4641, 10355, 23208, 52434, 119238, 272302, 624715, 1439586, 3329035, 7723353, 17974332, 41946862, 98136543, 230136785, 540859834, 1273645217, 3004841530, 7101517437, 16810654036, 39854525348, 94621732620, 224950428542, 535465514846, 1276128784737, 3044720948720, 7272175007841, 17386856915967, 41609801199087, 99670684111928, 238955727084254, 573360774386420, 1376837089630860, 3308764126191171, 7957256958283794, 19149694328506228, 46115679249370442, 111125023198040762, 267941826636175204, 646433889341639098, 1560460708781195478, 3768924211315976522, 9107707171651752850, 22020112055371883983, 53264882178765929245, 128903877018767159586, 312095509354848105514, 755960176581781422429, 1831866563301055461943, 4440851470520127715854, 10769869026085811825291, 26128886252691178913799, 63415069809483920442764, 153964356091782976724423, 373937747991975936017966, 908500784048788194813896, 2207973525844179375897501, 5367855788092394722780239, 13053960160887471570162186, 31755167086049496587497855, 77270472173378202155837143, 188077261542308295134223872, 457909065550704597273291014, 1115165194681535923031110438, 2716521249676194665104160483, 6619085378281378272215833397, 16132103128989793305097961362, 39326870592610533254429937069, 95893803896808211685222661778, 233879375049458391590317811366, 570546544663074707446614357597, 1392148793303975476592593878598, 3397608858830076013984401770568, 8293771449881197142382854212644, 20249746445904137513663328249672, 49450867782510794412659596235458, 120785019511666197733598404565551, 295076860334680412091808688807665, 721005064284596647207054443970788, 1762060340462307761691050403140135, 4307058934161988158319017861316542, 10529718506027909698423650877921477, 25747021917792169515323378704085982, 62966561145792275331841952840077732, 154015405894044114004769528686025159] ------------------------------------------------------------ Theorem Number, 117 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 3+ H H + H H H H + HH HH HH HH 2.5+ H H H H + H H H H + H H H H + H H H H 2+ H H H H H + H H HH H H + H H H H H H 1.5+ HH HH HH HH HH HH + H H H H H H + H H H H H H 1+ H H H H H H + H H H H H H + H H H H H H + H H H H H H 0.5+ H H H H H H + H H H H H H +H HH HH H -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 2 3 2 1 + (x + 2 x + 2 x - x - 3 x + 1) P(x) + (x + 3 x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence (3 n - 10) a(n - 1) (3 n - 8) a(n - 2) 2 (4 n - 15) a(n - 3) a(n) = ------------------- + ------------------ - --------------------- n - 2 n - 2 n - 2 (5 n - 18) a(n - 4) 2 (2 n - 9) a(n - 5) 5 (n - 4) a(n - 6) - ------------------- + -------------------- + ------------------ n - 2 n - 2 n - 2 3 (n - 4) a(n - 7) + ------------------ n - 2 subject to the initial conditions [a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 3, a(6) = 6, a(7) = 11] Just for fun, a(1000), equals 1145247889784214898460130876069454497699198945209860899382804351131772528282\ 133731708344230812745657626667920291158213887747665601298356727927944776\ 030647389055090332321758191672696872867145930254564580810357005416798425\ 235280336717321020337663125309372117332677581781296695840911474312424353\ 109238551195056650414827517078765491770489867998868102973083917330485510\ 399043443105952336355540946876142869551090788288618966120768479952859514\ 795957207874664550283182756465317216 For the sake of the OEIS here are the first, 100, terms. [1, 0, 1, 1, 2, 3, 6, 11, 23, 49, 111, 260, 631, 1571, 3998, 10351, 27183, 72222, 193765, 524152, 1427918, 3913839, 10785131, 29860663, 83023346, 231707013, 648868066, 1822699373, 5134485859, 14501047305, 41051801922, 116470558685, 331116492750, 943112497105, 2690965323833, 7690669891937, 22013351910519, 63100511237332, 181120160013138, 520539572163698, 1497829936462364, 4314837199837584, 12443230444884596, 35920670606415061, 103794869271147607, 300197286521208002, 868996122922789762, 2517626225693246437, 7299790270596987425, 21181646928330883083, 61507036728946177581, 178727788506850517912, 519695401194125167769, 1512109642114056218557, 4402345591213388938044, 12824513841726472678399, 37380380499502693529327, 109014237042999565846414, 318090431309338867876386, 928618961502649280575439, 2712290291874573897385879, 7925725232544474142754753, 23170692019202436270094753, 67768744210089068725511744, 198291428265081823208297909, 580438829259492218057110181, 1699735035454079974628644198, 4979351238773816135366226285, 14592370164402130306544769983, 42779476843772237493471224940, 125457578082203857583718106213, 368048952058834293418581033508, 1080083138285505007796767563496, 3170644490401645406403850948413, 9310493512451363834365498930284, 27348205767536943373804271403750, 80354890729020603260436860850253, 236167229309395496479949093808441, 694300446923883916492764288370007, 2041702813338529207790328193243177, 6005536220991345043954472164788529, 17669418816060756131445534993099489, 51999730668275388400877829723757133, 153068396153069627851719888972166535, 450684823412170479752466953309262686, 1327274512101663332984053719899520102, 3909727981272968899078067938135412974, 11519348937895569170200408275308392163, 33947097692236579161442595800013425794, 100061841976125938681066186451886631053, 295000900653572234496534231947047618528, 869891621951872165451134841310889932774, 2565617645671780764466166332700650893684, 7568360110286842554479538057942198692096, 22330212873582761969352565997259163646634, 65896655484064735791290558939780113001569, 194496409912782516835188524447856832105677, 574163843122884606850063550924349788068566, 1695252941486902059890063951026991901471039, 5006174931932292551782715214924205548964976, 14785937476305642720296210359047476219203690] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 118 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 3+ H H + H H H H + HH HH HH HH 2.5+ H H H H + H H H H + H H H H + H H H H 2+ H H H H H + H H H H HH + H H H H H H 1.5+ HH HH HH HH HH HH + H H H H H H + H H H H H H 1+ H H H H H H + H H H H H H + H H H H H H + H H H H H H 0.5+ H H H H H H + H H H H H H +H HH HH H -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-*+-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 7 6 5 4 3 2 2 4 3 2 1 + (x + 2 x + 2 x + x - x - 3 x + 1) P(x) + (x + x + 3 x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 1, 1, 2, 3, 5, 9, 16, 30, 58, 116, 239, 506, 1097, 2427, 5463, 12477, 28848, 67394, 158836, 377178, 901495, 2166867, 5234213, 12699051, 30930290, 75598789, 185359855, 455784844, 1123668716, 2776873715, 6877498572, 17068229197, 42439024789, 105707007672, 263725516660, 658967894157, 1648915263775, 4131583829601, 10365363804074, 26035798398589, 65470688357808, 164811011072201, 415302561925389, 1047513699571611, 2644544397931547, 6682191535862199, 16898420509896319, 42767701945366204, 108321066048548465, 274550837288738769, 696357117682632614, 1767372462308960760, 4488478712318269104, 11406021832716039226, 29001653107017572484, 73782782204311220057, 187811324578820557484, 478315299598201141936, 1218777069501669912391, 3107019537324250284192, 7924393223792736964588, 20220103680933308897535, 51616603776023631799277, 131818952594264610824282, 336777197104868241753975, 860752297305782872714291, 2200791837469090360102192, 5629111296473102966453270, 14403096346209760209248825, 36865688736322206868993176, 94391915584188999964681518, 241762486800264653818633439, 619413388059785426667077461, 1587471328993104064922016395, 4069687452247926398792681845, 10436202123998847589674392197, 26769897363547185608920104004, 68686345920426647296144500475, 176282993708252565466531022075, 452546969785698724707339960583, 1162056396731017551043333923461, 2984683088317573075660756014720, 7667856918842597699990931757657, 19703887367936868082006875399023, 50644169471185007417823062512338, 130197961586413959569843116196477, 334790997742176319121603071307000, 861065043064209575846281411734462, 2215075682539561817347226211311255, 5699404812652210175645233057118791, 14667521630026537828113104987883985, 37754463990531766806166131082876284, 97199118700228488192153500861825564, 250286233847847220249286755657934286, 644600218742256194485115463829505246, 1660431912815134961187878791724614232, 4277866004858347047255111494171913904, 11023186660234240231599180829063203219, 28409230651584433392323145563646776771] ------------------------------------------------------------ Theorem Number, 119 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 4 3 2 1 + (x + x + x + x - x - x - x + 1) P(x) + (x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 9, 20, 43, 94, 216, 505, 1198, 2888, 7046, 17359, 43137, 107986, 272046, 689198, 1754652, 4486913, 11519069, 29677863, 76709558, 198858270, 516900481, 1346928260, 3517812058, 9206982999, 24144187305, 63430675750, 166926209959, 439987494362, 1161457799001, 3070255463861, 8126765806794, 21537753472145, 57146877005927, 151798160546375, 403642671170599, 1074387128983240, 2862437766174048, 7633132414976498, 20372410447459630, 54417377540703801, 145469599771072044, 389162759164454635, 1041839048797994230, 2791046978146480666, 7482013275489230337, 20069799089461878199, 53867882389413538721, 144666793314662499979, 388731932228847260560, 1045116905170070363120, 2811280699994517254169, 7565891179843013798830, 20371584272556101353559, 54877087351154823046446, 147894461371360500600404, 398750098993789333077497, 1075552191594669068647971, 2902271444012572374471826, 7834562869658274177153008, 21157109326212823185371492, 57155440767212455161258473, 154459142393974795917404387, 417560856723240437282759617, 1129201890148987098172382917, 3054673524396563025177593345, 8265998275897394297578700464, 22374802833318826872122652538, 60583279272141959914982521230, 164086330649285086653717627671, 444543898955742904386559063447, 1204692560613252897219505954158, 3265532089955788106865668972679, 8854107516472847109120249657859, 24012966115006271427327717275224, 65140965498791000199417379317401, 176753178774204821202519803270294, 479713926425134540506383824683065, 1302257781851035311627912843109992, 3535971195076524443101782142887928, 9603184357286987364921458068688114, 26086412423311750697378497706485168, 70876753808269982532179552022657879, 192611209804903641890607349672943792, 523534801206376171916739816450584025, 1423291500980846384191200772841684210, 3870121247506880066064768564504438295, 10525333776394036585049608115707480862, 28630307203934076060493508135753907692, 77892072069382575356484169897529778150, 211951244648000788109732632639813482287, 576836257741440102366311224036503009129, 1570151092397116337614826283811806160336, 4274655842190291606027844557863134575971] ------------------------------------------------------------ Theorem Number, 120 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 1 + (x + 2 x + x + x - 2 x - x - x + 1) P(x) 4 3 2 + (2 x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 5, 9, 15, 27, 52, 99, 191, 380, 767, 1561, 3212, 6676, 13980, 29462, 62457, 133095, 284904, 612325, 1320835, 2858528, 6204794, 13504896, 29467046, 64443065, 141232079, 310127824, 682241562, 1503390100, 3318125585, 7334286838, 16234066006, 35980324422, 79843282903, 177385141221, 394524626312, 878382600262, 1957594535824, 4366863711975, 9749996633366, 21787566703548, 48726502251068, 109058113252455, 244271064473613, 547511721413636, 1228030388231890, 2756175501803298, 6189779845643445, 13909237626615638, 31273831159967158, 70355639977282178, 158360839466453771, 356631087517226215, 803535502075715446, 1811330859313674816, 4084981786161499396, 9216695405000005915, 20803989056321610682, 46978386063151976380, 106126487292393541621, 239838068767767185118, 542220278033115896654, 1226285592006812766558, 2774347696927967271630, 6278832855959591570828, 14214818277513329553301, 32191694467833655881358, 72926011196323997616532, 165254507682930495490441, 374586998426594973124882, 849331174613651352346800, 1926295937143221477067260, 4370058634352064414375042, 9916689051600915661994716, 22509108319009393216650851, 51104501330215476319608970, 116055707226039146541380078, 263619574451311192836750119, 598949259458766324199929106, 1361134842968359225485601500, 3093916712876970178692847268, 7034125719967269865497972628, 15995707321136515987498854278, 36381989171910295666778485227, 82766964402142036623410315006, 188327268860574074593683978712, 428600872006951760458663393481, 975606498851851793659981066972, 2221141899698881971264987257058, 5057736078329066998909302286110, 11518941426334046103018614433700, 26238795102693120770431186038224, 59778984596015548358887856377059, 136215011845586736829636505040690] ------------------------------------------------------------ Theorem Number, 121 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H H H + HH H HH H HH H + H HH H HH H H + HH HHH HHH HH 4+ H H H H H + H H HHH + HH HH H HH + H H HH H 3+ H H H + HH HH + H H + HH HH 2+ H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 6 5 3 2 3 6 5 3 2 2 1 + (x - 2 x - x + x + x - 1) P(x) + (2 x + x - 2 x - 2 x + 3) P(x) 3 2 + (x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 5, 11, 28, 76, 211, 603, 1765, 5264, 15937, 48839, 151202, 472225, 1485993, 4706844, 14994745, 48013004, 154436696, 498783089, 1616850496, 5258677510, 17155495909, 56122636054, 184069825882, 605132075407, 1993718666131, 6581959257798, 21770219454604, 72132609144399, 239393313735202, 795717620574009, 2648693476182953, 8828624078450717, 29465173088811459, 98457437552634323, 329369272090148464, 1103027614595877015, 3697721671613957100, 12408068651321649605, 41674994654944877763, 140097075013073845341, 471353613687260606177, 1587128931879860902479, 5348224467495188034491, 18035349882379680910115, 60861612575446205158642, 205519739753068972678883, 694453035266610905992136, 2348011630987864114638090, 7943557937841427302485261, 26889170203032263300230997, 91070528550580309538274714, 308608282745862412163668173, 1046304728325015826334785857, 3549128824657861954759780787, 12044556132054484371730517178, 40893867252310281572689219400, 138904770482112069033827176260, 472021002594407564000257953774, 1604665713022310408351788221724, 5457341266348644083436776116764, 18567157426246872236210550366105, 63193474271403094358924753152457, 215157522081650694011677394062788, 732813623056840252859387665367371, 2496770705583635702188829698635748, 8509568535300028664477282084567852, 29011888168716459915175295573805305, 98941849113584962386331971475792390, 337532653082144472910275106796152219, 1151806792788709943749157992804603915, 3931588267430959334859999826417811290, 13423865794966053710448317354049404630, 45846386205274160315105968060094314811, 156620078987872854793233932094448257435, 535182127030444009092860857528364660887, 1829214861034953381096946003245244760703, 6253656635830852640527004901838412684886, 21384882714487230010667560715941555639300, 73144332311382758073193270954887001912118, 250237841171995850710686988931368133141850, 856291055935496524320477080966931768867563, 2930782940632295758122099872732927105983101, 10033154805261385069977353492775158958494793, 34354279676122797433331027757543454548418875, 117655326756676047434274855936575922349640998, 403021081540695451691200672776973956592503536, 1380789361838233623157088485298286083558419854, 4731607803480864352462590028891014127013768721, 16216976478813348552294470551221507198158908641, 55591595066663998204221299846137505201070752220, 190600836132803636720816502467271281512353886715, 653604834893454714182973008142627835759411284024, 2241707379109917718335468588579714695453663057368, 7689786265286437722107208751585796001746302957504, 26382735148028890878132386512999786745614107129861] ------------------------------------------------------------ Theorem Number, 122 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH H + HH H HH H + H HH H H + HH HHH HH 4+ H H H HH + H H HH H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 4 3 2 1 + (x + x + x + x - x - x - x + 1) P(x) + (x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 9, 20, 43, 94, 216, 505, 1198, 2888, 7046, 17359, 43137, 107986, 272046, 689198, 1754652, 4486913, 11519069, 29677863, 76709558, 198858270, 516900481, 1346928260, 3517812058, 9206982999, 24144187305, 63430675750, 166926209959, 439987494362, 1161457799001, 3070255463861, 8126765806794, 21537753472145, 57146877005927, 151798160546375, 403642671170599, 1074387128983240, 2862437766174048, 7633132414976498, 20372410447459630, 54417377540703801, 145469599771072044, 389162759164454635, 1041839048797994230, 2791046978146480666, 7482013275489230337, 20069799089461878199, 53867882389413538721, 144666793314662499979, 388731932228847260560, 1045116905170070363120, 2811280699994517254169, 7565891179843013798830, 20371584272556101353559, 54877087351154823046446, 147894461371360500600404, 398750098993789333077497, 1075552191594669068647971, 2902271444012572374471826, 7834562869658274177153008, 21157109326212823185371492, 57155440767212455161258473, 154459142393974795917404387, 417560856723240437282759617, 1129201890148987098172382917, 3054673524396563025177593345, 8265998275897394297578700464, 22374802833318826872122652538, 60583279272141959914982521230, 164086330649285086653717627671, 444543898955742904386559063447, 1204692560613252897219505954158, 3265532089955788106865668972679, 8854107516472847109120249657859, 24012966115006271427327717275224, 65140965498791000199417379317401, 176753178774204821202519803270294, 479713926425134540506383824683065, 1302257781851035311627912843109992, 3535971195076524443101782142887928, 9603184357286987364921458068688114, 26086412423311750697378497706485168, 70876753808269982532179552022657879, 192611209804903641890607349672943792, 523534801206376171916739816450584025, 1423291500980846384191200772841684210, 3870121247506880066064768564504438295, 10525333776394036585049608115707480862, 28630307203934076060493508135753907692, 77892072069382575356484169897529778150, 211951244648000788109732632639813482287, 576836257741440102366311224036503009129, 1570151092397116337614826283811806160336, 4274655842190291606027844557863134575971] ------------------------------------------------------------ Theorem Number, 123 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H H HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 16 15 14 13 12 11 10 9 8 7 1 + (x - x + 5 x - 4 x + 9 x - 10 x + 11 x - 15 x + 10 x - 9 x 6 5 4 3 2 4 13 12 11 + 11 x - 3 x + 3 x - 4 x + x - x + 1) P(x) + (x - 2 x + 9 x 10 9 8 7 6 5 4 3 2 - 9 x + 17 x - 16 x + 16 x - 24 x + 7 x - 8 x + 13 x - 3 x + 3 x 3 - 4) P(x) 9 8 7 6 5 4 3 2 2 + (-x + 6 x - 7 x + 13 x - 5 x + 7 x - 14 x + 3 x - 3 x + 6) P(x) 5 4 3 2 + (x - 2 x + 5 x - x + x - 4) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 4, 8, 19, 45, 109, 274, 706, 1853, 4935, 13310, 36294, 99887, 277080, 773890, 2174564, 6142994, 17435967, 49699752, 142207256, 408311561, 1176056410, 3397136562, 9838815780, 28564499121, 83115843915, 242349891944, 708010477632, 2072130084832, 6074676800886, 17836576344636, 52449296429420, 154443347079891, 455370624288193, 1344296224042833, 3973101538881707, 11755524179079394, 34818258116193514, 103229008060692388, 306340787185880424, 909905907985941303, 2704944605526137569, 8047730208573057475, 23962142966274349035, 71400333638264115197, 212903724121787138250, 635274875582057982342, 1896807634533229819937, 5667044361896249801975, 16941461018536018552034, 50675247253133035903840, 151663852459302068188567, 454151178363641840946220, 1360637463611326749125506, 4078495115483496523670002, 12231095766329072899789213, 36697074224144886683988623, 110151723217308796647530048, 330779364211208261048905057, 993725827356903839035757398, 2986550802827264657190871788, 8979306222659152141008830522, 27007218475295484991166837452, 81259853248056922960743254955, 244583100660660153829137673619, 736421486674089753064633215785, 2218051427528819913261839518920, 6682787989701003052006455028430, 20140980458431291139609889355048, 60720662190831894650419213659464, 183114035391954655906522002346728, 552373000705795000525835783987615, 1666730964015327379511768432941102, 5030573883836617758315970625139016, 15187467902598696614512679454444238, 45863373600840634895077218659510962, 138534037449538073999509058682449166, 418556465258795426215264715221405589, 1264899413596602083001491538207902240, 3823487778569295944720069873113936211, 11560129540993990762501709865572681491, 34959286970484081744420684201522549420, 105744303499391353349596937150871992058, 319921705303491560691981225089334416683, 968100804279936279532054055475556884777, 2930120764108356154566020492379108770732, 8870263262264521475883094855050261705051, 26857872251900514412080420502807689627855, 81337144677435763254775297744856216182180, 246369333028750642755806698872490495383761, 746385203567840868272929012734727819752572, 2261602845499645031050459062952094686254883, 6854013617806049203504746064284314278331466, 20775299351072062541291263978452724295707271, 62982773259528325110667531595783021832436437, 190970785698740806851338590811851515473877661] ------------------------------------------------------------ Theorem Number, 124 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 13 12 11 10 9 8 7 6 4 3 2 1 + (x + x + 2 x + x + x - 3 x - x - 4 x + x + 2 x + x - 1) 3 9 8 7 6 5 4 3 2 2 P(x) + (x + 3 x + x + 4 x - x - 2 x - 4 x - 2 x + 3) P(x) 5 4 3 2 + (x + x + 2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 6, 10, 20, 38, 71, 141, 287, 583, 1202, 2521, 5325, 11323, 24282, 52406, 113664, 247757, 542480, 1192313, 2629716, 5818748, 12912436, 28729813, 64079490, 143246101, 320883803, 720195496, 1619321024, 3647055270, 8226810087, 18584803194, 42041964361, 95229445722, 215968599608, 490356105984, 1114569328616, 2536017260869, 5775957109341, 13167432219753, 30044425752651, 68611060035416, 156810360349206, 358665774457819, 820965478557937, 1880467407759439, 4310223033012889, 9885862452845120, 22688139458714478, 52100512493277179, 119710934944822697, 275210416296780580, 633032733279400641, 1456831279717572075, 3354332676055975953, 7726969805373657763, 17807843238326490566, 41058742911104314735, 94707794486141835055, 218547278951873016885, 504520534310020995281, 1165146383227358880327, 2691813944679198933254, 6221103655060471930962, 14382781044804077557878, 33263407167250290261739, 76954607918587159600611, 178091158645065231338585, 412273954575974881103440, 954687857089931041054750, 2211389126498755280117222, 5123817785575377937814225, 11875270986534273727976053, 27530333413891900687626772, 63840220572849222278104788, 148077560084497124425798169, 343552566984827422999062338, 797266374037095566105098803, 1850620324705013667862682500, 4296672645787599932254592314, 9978052601349627898050764057, 23176908557720826553542054224, 53846703494421872456717997242, 125127973383987991144787652593, 290830021503801381930721105937, 676100921983193825969329450941, 1572060634527419283928666538120, 3656037156311472113594424613186, 8504202409975272430583173001536, 19785018164064980155520108020417, 46038106439011476416261131849967, 107145737836783124653205894881028, 249406152369906834599933074660180, 580647632830318254723094356140953, 1352040998078175397319979202051475] ------------------------------------------------------------ Theorem Number, 125 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ H HH HH + HH H HH HH + H HH H H + HH H HH HH 4+ H HH H H + H HH H H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 6 5 3 2 3 1 + (x - 3 x - x + 2 x + x - 1) P(x) 6 5 3 2 2 3 2 + (3 x + x - 4 x - 2 x + 3) P(x) + (2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 24, 50, 100, 207, 447, 965, 2097, 4641, 10355, 23208, 52434, 119238, 272302, 624715, 1439586, 3329035, 7723353, 17974332, 41946862, 98136543, 230136785, 540859834, 1273645217, 3004841530, 7101517437, 16810654036, 39854525348, 94621732620, 224950428542, 535465514846, 1276128784737, 3044720948720, 7272175007841, 17386856915967, 41609801199087, 99670684111928, 238955727084254, 573360774386420, 1376837089630860, 3308764126191171, 7957256958283794, 19149694328506228, 46115679249370442, 111125023198040762, 267941826636175204, 646433889341639098, 1560460708781195478, 3768924211315976522, 9107707171651752850, 22020112055371883983, 53264882178765929245, 128903877018767159586, 312095509354848105514, 755960176581781422429, 1831866563301055461943, 4440851470520127715854, 10769869026085811825291, 26128886252691178913799, 63415069809483920442764, 153964356091782976724423, 373937747991975936017966, 908500784048788194813896, 2207973525844179375897501, 5367855788092394722780239, 13053960160887471570162186, 31755167086049496587497855, 77270472173378202155837143, 188077261542308295134223872, 457909065550704597273291014, 1115165194681535923031110438, 2716521249676194665104160483, 6619085378281378272215833397, 16132103128989793305097961362, 39326870592610533254429937069, 95893803896808211685222661778, 233879375049458391590317811366, 570546544663074707446614357597, 1392148793303975476592593878598, 3397608858830076013984401770568, 8293771449881197142382854212644, 20249746445904137513663328249672, 49450867782510794412659596235458, 120785019511666197733598404565551, 295076860334680412091808688807665, 721005064284596647207054443970788, 1762060340462307761691050403140135, 4307058934161988158319017861316542, 10529718506027909698423650877921477, 25747021917792169515323378704085982, 62966561145792275331841952840077732, 154015405894044114004769528686025159] ------------------------------------------------------------ Theorem Number, 126 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ H HH HH + HH H HH HH + H HH H H + HH H HH HH 4+ H HH H H + H HH H H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 1 + (x + 2 x + x + x - 2 x - x - x + 1) P(x) 4 3 2 + (2 x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 5, 9, 15, 27, 52, 99, 191, 380, 767, 1561, 3212, 6676, 13980, 29462, 62457, 133095, 284904, 612325, 1320835, 2858528, 6204794, 13504896, 29467046, 64443065, 141232079, 310127824, 682241562, 1503390100, 3318125585, 7334286838, 16234066006, 35980324422, 79843282903, 177385141221, 394524626312, 878382600262, 1957594535824, 4366863711975, 9749996633366, 21787566703548, 48726502251068, 109058113252455, 244271064473613, 547511721413636, 1228030388231890, 2756175501803298, 6189779845643445, 13909237626615638, 31273831159967158, 70355639977282178, 158360839466453771, 356631087517226215, 803535502075715446, 1811330859313674816, 4084981786161499396, 9216695405000005915, 20803989056321610682, 46978386063151976380, 106126487292393541621, 239838068767767185118, 542220278033115896654, 1226285592006812766558, 2774347696927967271630, 6278832855959591570828, 14214818277513329553301, 32191694467833655881358, 72926011196323997616532, 165254507682930495490441, 374586998426594973124882, 849331174613651352346800, 1926295937143221477067260, 4370058634352064414375042, 9916689051600915661994716, 22509108319009393216650851, 51104501330215476319608970, 116055707226039146541380078, 263619574451311192836750119, 598949259458766324199929106, 1361134842968359225485601500, 3093916712876970178692847268, 7034125719967269865497972628, 15995707321136515987498854278, 36381989171910295666778485227, 82766964402142036623410315006, 188327268860574074593683978712, 428600872006951760458663393481, 975606498851851793659981066972, 2221141899698881971264987257058, 5057736078329066998909302286110, 11518941426334046103018614433700, 26238795102693120770431186038224, 59778984596015548358887856377059, 136215011845586736829636505040690] ------------------------------------------------------------ Theorem Number, 127 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 7+ HHH + H HH + HH HH 6+ H H + H H + HH HH 5+ HHH HH HH + H HH H H + HH HHH HH 4+ H H H + HH HH 3+ HH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 13 12 11 10 9 8 7 6 4 3 2 1 + (x + x + 2 x + x + x - 3 x - x - 4 x + x + 2 x + x - 1) 3 9 8 7 6 5 4 3 2 2 P(x) + (x + 3 x + x + 4 x - x - 2 x - 4 x - 2 x + 3) P(x) 5 4 3 2 + (x + x + 2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 6, 10, 20, 38, 71, 141, 287, 583, 1202, 2521, 5325, 11323, 24282, 52406, 113664, 247757, 542480, 1192313, 2629716, 5818748, 12912436, 28729813, 64079490, 143246101, 320883803, 720195496, 1619321024, 3647055270, 8226810087, 18584803194, 42041964361, 95229445722, 215968599608, 490356105984, 1114569328616, 2536017260869, 5775957109341, 13167432219753, 30044425752651, 68611060035416, 156810360349206, 358665774457819, 820965478557937, 1880467407759439, 4310223033012889, 9885862452845120, 22688139458714478, 52100512493277179, 119710934944822697, 275210416296780580, 633032733279400641, 1456831279717572075, 3354332676055975953, 7726969805373657763, 17807843238326490566, 41058742911104314735, 94707794486141835055, 218547278951873016885, 504520534310020995281, 1165146383227358880327, 2691813944679198933254, 6221103655060471930962, 14382781044804077557878, 33263407167250290261739, 76954607918587159600611, 178091158645065231338585, 412273954575974881103440, 954687857089931041054750, 2211389126498755280117222, 5123817785575377937814225, 11875270986534273727976053, 27530333413891900687626772, 63840220572849222278104788, 148077560084497124425798169, 343552566984827422999062338, 797266374037095566105098803, 1850620324705013667862682500, 4296672645787599932254592314, 9978052601349627898050764057, 23176908557720826553542054224, 53846703494421872456717997242, 125127973383987991144787652593, 290830021503801381930721105937, 676100921983193825969329450941, 1572060634527419283928666538120, 3656037156311472113594424613186, 8504202409975272430583173001536, 19785018164064980155520108020417, 46038106439011476416261131849967, 107145737836783124653205894881028, 249406152369906834599933074660180, 580647632830318254723094356140953, 1352040998078175397319979202051475] ------------------------------------------------------------ Theorem Number, 128 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 4 3 2 2 1 + (x + 2 x + 2 x + x - 2 x - x - x + 1) P(x) 5 4 3 2 + (x + 2 x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 4, 7, 12, 19, 32, 57, 102, 184, 338, 631, 1193, 2278, 4386, 8512, 16640, 32727, 64698, 128497, 256288, 513091, 1030652, 2076551, 4195389, 8497633, 17251501, 35097685, 71545640, 146109780, 298889934, 612391857, 1256577012, 2581964630, 5312226040, 10942958613, 22568085609, 46593874760, 96296775298, 199214136675, 412508168198, 854927060291, 1773336983575, 3681307349391, 7647959447370, 15900359496772, 33080520927256, 68869800315583, 143471049030427, 299066191297844, 623774018719246, 1301769563196978, 2718180873152235, 5678728121019830, 11869812600023930, 24822684507644365, 51934768256973512, 108708894175429681, 227647361551092841, 476918760095371805, 999549561565926862, 2095736135278881599, 4395777411189686495, 9223512701958825628, 19360365984653297116, 40652059395072526234, 85388375869205205243, 179414611502079962724, 377099148730245402023, 792844459482558248785, 1667444254566230073271, 3507856998483416898418, 7381696118116215545446, 15537845884622259716287, 32714673959131873020032, 68898287436832289381615, 145139401140260625493636, 305823205702351606859661, 644556519941560063235675, 1358796405023640572966082, 2865154456516923094879448, 6042817833843456908071545, 12747540906403185592012971, 26897164309201694344507690, 56764595715167086292090332, 119822245028973861534159847, 252978817559414921672077464, 534214521684417321803286560, 1128314411252867525432280150, 2383557588085094296518911183, 5036170748773190613622743871, 10642723994868396804864305247, 22494743116136658770993174434, 47553615795218186780095160580, 100544600599431729835469608976] ------------------------------------------------------------ Theorem Number, 129 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} To make it crystal clear here is such a path of semi-length 8 6+ H H + HHHH HHHH + H H H H 5+ HH HHH HH H + HH H H HH + H H HH H + HH HHH HH 4+ H H H + H H + HH HH 3+ H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 2 P(x) - 3 P(x) + x + 2 = 0 The sequence a(n) satisfies the linear recurrence 2 (2 n - 3) a(n - 1) a(n) = -------------------- n subject to the initial conditions [a(1) = 1] Just for fun, a(1000), equals 5122940537742595583629721118011068145063594016961973571336624906632686808909\ 664221683174072492771901454389110355172645553815612301161892926508373060\ 953630761788426454813208221982269944853718139764096763670323818312854111\ 522472840281253967424056279986385037883682593079202362580278000997717513\ 916176050889240333946302308060371780217225686149459455971582278174881316\ 427808815517028766512349295334236903877354174181211626901986763826561956\ 922125192308041887962723728737463807731411173669284884156264596304465980\ 743324500384028661550630231750062292424477513997778655003357934700239897\ 72130248615305440 For the sake of the OEIS here are the first, 100, terms. [1, 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, 18367353072152, 69533550916004, 263747951750360, 1002242216651368, 3814986502092304, 14544636039226909, 55534064877048198, 212336130412243110, 812944042149730764, 3116285494907301262, 11959798385860453492, 45950804324621742364, 176733862787006701400, 680425371729975800390, 2622127042276492108820, 10113918591637898134020, 39044429911904443959240, 150853479205085351660700, 583300119592996693088040, 2257117854077248073253720, 8740328711533173390046320, 33868773757191046886429490, 131327898242169365477991900, 509552245179617138054608572, 1978261657756160653623774456, 7684785670514316385230816156, 29869166945772625950142417512, 116157871455782434250553845880, 451959718027953471447609509424, 1759414616608818870992479875972, 6852456927844873497549658464312, 26700952856774851904245220912664, 104088460289122304033498318812080, 405944995127576985730643443367112, 1583850964596120042686772779038896, 6182127958584855650487080847216336, 24139737743045626825711458546273312, 94295850558771979787935384946380125, 368479169875816659479009042713546950, 1440418573150919668872489894243865350, 5632681584560312734993915705849145100, 22033725021956517463358552614056949950, 86218923998960285726185640663701108500, 337485502510215975556783793455058624700, 1321422108420282270489942177190229544600, 5175569924646105559418940193995065716350, 20276890389709399862928998568254641025700, 79463489365077377841208237632349268884500, 311496878311103321137536291518809134027240, 1221395654430378811828760722007962130791020, 4790408930363303911328386208394864461024520, 18793142726809884575211361279087545193250040, 73745243611532458459690151854647329239335600, 289450081175264899454283846029490767264392230, 1136359577947336271931632877004667456667613940, 4462290049988320482463241297506133183499654740, 17526585015616776834735140517915655636396234280, 68854441132780194707888052034668647142985206100, 270557451039395118028642463289168566420671280440, 1063353702922273835973036658043476458723103404520, 4180080073556524734514695828170907458428751314320, 16435314834665426797069144960762886143367590394940, 64633260585762914370496637486146181462681535261000, 254224158304000796523953440778841647086547372026600, 1000134600800354781929399250536541864362461089950800, 3935312233584004685417853572763349509774031680023800, 15487357822491889407128326963778343232013931127835600, 60960876535340415751462563580829648891969728907438000, 239993345518077005168915776623476723006280827488229600, 944973797977428207852605870454939596837230758234904050, 3721443204405954385563870541379246659709506697378694300, 14657929356129575437016877846657032761712954950899755100, 57743358069601357782187700608042856334020731624756611000, 227508830794229349661819540395688853956041682601541047340] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 130 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H H + H H H H H H + H HH H HH HH H + H H H H H H + H HH H H H 2+ H H H H H + H H H H + H H H H + HH HH H HH + H H H H 1+ H H H + H H + H H + HH HH +H H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 2 2 (x + 2 x + 2 x) P(x) + (-2 x - 3 x - 1) P(x) + x + 1 = 0 The sequence a(n) satisfies the linear recurrence (n - 2) a(n - 1) (3 n - 5) a(n - 2) (8 n - 13) a(n - 3) a(n) = ---------------- + 3/2 ------------------ + 1/2 ------------------- n + 1 n + 1 n + 1 (n - 2) a(n - 4) + 3/2 ---------------- n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1] Just for fun, a(1000), equals 7343404681614865434684763507210347427020178928606879284591986851267868435212\ 249556011530055635260692642873173384899274398440091540022367641511462811\ 008588715366229430034654248374874429801785393016846152183226987120054638\ 377243219486082547546109186858130940779431142925904831163236883434548069\ 717001456259229640670551010840014168855155974450769022681600038685269854\ 610212079351914130074608602819585260369730841413939433497804288630925688\ 421813161058031288831930158762023521 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 3, 7, 17, 43, 110, 286, 753, 2003, 5376, 14540, 39589, 108427, 298512, 825664, 2293271, 6393539, 17885835, 50191175, 141247519, 398537101, 1127203038, 3195229662, 9076078057, 25830193513, 73643406563, 210312889095, 601558143229, 1723166250419, 4942839351320, 14196809911512, 40826123674963, 117541006104195, 338779909115505, 977457947592893, 2822979976721987, 8160665572497041, 23611944396102714, 68376681868569674, 198169750797798271, 574782976348372753, 1668371611006438146, 4846075456079115622, 14085849428228221307, 40969362561239301701, 119235949321876092360, 347229239641557994136, 1011757164052160972473, 2949698621922111594973, 8604216545991491941581, 25111278043912438342801, 73323304601568480731897, 214201905567546134575859, 626044889720284349690094, 1830549990322901795312942, 5354823514457078459010313, 15670757112005494774639051, 45878599087262878735384752, 134369299666916034362964172, 393689966411509441770005261, 1153899343965725112108795443, 3383264305703491359375922560, 9923244697941619216569794624, 29115000903164573472390932419, 85451758118279984454158890195, 250877793948601314014941971921, 736777877691289303907872561885, 2164414038110594458093425771443, 6360187870629936233099648197313, 18694848840332640943329103122570, 54965868576235609763116372140890, 161651671060236413178711527166019, 475532533517495969396124008055577, 1399233799660317951564059213015592, 4118201121063760930956059935817164, 12123536696685281838476884788352799, 35698751359647955775282558186295793, 105141975845442998724664146494190416, 309739251405543919680997857296802976, 912664235708260313995444264873081893, 2689789263907547466230267373821274617, 7928951321295704744188805383739325609, 23377681170924021285051888390957923933, 68940317392992307735795082187711452173, 203343055362393936480696319767811457959, 599884647429347603580336902386533043922, 1770054610915592642421723575150356117650, 5223773514153323773797895768878478839271, 15419101377748600955972333826656656984531, 45520727173998870390619930777957739468395, 134410477602615207286484828173887369300119, 396944068749335739220874303030506094827475, 1172455237187575683372434195700210272843517, 3463638304632421047516261496691878570034312, 10233794644880525218408203904987488653900680, 30241783348700004437279186446509997511733421, 89380608824471922293587992331001273395222029] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 131 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H H H + H HH HHH HHH + HH H H HH H HH + H HH H H HH H 3+ H H H H H + HH H HH HH + H HH H H + HH HHH HH 2+ H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 3 2 3 3 2 2 (x - 2 x + 3 x - 2 x + 1) P(x) + (2 x - 6 x + 6 x - 4) P(x) 2 + (3 x - 5 x + 5) P(x) + x - 2 = 0 The sequence a(n) satisfies the linear recurrence 2 (784 n - 4175 n + 5430) a(n - 1) a(n) = -1/6 --------------------------------- n (n - 20) 3 2 (806 n - 6265 n + 15085 n - 11270) a(n - 2) + 1/2 --------------------------------------------- (n - 20) n (n - 1) 3 2 (775 n - 9483 n + 37640 n - 46200) a(n - 3) + 1/3 --------------------------------------------- (n - 20) n (n - 1) 3 2 (2338 n - 24703 n + 83195 n - 84510) a(n - 4) - 1/2 ----------------------------------------------- (n - 20) n (n - 1) 3 2 (13547 n - 160893 n + 589450 n - 604980) a(n - 5) + 1/3 --------------------------------------------------- (n - 20) n (n - 1) 2 (n - 5) (23120 n - 168299 n + 216750) a(n - 6) - 1/6 ----------------------------------------------- (n - 20) n (n - 1) (802 n - 1345) (n - 5) (n - 6) a(n - 7) + 23/6 --------------------------------------- (n - 20) n (n - 1) subject to the initial conditions [a(1) = 1, a(2) = 1, a(3) = 2, a(4) = 5, a(5) = 12, a(6) = 30, a(7) = 81] Just for fun, a(1000), equals FAIL[1000] For the sake of the OEIS here are the first, 100, terms. [1, 1, 1, 2, 5, 12, 30, 81, 229, 662, 1948, 5831, 17710, 54419, 168833, 528155, 1664213, 5277256, 16827858, 53926063, 173576588, 560936415, 1819285007, 5919833795, 19320385526, 63228230571, 207443469767, 682177801649, 2248171612205, 7423837365735, 24560350107571, 81394262559556, 270182631898549, 898217066339078, 2990372473872432, 9969027651170351, 33275945950334338, 111205809076329913, 372062450716585111, 1246147908447295661, 4177965051016674116, 14021011566810825649, 47096943775584809319, 158338272822270193117, 532771736317317709107, 1794081031969168870847, 6046069620444553359431, 20390133033818507041768, 68812816820455499805574, 232385138864995941107851, 785281303758181049919031, 2655271411885935199895101, 8983567960720145181101627, 30411314877899936755434201, 103005075976443659262040713, 349068249999754776625254132, 1183537570562806527314290693, 4014817961334241947551433417, 13625558839596423774916375743, 46263701413779054058541936532, 157151058003782357858755485575, 534046102763843232485954101880, 1815593591097072181893422226844, 6174918323442572324039618112447, 21009271327001690661407916437429, 71507686297257733068586785592808, 243473352912509733096516091480434, 829282227007395032195600984125183, 2825536621845165696860641062544160, 9630366970485697588071779896672647, 32834006993756818115591952855878967, 111979908830449148216631580041483343, 382021401817036743662791711928588378, 1303655934802611838953451304395971225, 4450025332458439029031005586983475661, 15194374443616234519888158067081669447, 51894434580369708393426302508607294107, 177285519217389069991239536808496071653, 605811181128998091111670708159025914981, 2070665674596829532527082350056760357124, 7079272150084604177375712025727064755828, 24208646340580505196888120163464510470975, 82804344693779747107614977000710374753661, 283291800762222662725968711157625925621617, 969417325702992920850057293357303761464115, 3318036475368752735752674264094357994062847, 11359074196872447783790061981782481726423585, 38895019257007873575802925204535284491890554, 133208600389876893412305839417838500907223451, 456305575539147863651253579693219428425657021, 1563372988782039801075232072048889631931692739, 5357360833210372527055319257073755767108049928, 18361952762323231986995172334810224271266626627, 62945512379293241106405162797616205629319743698, 215817641682835340151787194936128973047345821576, 740088660683565258849948110822952227672655645783, 2538362384191128104441030871052512322129732425414, 8707531186263942524078713680230936247912662583419, 29874904075524665800762954407332809563174993694615, 102514815648188524910760889830801959846415911697437, 351830914795587916846468105997185262724999878660635] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 132 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 2 1 + P(x) x + P(x) x + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence 2 (518 n - 1031 n + 780) a(n - 1) a(n) = -1/195 -------------------------------- n + 2 3 2 (1036 n - 4335 n + 4367 n - 1170) a(n - 2) - 1/195 -------------------------------------------- (n + 2) (n + 1) 3 2 (2590 n - 11115 n + 11939 n - 1812) a(n - 3) + 1/195 ---------------------------------------------- (n + 2) (n + 1) 2 (n - 3) (1036 n - 1865 n + 1074) a(n - 4) + 1/65 ------------------------------------------ (n + 2) (n + 1) 161 (74 n - 55) (n - 3) (n - 4) a(n - 5) + --- ------------------------------------ 195 (n + 2) (n + 1) subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 1456286415952438964652364581682787386720721652723427223868652268704217615734\ 013661511014559584083968554107289545538396109682528052843351355027899510\ 336875839145124807012715538480932028100745003225030603205050905923754789\ 120913436417646081847745149534148691270052401108733311860953962982834165\ 142352139116820596104418788220221329115699403613078628786955744338071295\ 785210326056171164845618097 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 4, 8, 13, 31, 71, 144, 318, 729, 1611, 3604, 8249, 18803, 42907, 98858, 228474, 528735, 1228800, 2865180, 6693712, 15676941, 36807239, 86584783, 204060509, 481823778, 1139565120, 2699329341, 6403500057, 15211830451, 36183117255, 86171536894, 205459894230, 490417795075, 1171809537516, 2802705573178, 6709741472212, 16077633792535, 38557470978465, 92543928713413, 222292252187223, 534346494991678, 1285375448785399, 3094086155360823, 7452774657572624, 17962853486458385, 43320517724198861, 104535253974750259, 252390609561764423, 609699521485999914, 1473608895007574938, 3563397472304118410, 8620933920694737411, 20866285270840567272, 50527739882573522706, 122405798503806040965, 296657476936886314309, 719256672330233630085, 1744545018773625965860, 4232966162296687137816, 10274640808649742568633, 24948409910226791411907, 60599469176019760272535, 147244830551438483529790, 357892285312932720768838, 870164978202711215226483, 2116332778149031726542808, 5148677470688468794716894, 12529484860432553111063488, 30499528076281485551927755, 74262935424150986850220725, 180870207357836188170341069, 440630810522439231975963839, 1073724026979876073732588658, 2617084008072054452326466527, 6380388743696091681799439061, 15558886191823487315141504508, 37949779254079097680017924283, 92584221613943667202286169191, 225922433438019608173638548805, 551409374869368431174631180753, 1346105758407373798214652723522, 3286793196606678831626181940722, 8026971507652658919500628768874, 19607185640982275440523729181033, 47902823795394726062462438796736, 117054324948067108888357885549374, 286083328086281443692788683942327, 699317934466646636669713105276839, 1709748244857899463255482766619759, 4180839106559551650627420507840427, 10225086357877339275440446600065760, 25011587559313816213376886296261184, 61190615947546730766871192612064833, 149725673741988363612845353038008205, 366415846156516055604065564079552535, 896845046441850691487935128438196159] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 133 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 2 2 (x + 2 x + 2 x) P(x) + (-2 x - 3 x - 1) P(x) + x + 1 = 0 The sequence a(n) satisfies the linear recurrence (n - 2) a(n - 1) (3 n - 5) a(n - 2) (8 n - 13) a(n - 3) a(n) = ---------------- + 3/2 ------------------ + 1/2 ------------------- n + 1 n + 1 n + 1 (n - 2) a(n - 4) + 3/2 ---------------- n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1] Just for fun, a(1000), equals 7343404681614865434684763507210347427020178928606879284591986851267868435212\ 249556011530055635260692642873173384899274398440091540022367641511462811\ 008588715366229430034654248374874429801785393016846152183226987120054638\ 377243219486082547546109186858130940779431142925904831163236883434548069\ 717001456259229640670551010840014168855155974450769022681600038685269854\ 610212079351914130074608602819585260369730841413939433497804288630925688\ 421813161058031288831930158762023521 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 3, 7, 17, 43, 110, 286, 753, 2003, 5376, 14540, 39589, 108427, 298512, 825664, 2293271, 6393539, 17885835, 50191175, 141247519, 398537101, 1127203038, 3195229662, 9076078057, 25830193513, 73643406563, 210312889095, 601558143229, 1723166250419, 4942839351320, 14196809911512, 40826123674963, 117541006104195, 338779909115505, 977457947592893, 2822979976721987, 8160665572497041, 23611944396102714, 68376681868569674, 198169750797798271, 574782976348372753, 1668371611006438146, 4846075456079115622, 14085849428228221307, 40969362561239301701, 119235949321876092360, 347229239641557994136, 1011757164052160972473, 2949698621922111594973, 8604216545991491941581, 25111278043912438342801, 73323304601568480731897, 214201905567546134575859, 626044889720284349690094, 1830549990322901795312942, 5354823514457078459010313, 15670757112005494774639051, 45878599087262878735384752, 134369299666916034362964172, 393689966411509441770005261, 1153899343965725112108795443, 3383264305703491359375922560, 9923244697941619216569794624, 29115000903164573472390932419, 85451758118279984454158890195, 250877793948601314014941971921, 736777877691289303907872561885, 2164414038110594458093425771443, 6360187870629936233099648197313, 18694848840332640943329103122570, 54965868576235609763116372140890, 161651671060236413178711527166019, 475532533517495969396124008055577, 1399233799660317951564059213015592, 4118201121063760930956059935817164, 12123536696685281838476884788352799, 35698751359647955775282558186295793, 105141975845442998724664146494190416, 309739251405543919680997857296802976, 912664235708260313995444264873081893, 2689789263907547466230267373821274617, 7928951321295704744188805383739325609, 23377681170924021285051888390957923933, 68940317392992307735795082187711452173, 203343055362393936480696319767811457959, 599884647429347603580336902386533043922, 1770054610915592642421723575150356117650, 5223773514153323773797895768878478839271, 15419101377748600955972333826656656984531, 45520727173998870390619930777957739468395, 134410477602615207286484828173887369300119, 396944068749335739220874303030506094827475, 1172455237187575683372434195700210272843517, 3463638304632421047516261496691878570034312, 10233794644880525218408203904987488653900680, 30241783348700004437279186446509997511733421, 89380608824471922293587992331001273395222029] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 134 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH H + HH H H HH + H HH HH H + HH H H HH 4+ H HH HH H + H HH HH H + HH HH H HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 2 2 1 + (x + x + x) P(x) + (-x - x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence 2 (n - 1) a(n - 1) (n - 1) a(n - 2) 2 (n - 1) a(n - 3) a(n) = ------------------ + ---------------- + ------------------ n + 1 n + 1 n + 1 (n - 3) a(n - 4) - ---------------- n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1] Just for fun, a(1000), equals 8234659334401657747875389407339732571279656787453168797588442711849374055534\ 399731682304762377542535497852520568576497017328122037389731286460846010\ 443044461555595775102715778648717413957811227465415071230974603488309732\ 107343673557325707026620173863825091561733935419104095484779003697596919\ 255210685186353505572823322729205796609579316204499517633378703540535080\ 3409820335280448356407307981663834920780317426749 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 2, 5, 10, 22, 50, 113, 260, 605, 1418, 3350, 7967, 19055, 45810, 110637, 268301, 653066, 1594980, 3907395, 9599326, 23643751, 58374972, 144442170, 358136905, 889671937, 2214015802, 5518884019, 13778312440, 34448765740, 86247503194, 216212219905, 542679337066, 1363662087715, 3430394504590, 8638347021316, 21774280683160, 54936700030009, 138729047625190, 350622839271715, 886878410779009, 2245035973560322, 5687260991374732, 14417497508837605, 36573842414782720, 92839581311917417, 235812199941976300, 599320573575117058, 1524059250959474845, 3877794297358396855, 9871859585738505998, 25144093549048182413, 64074878999855307016, 163360408628999551420, 416683052591437311350, 1063305463526486150711, 2714544416605796805210, 6932923285598087446149, 17713749965177310901250, 45276590676999361770020, 115771243190730143287029, 296132708151904856692605, 757749959340385997844066, 1939611300690566481202245, 4966475689992131813080275, 12721037219231181524918736, 32593672118747960232345768, 83536591670236848918980793, 214165343693311340487787210, 549220913503729937675166115, 1408857141116990223897481488, 3614977486126526634363849600, 9278116932131800430461411537, 23819172672221244289361500615, 61165029262413086366062908710, 157103765800777960498332811967, 403621422467675998133976904567, 1037202229331537017532021569084, 2665948789464832385353926176270, 6853886233863366236936936795735, 17624483272650789398585868641894, 45330245872305860180937387609005, 116613755674453256884710203714666, 300054011066011799837634994176260, 772209302119215355747708116143270, 1987717304715678281668936564065731, 5117482079734060283987442566130410, 13177661616778411036873823419266269, 33938991026722940778160525149248795, 87425144864325959468794760227423610, 225241810423464354245477849720958404, 580410529842003323164622447324796887, 1495869269272260873776735256040173482, 3855871824791709224683110411230106875, 9940785017959226569084248730088041700, 25632237357502162050592459300023620282, 66102628816394659298417014543745019551, 170496711589999877722570711772593704641] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 135 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 7+ HHH + H HH + HH HH 6+ HH H H + H HH H H + HH HH HH HH 5+ HH H HH + H H + HH HH 4+ H H + HH HH 3+ HH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 2 2 (x + x + 2 x) P(x) + (-x - 3 x - 1) P(x) + x + 1 = 0 The sequence a(n) satisfies the linear recurrence (n - 1) a(n - 1) (n - 2) a(n - 2) (7 n - 17) a(n - 3) a(n) = 3/2 ---------------- + 3/2 ---------------- + 1/2 ------------------- n + 1 n + 1 n + 1 (3 n - 13) a(n - 4) (5 n - 22) a(n - 5) (n - 5) a(n - 6) + 3/2 ------------------- + 1/2 ------------------- + 3/2 ---------------- n + 1 n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 2, a(6) = 5] Just for fun, a(1000), equals 3110044087209480005117630829262690785178962570065949042859259790317737040379\ 759701100163072347133061628900994344134390606794155733990395791805633590\ 537024931819516117069440621896922710132357543951542671730020154374140252\ 481152668325136984199423840335486951160089481484427605735584117701225360\ 185443273546843074814776347936817145348029484175062959758647141754841797\ 753245545859806563436028413641584301856998781416920512719059909064340976\ 0 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 2, 5, 11, 25, 59, 141, 341, 834, 2059, 5124, 12840, 32371, 82050, 208966, 534479, 1372334, 3535944, 9139675, 23692836, 61582898, 160460533, 419044485, 1096638614, 2875503146, 7553584753, 19876005353, 52383517030, 138263312497, 365449260129, 967207210893, 2563018745281, 6799767407380, 18060050175493, 48017685199887, 127796083238768, 340444518042078, 907752844356051, 2422494371813441, 6470152610896746, 17294449355027437, 46261996007121913, 123837829563019809, 331726717724960709, 889187531069931542, 2384956488565065577, 6400749401946081761, 17188349671696188130, 46182861841598432798, 124154272173533996975, 333939679211499088158, 898652419534228226604, 2419497096058108744166, 6517195853819678326088, 17562721011645909868766, 47349085933302136830686, 127706762787997722778519, 344582115297767116357518, 930127395634033290690560, 2511641225135101661616031, 6784737997508785947833423, 18334284135645956344807060, 49561625230123678806569948, 134021151295179820442198667, 362529278870271414806465126, 980958575089081220186575686, 2655168958539467728055683765, 7188922641889272855844422284, 19469823265240457710850266212, 52745243719287993339338899111, 142930323501759518804118082079, 387420007592312848246718803936, 1050396463254153780076312101790, 2848623504656927168323648787217, 7727244004300524919097211948918, 20966177067251074388309237930728, 56900529025354369049987241325696, 154459018041216523663673725397056, 419380000231229978551289480086232, 1138930702193997869259515535996900, 3093711169515805375308283798382397, 8405294433076864048756502355472832, 22840977954137951310899630769331300, 62081617647281364849542856240229601, 168770275549107517139128210128571621, 458893179504305904371260921975082624, 1247981426695721573015941182507884778, 3394561989992914077350719681100010261, 9234996898972555075219612487910040538, 25128437916389102357822582745659397424, 68386180935302267072806640040633043691, 186141742299768525598859015802344525816, 506745913283408122311836317282617805046, 1379768832938553665690210141896566743301, 3757427285946611381207676416183901480105, 10233911186736146182371403793350552546986, 27877779223509090151328083619029560684242] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 136 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH + HH H + H HH + HH H 4+ H HH HH + H HH HH H + HH HH HH HH 3+ H H H H + HH H HH HH + HH HH H HH 2+ HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 1 + (x + x) P(x) + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 1) a(n - 1) (2 n - 1) a(n - 2) 3 (2 n - 5) a(n - 3) a(n) = ------------------ - ------------------ + -------------------- n + 1 n + 1 n + 1 (n - 2) a(n - 4) 2 (2 n - 7) a(n - 5) - ---------------- + -------------------- n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 3134627918704846396589045582068293464668313386749891925759492579301366817419\ 266879466277019895299966556024437351904982229786354769359885950645415854\ 487762568754124509622513897418551096745213564746196714896494879980931898\ 708502918937008296833152741264657083164783324054454272242916859440693717\ 64365001116217342900388877265443604205502181219138765202835544779530 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 42, 84, 170, 354, 740, 1549, 3269, 6945, 14811, 31711, 68177, 147091, 318313, 690837, 1503351, 3279445, 7169907, 15708485, 34482475, 75830981, 167042763, 368548926, 814341362, 1801867812, 3992172298, 8855912464, 19668236110, 43729885814, 97330067640, 216844435346, 483571453768, 1079354856828, 2411240775366, 5391038526802, 12062679137620, 27010935099196, 60526554883902, 135721815227704, 304535849636366, 683755276072670, 1536121565708096, 3453050076522008, 7766456476976834, 17477390474177854, 39350957376284872, 88644320911742304, 199781711595784890, 450466438264400446, 1016164744759318568, 2293264805607373804, 5177575503207130558, 11694317639185843274, 26423715558659122260, 59728010995453379421, 135058602016778295285, 305507289981338348257, 691307640696528081979, 1564829833276593856647, 3543272754560789200681, 8025639916478510155339, 18183959683820504446497, 41212333247384488368873, 93431281455612601484243, 211875294974960472423745, 480604617462452614426815, 1090465639820281676643073, 2474853003558648387670983, 5618201029263006344110809, 12757123617698163352573711, 28974313940315478259062979, 65822724585319538752564981, 149567876677718197286698671, 339936780414026450535924453, 772774892529317395366017603, 1757116730487566683312107729, 3996122690354078859729209647, 9090033908542877936842438557, 20681339112467629598326301035, 47062642985297219558085765929, 107116551464339266855499450015, 243847122183568901454753070765, 555210537090801909689698475727, 1264372572489361483075032316117, 2879838126594606020044275873923, 6560472630207720552639437125041, 14947710059210582396858604974621, 34063179534729044501719950750975, 77636370512116064921945474111629, 176975591352231844422902101233579, 403485801251068480867972873270661, 920043891167408188870554973636539] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 137 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H H + HHHH HHHH + H H HH H 5+ HH HHH HH + HH HH + H H + HH HH 4+ H H + H H + HH HH 3+ H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HHH HH HH + H H H H +HH HHHH HH -*-+-+-+-+*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 3 2 3 3 2 2 (x - 2 x + 3 x - 2 x + 1) P(x) + (2 x - 6 x + 6 x - 4) P(x) 2 + (3 x - 5 x + 5) P(x) + x - 2 = 0 The sequence a(n) satisfies the linear recurrence 2 (784 n - 4175 n + 5430) a(n - 1) a(n) = -1/6 --------------------------------- n (n - 20) 3 2 (806 n - 6265 n + 15085 n - 11270) a(n - 2) + 1/2 --------------------------------------------- (n - 20) n (n - 1) 3 2 (775 n - 9483 n + 37640 n - 46200) a(n - 3) + 1/3 --------------------------------------------- (n - 20) n (n - 1) 3 2 (2338 n - 24703 n + 83195 n - 84510) a(n - 4) - 1/2 ----------------------------------------------- (n - 20) n (n - 1) 3 2 (13547 n - 160893 n + 589450 n - 604980) a(n - 5) + 1/3 --------------------------------------------------- (n - 20) n (n - 1) 2 (n - 5) (23120 n - 168299 n + 216750) a(n - 6) - 1/6 ----------------------------------------------- (n - 20) n (n - 1) (802 n - 1345) (n - 5) (n - 6) a(n - 7) + 23/6 --------------------------------------- (n - 20) n (n - 1) subject to the initial conditions [a(1) = 1, a(2) = 1, a(3) = 2, a(4) = 5, a(5) = 12, a(6) = 30, a(7) = 81] Just for fun, a(1000), equals FAIL[1000] For the sake of the OEIS here are the first, 100, terms. [1, 1, 1, 2, 5, 12, 30, 81, 229, 662, 1948, 5831, 17710, 54419, 168833, 528155, 1664213, 5277256, 16827858, 53926063, 173576588, 560936415, 1819285007, 5919833795, 19320385526, 63228230571, 207443469767, 682177801649, 2248171612205, 7423837365735, 24560350107571, 81394262559556, 270182631898549, 898217066339078, 2990372473872432, 9969027651170351, 33275945950334338, 111205809076329913, 372062450716585111, 1246147908447295661, 4177965051016674116, 14021011566810825649, 47096943775584809319, 158338272822270193117, 532771736317317709107, 1794081031969168870847, 6046069620444553359431, 20390133033818507041768, 68812816820455499805574, 232385138864995941107851, 785281303758181049919031, 2655271411885935199895101, 8983567960720145181101627, 30411314877899936755434201, 103005075976443659262040713, 349068249999754776625254132, 1183537570562806527314290693, 4014817961334241947551433417, 13625558839596423774916375743, 46263701413779054058541936532, 157151058003782357858755485575, 534046102763843232485954101880, 1815593591097072181893422226844, 6174918323442572324039618112447, 21009271327001690661407916437429, 71507686297257733068586785592808, 243473352912509733096516091480434, 829282227007395032195600984125183, 2825536621845165696860641062544160, 9630366970485697588071779896672647, 32834006993756818115591952855878967, 111979908830449148216631580041483343, 382021401817036743662791711928588378, 1303655934802611838953451304395971225, 4450025332458439029031005586983475661, 15194374443616234519888158067081669447, 51894434580369708393426302508607294107, 177285519217389069991239536808496071653, 605811181128998091111670708159025914981, 2070665674596829532527082350056760357124, 7079272150084604177375712025727064755828, 24208646340580505196888120163464510470975, 82804344693779747107614977000710374753661, 283291800762222662725968711157625925621617, 969417325702992920850057293357303761464115, 3318036475368752735752674264094357994062847, 11359074196872447783790061981782481726423585, 38895019257007873575802925204535284491890554, 133208600389876893412305839417838500907223451, 456305575539147863651253579693219428425657021, 1563372988782039801075232072048889631931692739, 5357360833210372527055319257073755767108049928, 18361952762323231986995172334810224271266626627, 62945512379293241106405162797616205629319743698, 215817641682835340151787194936128973047345821576, 740088660683565258849948110822952227672655645783, 2538362384191128104441030871052512322129732425414, 8707531186263942524078713680230936247912662583419, 29874904075524665800762954407332809563174993694615, 102514815648188524910760889830801959846415911697437, 351830914795587916846468105997185262724999878660635] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 138 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 2 2 (x + x + 2 x) P(x) + (-x - 3 x - 1) P(x) + 1 + x = 0 The sequence a(n) satisfies the linear recurrence (n - 1) a(n - 1) (n - 2) a(n - 2) (7 n - 17) a(n - 3) a(n) = 3/2 ---------------- + 3/2 ---------------- + 1/2 ------------------- n + 1 n + 1 n + 1 (3 n - 13) a(n - 4) (5 n - 22) a(n - 5) (n - 5) a(n - 6) + 3/2 ------------------- + 1/2 ------------------- + 3/2 ---------------- n + 1 n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 2, a(6) = 5] Just for fun, a(1000), equals 3110044087209480005117630829262690785178962570065949042859259790317737040379\ 759701100163072347133061628900994344134390606794155733990395791805633590\ 537024931819516117069440621896922710132357543951542671730020154374140252\ 481152668325136984199423840335486951160089481484427605735584117701225360\ 185443273546843074814776347936817145348029484175062959758647141754841797\ 753245545859806563436028413641584301856998781416920512719059909064340976\ 0 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 2, 5, 11, 25, 59, 141, 341, 834, 2059, 5124, 12840, 32371, 82050, 208966, 534479, 1372334, 3535944, 9139675, 23692836, 61582898, 160460533, 419044485, 1096638614, 2875503146, 7553584753, 19876005353, 52383517030, 138263312497, 365449260129, 967207210893, 2563018745281, 6799767407380, 18060050175493, 48017685199887, 127796083238768, 340444518042078, 907752844356051, 2422494371813441, 6470152610896746, 17294449355027437, 46261996007121913, 123837829563019809, 331726717724960709, 889187531069931542, 2384956488565065577, 6400749401946081761, 17188349671696188130, 46182861841598432798, 124154272173533996975, 333939679211499088158, 898652419534228226604, 2419497096058108744166, 6517195853819678326088, 17562721011645909868766, 47349085933302136830686, 127706762787997722778519, 344582115297767116357518, 930127395634033290690560, 2511641225135101661616031, 6784737997508785947833423, 18334284135645956344807060, 49561625230123678806569948, 134021151295179820442198667, 362529278870271414806465126, 980958575089081220186575686, 2655168958539467728055683765, 7188922641889272855844422284, 19469823265240457710850266212, 52745243719287993339338899111, 142930323501759518804118082079, 387420007592312848246718803936, 1050396463254153780076312101790, 2848623504656927168323648787217, 7727244004300524919097211948918, 20966177067251074388309237930728, 56900529025354369049987241325696, 154459018041216523663673725397056, 419380000231229978551289480086232, 1138930702193997869259515535996900, 3093711169515805375308283798382397, 8405294433076864048756502355472832, 22840977954137951310899630769331300, 62081617647281364849542856240229601, 168770275549107517139128210128571621, 458893179504305904371260921975082624, 1247981426695721573015941182507884778, 3394561989992914077350719681100010261, 9234996898972555075219612487910040538, 25128437916389102357822582745659397424, 68386180935302267072806640040633043691, 186141742299768525598859015802344525816, 506745913283408122311836317282617805046, 1379768832938553665690210141896566743301, 3757427285946611381207676416183901480105, 10233911186736146182371403793350552546986, 27877779223509090151328083619029560684242] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 139 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H H + HH H H HH H HH + H H HH H HH H + HH HHH HHH HH 2+ H H H H + H H + HH HH + H H 1+ H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-+*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 5 4 3 2 4 (x - 4 x + 10 x - 16 x + 19 x - 16 x + 10 x - 4 x + x) P(x) 8 7 6 5 4 3 2 3 + (3 x - 8 x + 16 x - 21 x + 21 x - 15 x + 6 x - x - 1) P(x) 7 6 5 4 3 2 2 + (3 x - 4 x + 9 x - 10 x + 7 x - 2 x - 2 x + 4) P(x) 6 4 3 3 2 + (x + 4 x - x + x - 5) P(x) + x + x + x + 2 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 5, 11, 24, 57, 143, 365, 945, 2491, 6670, 18072, 49440, 136413, 379225, 1061150, 2986375, 8447363, 24003522, 68486632, 196128871, 563552103, 1624258570, 4694536085, 13603417562, 39512464267, 115020074742, 335503993954, 980493106164, 2870507676310, 8417635991050, 24722564252548, 72715614008229, 214168273978413, 631599318260502, 1864901332635111, 5512758393041127, 16313774254592883, 48326782324732806, 143300153857490290, 425314166096320854, 1263449918919230716, 3756411412497278241, 11177338176779037572, 33284195562683696201, 99187631244999556979, 295789695059160150629, 882676935845484829320, 2635734609760513787208, 7875378096137600453313, 23545088322296198905215, 70433354504043216513947, 210812371763328765105455, 631312737440837259051895, 1891538996557903110923232, 5670225702168751669909489, 17005606493423207774151623, 51025052843263963615836352, 153167827735731209184093120, 459978922059315778620793943, 1381937662222843097703294747, 4153490993800100480920579500, 12488400741845778490011022688, 37563306146081444322155538845, 113026210383971614012317530558, 340210780084458065388161676627, 1024391376904080613513752407831, 3085519477201054238151005193120, 9296746603343207911191637723544, 28020119847926500788684886641933, 84477570891028505485009572116976, 254766067325913912733314462555107, 768540822841117030156114993675315, 2319071243486683137669060865200833, 6999705504995474528441148971080382, 21132984791138027612296558822606062, 63819631911012825373065768469189695, 192777896438174883966309186639179353, 582461040681096573679327378303466951, 1760275116287171703510922767669313592, 5321029425636963857926177864160245707, 16088284191743052639725274041688101945, 48654202756852952216458483388829914173, 147172018254361700820934684726414485398, 445268689103239014293246978961264189120, 1347438479207895702050010049126051790741, 4078341008031071771324078745101161132883, 12346501840682220866878737290114744598405, 37384207389334237897400451650707981973144, 113217724087608095731211245015489261930089, 342942133743587420755376121569364839051741, 1038976448256873089794031841259145669311175, 3148236132818936733136404203299493127826447, 9541221867002690493668185241916253853086385, 28921059779991299069231083522844516729746338, 87679161254655821272720099177356620722906477, 265857585240028205033057254699954947310982154, 806251928850455145587442445429262904614369824, 2445457498700289531488315690260484221080250702] ------------------------------------------------------------ Theorem Number, 140 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 7+ HH + H HH + HH HH 6+ H HH + H HH + HH HH 5+ HH HH + H HH + HH HH 4+ H H + HH HH 3+ HH HH H + HH HH H HH + HH HH HH HH 2+ HH HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 3 2 3 1 + P(x) x + (-x + x) P(x) + (x - x - 1) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 45, 90, 187, 401, 851, 1815, 3925, 8517, 18528, 40542, 89084, 196294, 433960, 962393, 2139725, 4768821, 10653090, 23847852, 53488512, 120189110, 270525714, 609872556, 1376944183, 3113165351, 7047902436, 15975594178, 36254691983, 82367121385, 187327128656, 426463015182, 971794420430, 2216459060360, 5059629805741, 11559371030889, 26429562274813, 60474319746292, 138472393707020, 317287830492994, 727493771194568, 1669087821473347, 3831706461817534, 8801529240735440, 20228664934673007, 46516855554565603, 107023436481333016, 246356874740823900, 567361687949009896, 1307246629320351536, 3013355124126816199, 6949151353923786831, 16032292599544117909, 37002915021440054034, 85437203538695002412, 197343801364604072373, 455995108480664878088, 1054028117510157798185, 2437220586009809344922, 5637462008726047404420, 13044104958896714457266, 30191362314092522717088, 69901256387316040143189, 161889000866928364911673, 375038792667449550811129, 869076638050116790409663, 2014464785302280486552292, 4670655362261462738855867, 10832019860018976585731355, 25127635081038417757157497, 58304423258510204427627370, 135318255823699614530347478, 314133072780421992235087867, 729408461607772465334189902, 1694046655330387389384834123, 3935274500031711929656356229, 9143606614506172522351127413, 21249596815155851886618287769, 49393797017310482802102109124, 114836688005465996914314854101, 267038248650994935249780214935, 621082017773923851102098581060, 1444791837130910209523805410400, 3361558483284669762056309772047, 7822641317082942993733966553444, 18207145325605714380902195077507, 42384238894815170678689367097517, 98682343013046975787558512672747, 229797628412303496855469928034966, 535206276701024419566301617271889, 1246708738543803536060775647603284, 2904528122457933390105864083112466, 6767863837948236105079296085122986] ------------------------------------------------------------ Theorem Number, 141 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ HH HH + HH HH + H H + HH HH 4+ HH H H + H HH H H + HH HH HH HH 3+ H H H H + HH HH HH HH + HH H HH HH 2+ HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 2 1 + P(x) x + P(x) x + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence 2 (518 n - 1031 n + 780) a(n - 1) a(n) = -1/195 -------------------------------- n + 2 3 2 (1036 n - 4335 n + 4367 n - 1170) a(n - 2) - 1/195 -------------------------------------------- (n + 2) (n + 1) 3 2 (2590 n - 11115 n + 11939 n - 1812) a(n - 3) + 1/195 ---------------------------------------------- (n + 2) (n + 1) 2 (n - 3) (1036 n - 1865 n + 1074) a(n - 4) + 1/65 ------------------------------------------ (n + 2) (n + 1) 161 (74 n - 55) (n - 3) (n - 4) a(n - 5) + --- ------------------------------------ 195 (n + 2) (n + 1) subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 1456286415952438964652364581682787386720721652723427223868652268704217615734\ 013661511014559584083968554107289545538396109682528052843351355027899510\ 336875839145124807012715538480932028100745003225030603205050905923754789\ 120913436417646081847745149534148691270052401108733311860953962982834165\ 142352139116820596104418788220221329115699403613078628786955744338071295\ 785210326056171164845618097 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 4, 8, 13, 31, 71, 144, 318, 729, 1611, 3604, 8249, 18803, 42907, 98858, 228474, 528735, 1228800, 2865180, 6693712, 15676941, 36807239, 86584783, 204060509, 481823778, 1139565120, 2699329341, 6403500057, 15211830451, 36183117255, 86171536894, 205459894230, 490417795075, 1171809537516, 2802705573178, 6709741472212, 16077633792535, 38557470978465, 92543928713413, 222292252187223, 534346494991678, 1285375448785399, 3094086155360823, 7452774657572624, 17962853486458385, 43320517724198861, 104535253974750259, 252390609561764423, 609699521485999914, 1473608895007574938, 3563397472304118410, 8620933920694737411, 20866285270840567272, 50527739882573522706, 122405798503806040965, 296657476936886314309, 719256672330233630085, 1744545018773625965860, 4232966162296687137816, 10274640808649742568633, 24948409910226791411907, 60599469176019760272535, 147244830551438483529790, 357892285312932720768838, 870164978202711215226483, 2116332778149031726542808, 5148677470688468794716894, 12529484860432553111063488, 30499528076281485551927755, 74262935424150986850220725, 180870207357836188170341069, 440630810522439231975963839, 1073724026979876073732588658, 2617084008072054452326466527, 6380388743696091681799439061, 15558886191823487315141504508, 37949779254079097680017924283, 92584221613943667202286169191, 225922433438019608173638548805, 551409374869368431174631180753, 1346105758407373798214652723522, 3286793196606678831626181940722, 8026971507652658919500628768874, 19607185640982275440523729181033, 47902823795394726062462438796736, 117054324948067108888357885549374, 286083328086281443692788683942327, 699317934466646636669713105276839, 1709748244857899463255482766619759, 4180839106559551650627420507840427, 10225086357877339275440446600065760, 25011587559313816213376886296261184, 61190615947546730766871192612064833, 149725673741988363612845353038008205, 366415846156516055604065564079552535, 896845046441850691487935128438196159] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 142 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 1 + (x + x) P(x) + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 1) a(n - 1) (2 n - 1) a(n - 2) 3 (2 n - 5) a(n - 3) a(n) = ------------------ - ------------------ + -------------------- n + 1 n + 1 n + 1 (n - 2) a(n - 4) 2 (2 n - 7) a(n - 5) - ---------------- + -------------------- n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 3134627918704846396589045582068293464668313386749891925759492579301366817419\ 266879466277019895299966556024437351904982229786354769359885950645415854\ 487762568754124509622513897418551096745213564746196714896494879980931898\ 708502918937008296833152741264657083164783324054454272242916859440693717\ 64365001116217342900388877265443604205502181219138765202835544779530 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 42, 84, 170, 354, 740, 1549, 3269, 6945, 14811, 31711, 68177, 147091, 318313, 690837, 1503351, 3279445, 7169907, 15708485, 34482475, 75830981, 167042763, 368548926, 814341362, 1801867812, 3992172298, 8855912464, 19668236110, 43729885814, 97330067640, 216844435346, 483571453768, 1079354856828, 2411240775366, 5391038526802, 12062679137620, 27010935099196, 60526554883902, 135721815227704, 304535849636366, 683755276072670, 1536121565708096, 3453050076522008, 7766456476976834, 17477390474177854, 39350957376284872, 88644320911742304, 199781711595784890, 450466438264400446, 1016164744759318568, 2293264805607373804, 5177575503207130558, 11694317639185843274, 26423715558659122260, 59728010995453379421, 135058602016778295285, 305507289981338348257, 691307640696528081979, 1564829833276593856647, 3543272754560789200681, 8025639916478510155339, 18183959683820504446497, 41212333247384488368873, 93431281455612601484243, 211875294974960472423745, 480604617462452614426815, 1090465639820281676643073, 2474853003558648387670983, 5618201029263006344110809, 12757123617698163352573711, 28974313940315478259062979, 65822724585319538752564981, 149567876677718197286698671, 339936780414026450535924453, 772774892529317395366017603, 1757116730487566683312107729, 3996122690354078859729209647, 9090033908542877936842438557, 20681339112467629598326301035, 47062642985297219558085765929, 107116551464339266855499450015, 243847122183568901454753070765, 555210537090801909689698475727, 1264372572489361483075032316117, 2879838126594606020044275873923, 6560472630207720552639437125041, 14947710059210582396858604974621, 34063179534729044501719950750975, 77636370512116064921945474111629, 176975591352231844422902101233579, 403485801251068480867972873270661, 920043891167408188870554973636539] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 143 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 3 2 3 1 + P(x) x + (-x + x) P(x) + (x - x - 1) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 45, 90, 187, 401, 851, 1815, 3925, 8517, 18528, 40542, 89084, 196294, 433960, 962393, 2139725, 4768821, 10653090, 23847852, 53488512, 120189110, 270525714, 609872556, 1376944183, 3113165351, 7047902436, 15975594178, 36254691983, 82367121385, 187327128656, 426463015182, 971794420430, 2216459060360, 5059629805741, 11559371030889, 26429562274813, 60474319746292, 138472393707020, 317287830492994, 727493771194568, 1669087821473347, 3831706461817534, 8801529240735440, 20228664934673007, 46516855554565603, 107023436481333016, 246356874740823900, 567361687949009896, 1307246629320351536, 3013355124126816199, 6949151353923786831, 16032292599544117909, 37002915021440054034, 85437203538695002412, 197343801364604072373, 455995108480664878088, 1054028117510157798185, 2437220586009809344922, 5637462008726047404420, 13044104958896714457266, 30191362314092522717088, 69901256387316040143189, 161889000866928364911673, 375038792667449550811129, 869076638050116790409663, 2014464785302280486552292, 4670655362261462738855867, 10832019860018976585731355, 25127635081038417757157497, 58304423258510204427627370, 135318255823699614530347478, 314133072780421992235087867, 729408461607772465334189902, 1694046655330387389384834123, 3935274500031711929656356229, 9143606614506172522351127413, 21249596815155851886618287769, 49393797017310482802102109124, 114836688005465996914314854101, 267038248650994935249780214935, 621082017773923851102098581060, 1444791837130910209523805410400, 3361558483284669762056309772047, 7822641317082942993733966553444, 18207145325605714380902195077507, 42384238894815170678689367097517, 98682343013046975787558512672747, 229797628412303496855469928034966, 535206276701024419566301617271889, 1246708738543803536060775647603284, 2904528122457933390105864083112466, 6767863837948236105079296085122986] ------------------------------------------------------------ Theorem Number, 144 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H H + HH H H HH + H HH HH H + HH H H HH 4+ H H H H + H HH HH H + HH H H HH + H HH H H 3+ H H H H + HH H HH HH + H HH H H + HH HHH HH 2+ H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 2 3 1 + P(x) x + (x - x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 1) a(n - 1) (n - 2) a(n - 2) (2 n - 7) a(n - 3) a(n) = ------------------ - ---------------- + ------------------ n + 1 n + 1 n + 1 2 (n - 5) a(n - 4) (n - 8) a(n - 6) + ------------------ - ---------------- n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 2] Just for fun, a(1000), equals 1176950714989472039879867097164787467731168548390670636801992136416332972086\ 898277348698087993919571204091110820742448542481636855988358314899690938\ 672843649138648682133884841687298042373689270639308856986357452480963423\ 383401026848945091071625435678738959251923453512607097256262920947479103\ 356615910327226686348752585420237437 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 22, 42, 80, 152, 292, 568, 1112, 2185, 4313, 8557, 17050, 34089, 68370, 137542, 277475, 561185, 1137595, 2311014, 4704235, 9593662, 19598920, 40103635, 82185653, 168666493, 346613232, 713200114, 1469254621, 3030218948, 6256281188, 12930039374, 26748697772, 55386529370, 114785051382, 238083048103, 494216315763, 1026681547651, 2134372036796, 4440242721757, 9243424565624, 19254704030249, 40133535117994, 83701671288887, 174665494666782, 364684302692317, 761824952311410, 1592257031239222, 3329531677118927, 6965586177249102, 14579064797995464, 30527584089316653, 63949861857983311, 134018617814709631, 280972131660117384, 589289169477022354, 1236390172104441711, 2595012857019532078, 5448483097227962922, 11443510685976418890, 24042863051171641274, 50530333059247995344, 106231480858589059892, 223401249061635751536, 469943442677589917028, 988848424941723500999, 2081299761445939780379, 4381845845937552096915, 9227711560622784969822, 19437589516267668693817, 40954291792943394506568, 86310235142767979004036, 181940086112454163969993, 383614626529901470057969, 809021619679751516526841, 1706557543555582414238756, 3600603520716009828550165, 7598380174796313007709308, 16038212465609702505254728, 33859308082950214419570804, 71496599403133792320146486, 150999659234389445825596123, 318968628398739129595019281, 673906439126292155812743214, 1424062795029526232414783635, 3009781004697296226116225432, 6362316258145714947705909656, 13451435620129537321656044592, 28444187117982772743078140610, 60157315325649798906283809598, 127248293230730230066048509796, 269204737445574334829247116243, 569612221084905567519367331211] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 145 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} To make it crystal clear here is such a path of semi-length 8 4+ H + HH + HH HH + H H + H H 3+ H H H H H H + H H H H H H H H H H + H HH H HH H HH HH H HH H + H H H H H H H H H H + H HH HH HH HH H 2+ H H H H H H + H H + H H + HH HH + H H 1+ H H + H H + H H + HH HH +H H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 3 2 2 2 1 + (x - 2 x + 4 x - 3 x + 1) P(x) + (-2 x + 3 x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence (7 n - 13) a(n - 1) 2 (8 n - 17) a(n - 2) 6 (3 n - 7) a(n - 3) a(n) = ------------------- - --------------------- + -------------------- n - 1 n - 1 n - 1 3 (3 n - 7) a(n - 4) 2 (2 n - 5) a(n - 5) - -------------------- + -------------------- n - 1 n - 1 subject to the initial conditions [a(1) = 1, a(2) = 1, a(3) = 2, a(4) = 5, a(5) = 13] Just for fun, a(1000), equals 2716559067884306025265354750689893774842105816701567123244169574053297657320\ 317317349567413239488429974413465747402908821830473562309228716494996176\ 595940291483287287411748068922464671018555193168779669983777644776280751\ 197274818375125087818631899370573740793761405516981153911153117758417146\ 556178026026973593726512794177954441763109485683153660518667691735931504\ 847347631339163616070710570105902273412448625192387687302812482186042841\ 850767559594482737042447155211937565663667893282217221251515879010243535\ 681978756839055406103401127469149255794185301041818760408669306765240568\ 84695559209136547 For the sake of the OEIS here are the first, 100, terms. [1, 1, 1, 2, 5, 13, 36, 106, 327, 1045, 3433, 11529, 39414, 136733, 480180, 1703807, 6099193, 22000823, 79890801, 291808480, 1071403389, 3952020216, 14638293671, 54424065467, 203034222400, 759790586108, 2851348853311, 10728474996286, 40463929597582, 152954346844120, 579360875031653, 2198706838568725, 8359108281762525, 31832900933458770, 121414387352480205, 463767198460804970, 1773894781707379869, 6793896919957722629, 26052030028473231638, 100015329095164501530, 384385629016815344827, 1478831105536752569398, 5695056470981344209128, 21952519511373386671936, 84695039144155916932505, 327039800428485491619329, 1263849346853594929350751, 4887944251701029342021731, 18918148640934696765014662, 73272101566469724180468725, 283983647534388214714032138, 1101363134547071249662837679, 4274042525751826295352081729, 16596155903083816977883306178, 64480207173576525986995652350, 250660591281133325641048903997, 974938932457463152083188142642, 3793953307300744259673956686432, 14771402095305943793533878108684, 57538563187123499224676355874275, 224231509151413051900656599230391, 874234120716633287861301617092121, 3409933014419308753323870067033561, 13305914974091951363590121422535042, 51941987339926099427008289180698045, 202843784590304456833386205315309093, 792444471220790226244990874714718572, 3096954908157832767764344154065645742, 12107514005991332947546748790637128179, 47350492565150461230598017913819879570, 185241811015541212937076396676028807406, 724927038400062602572950065255074139254, 2837829450682614815974136073288339363699, 11112483252559631561693102928278024985673, 43527634610442419276320586793769477999025, 170547192049131276452772908381867207527027, 668415053071164121697984739272600555219868, 2620394831415671278102571376957555897354893, 10275499961813481392473386612147094869264756, 40304346200680840607164432177861940114537445, 158128656976081233145797059954287281345378829, 620549272392996527969861499028241484683553556, 2435825959661549123079884612015819570696569426, 9563523807152463365507878825171338579431369007, 37556822486237882261733164483468588779925791880, 147521916009814089749245937909392456821746691008, 579587130778529235837383492121798439354374354532, 2277577552383928485646161832913067788479659918837, 8951951217127583545340406069758233391313801242839, 35192510622059707872431236884719519585277356466165, 138378555557421031541883664034976143790863345171281, 544216107369896298264914135651495399112175924242624, 2140701770707351382756129069050804292314776393683039, 8422117716719255995837920328495013558266106497978818, 33140957549002623428605516280328181391986633550353865, 130432465749516068521174775110284252020382580429707777, 513430446233345877626937748420955976323699471178212068, 2021395071060687557510093127508831866952506010653438058, 7959630606604663061200142964162158677921005371935221891, 31347669318417990077309634950839809028292963386018384900, 123477203294324705006709282110510356464796912845199542184] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 146 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H H + HH H HH H + H HH H H + HH H HH HH 4+ H H H H H + H HH H H HHH + HH H HH HH H HH + H HH H H HH H 3+ H H H H + HH HH + H H + HH HH 2+ H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 2 2 2 1 + (x + x - x + 1) P(x) + (-x + x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence 3 (n - 2) a(n - 1) 6 a(n - 2) (n - 7) a(n - 3) (2 n - 11) a(n - 4) a(n) = ------------------ - ---------- - ---------------- + ------------------- n - 1 n - 1 n - 1 n - 1 (2 n - 5) a(n - 5) 3 (n - 4) a(n - 6) + ------------------ + ------------------ n - 1 n - 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 2, a(6) = 5] Just for fun, a(1000), equals 2878856782144216993304909072623096791238839518518732785727996575633444455066\ 956567407176766906801478386293066728857038629471924993382928181488681958\ 710041927681409642009723491606649261960519836795925263537967363891891877\ 002397732039295842231821680571756238560688066459575319018806752729651146\ 307687782271404813967049306258778771027776415891416409801808828208302403\ 754056299604571215754189083212255788144789217206287683307324816030984715\ 912400078786930743617163678096273514 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 2, 5, 11, 26, 64, 160, 408, 1057, 2773, 7354, 19684, 53108, 144285, 394395, 1083893, 2993169, 8301373, 23113093, 64579846, 181021143, 508903147, 1434529326, 4053774130, 11481626162, 32588767361, 92680384675, 264061121393, 753641895385, 2154377477842, 6167816874173, 17682947621645, 50763953628170, 145915335052953, 419914634020438, 1209784305460203, 3489122437332586, 10073078554564695, 29108790529242620, 84194127583139219, 243734132718405830, 706171736309540520, 2047610949222803792, 5941724010837544122, 17254056039415814789, 50138381671889842151, 145793214118265385208, 424209581629997008364, 1235062051960423283060, 3597922845367850458123, 10487208390879264918883, 30584663463760785497977, 89243080822651069459771, 260533876350475763594648, 760965285519087098174307, 2223663640882079998345485, 6500846940194260255293338, 19013441322241949351689809, 55633340339776650260833834, 162849771289646446389856459, 476882138630038856845066186, 1397016057262264292104843536, 4094048498713833927119509096, 12002199935911865573364250368, 35198186649823939053399904897, 103258726164565685925228187639, 303023253756858648014378213498, 889536432693308266926378432142, 2612078093684686900158388271554, 7672544560379181785324320290350, 22543424908109986923445574304695, 66255831375073245682145803993412, 194781951077045009504532440141396, 572783503254869268864047217594022, 1684792405001414647348856645732013, 4956937407787336387901741593452936, 14587765426674530465484714813300616, 42940753950790728299148874663609176, 126430973352845019344790249089690895, 372338273101134145496109926981954000, 1096780740183236388272441528847245066, 3231450663204362980554731118035642044, 9522885200305441053069920474784222079, 28069241670114708661852669209760908380, 82752629977973030745221962129345214936, 244016886093782321888697308342626742333, 719685797597200359808440118040501049515, 2122995392809778717115751666984513854737, 6263778537286879755046906861656257371389, 18484307155950499152062033476773319732083, 54556647215785861139168869263297189076433, 161052775284673849736973509097646627234736, 475513860462577865152939584703040055308473, 1404206597899854279669242235315364776065237, 4147345179697928605850235606984347636322434, 12251216958902729677442285550386020013605594, 36195679054955691421660475357757577042688188] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 147 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 3+ H H + H H H H + HH HH HH HH 2.5+ H H H H + H H H H + H H H H + H HH H 2+ H H H + H H + H H 1.5+ HH HH + H H + H H 1+ H H H H H H + HH H H H H H H H H + H H H H H H H H H H + H H H H H H H H H H 0.5+ H H H H H H H H H H + H H H H H H H H H H +H HH HH HH HH H -*-+-+-+-+*-+-+-+-+*-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-*+-+-+-+-*+-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 3 -1 + (x - 3 x + 6 x - 7 x + 7 x - 4 x + 1) P(x) 4 3 2 2 2 + (-3 x + 5 x - 9 x + 8 x - 3) P(x) + (2 x - 4 x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 5, 12, 29, 74, 199, 554, 1580, 4599, 13629, 41007, 124950, 384808, 1196010, 3747027, 11821256, 37523229, 119754660, 384047582, 1236976421, 3999780023, 12979240458, 42253474511, 137960509242, 451669185680, 1482402326444, 4876508223150, 16075907075992, 53100657695590, 175720809546355, 582495048581828, 1934011326455707, 6431021929933012, 21414877019549000, 71405175936331293, 238390696364328014, 796827183622750392, 2666406700941573674, 8932035228863731542, 29951087039643233174, 100528914233734415521, 337726663331342107359, 1135577613895114232222, 3821447399021126861689, 12870108500369910092496, 43377502675993578865587, 146305498040330514857658, 493806920018551613984013, 1667790945150358726863700, 5636405421129196150426245, 19060222673631623909309669, 64492388315933639743827121, 218340300664204189679262424, 739596870103919458535559424, 2506589512129250439904929232, 8499436166773644383255497931, 28834203095531408174101316777, 97865435244147141643724580885, 332312929827145883877564896974, 1128898818275654051949507939112, 3836598428445000055724268993497, 13044132578449189905222612800257, 44366572070761163512125173414595, 150960354073285919729699123970279, 513843778173627271234045629372531, 1749667435722230424981844473882405, 5959796016071331726800894082927865, 20307396272010013338048407694338611, 69218109534357021809344240032151246, 236006375173418561456532818720534102, 804937742592748279543019011219757331, 2746196546755413745801079650875838825, 9371909379169374468353399703869545521, 31992501744803496108426994087772295442, 109241748360914014259847741134534865867, 373117994182316404320539025462855575157, 1274728742041129800013073990663875472910, 4356126017486989008745040937927098462749, 14889884127520776439199122505070866766943, 50908193937206048172852892645702958973655, 174095247164727931352474741478183236910534, 595506430413343673847234697689138746032752, 2037435488027457279179217158357126956782541, 6972311652168179652233080636088514515646073, 23865083446147062045271868163117891384795476, 81703408011380205708376672572522315362235932, 279773335179699265305006937822281020199174023, 958207041437321309580759936460430793948328530, 3282443900489104842728265442779956311398980686, 11246523018497041223979591658610452594266227547, 38540771747913170357268601376105839730333735506, 132099727203511387502602637549167246847556748004, 452857008965978701094742813830497985574565158089, 1552731118955646347123917939554121904818246830295, 5324830582291895640932544203205749100761200265373, 18263673804706504903253280721851321222705981935308, 62652987313369405584985020744074808260772456463574, 214963759678046363792947428718327637564925230657702] ------------------------------------------------------------ Theorem Number, 148 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H H + HH H H HH + H HH HH H + HH H H HH 4+ H H H H + H HH HH H + HH H H HH + H HH H H 3+ H H H H + HH H HH HH + H HH H H + HH HHH HH 2+ H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 3 4 3 2 2 -1 + (x + 2 x - x + x - x + 1) P(x) + (-2 x + x - 2 x + 2 x - 3) P(x) 2 + (x - x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 46, 94, 199, 439, 961, 2113, 4728, 10638, 24004, 54567, 124738, 286100, 658902, 1523561, 3533694, 8219807, 19175241, 44844787, 105117745, 246938893, 581271855, 1370801044, 3238379640, 7662884761, 18160172211, 43099390904, 102426035307, 243726755578, 580655580178, 1384932709103, 3306794777897, 7903693118041, 18909278995761, 45281554377074, 108530027712894, 260341203346223, 625005317744160, 1501608816996828, 3610336668014877, 8686460115399371, 20913632597618146, 50384354111881549, 121459160713524172, 292969666722436726, 707070780696631056, 1707424831884850449, 4125246497394886642, 9971935177734506502, 24116945055530422878, 58354041165562962555, 141259692927959905224, 342103485342872695337, 828862025018976189145, 2009029841925232601585, 4871515952686337473281, 11817073539321134031521, 28675998351756531656835, 69612131191902560435320, 169045763869085582882160, 410650213529812223368770, 997892674377742909288876, 2425690847473912236264926, 5898246126299524653958956, 14346379729918995608354591, 34905194678328946279956726, 84949845555910352980918162, 206802853409472003358290619, 503580166605875837883509918, 1226579626343965688411938881, 2988373791468938272294751313, 7282546274134082391128998885, 17751619375203540121720680747, 43280914162805330098090017988, 105549463086185917149368841094, 257462696364279602325677747884, 628158000160191822477935767084, 1532912888447769419915804392627, 3741604135787909331331389618256, 9134564337311934498324607449420, 22305163745068650394954160327031, 54476421551688686284866768832516, 133074664291962641673573493873036, 325135188262952797925515855558143, 794534125103762472134490936108882, 1941955965184663382273487480183164, 4747257141764020796262302049059161, 11607028692460250300264994515896655, 28383941376456748393597046710009226, 69421845442964953819449619451462120, 169820428896576356561050206615538112, 415482369434004891765163304919548259] ------------------------------------------------------------ Theorem Number, 149 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H + HH + HH HH + H H + H H 3+ H H H H + H H H H H H + H HH H HH HH H + H H H H H H + H H H HH H 2+ H H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 2 2 2 1 + (x + x - x + 1) P(x) + (-x + x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence 3 (n - 2) a(n - 1) 6 a(n - 2) (n - 7) a(n - 3) (2 n - 11) a(n - 4) a(n) = ------------------ - ---------- - ---------------- + ------------------- n - 1 n - 1 n - 1 n - 1 (2 n - 5) a(n - 5) 3 (n - 4) a(n - 6) + ------------------ + ------------------ n - 1 n - 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 2, a(6) = 5] Just for fun, a(1000), equals 2878856782144216993304909072623096791238839518518732785727996575633444455066\ 956567407176766906801478386293066728857038629471924993382928181488681958\ 710041927681409642009723491606649261960519836795925263537967363891891877\ 002397732039295842231821680571756238560688066459575319018806752729651146\ 307687782271404813967049306258778771027776415891416409801808828208302403\ 754056299604571215754189083212255788144789217206287683307324816030984715\ 912400078786930743617163678096273514 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 2, 5, 11, 26, 64, 160, 408, 1057, 2773, 7354, 19684, 53108, 144285, 394395, 1083893, 2993169, 8301373, 23113093, 64579846, 181021143, 508903147, 1434529326, 4053774130, 11481626162, 32588767361, 92680384675, 264061121393, 753641895385, 2154377477842, 6167816874173, 17682947621645, 50763953628170, 145915335052953, 419914634020438, 1209784305460203, 3489122437332586, 10073078554564695, 29108790529242620, 84194127583139219, 243734132718405830, 706171736309540520, 2047610949222803792, 5941724010837544122, 17254056039415814789, 50138381671889842151, 145793214118265385208, 424209581629997008364, 1235062051960423283060, 3597922845367850458123, 10487208390879264918883, 30584663463760785497977, 89243080822651069459771, 260533876350475763594648, 760965285519087098174307, 2223663640882079998345485, 6500846940194260255293338, 19013441322241949351689809, 55633340339776650260833834, 162849771289646446389856459, 476882138630038856845066186, 1397016057262264292104843536, 4094048498713833927119509096, 12002199935911865573364250368, 35198186649823939053399904897, 103258726164565685925228187639, 303023253756858648014378213498, 889536432693308266926378432142, 2612078093684686900158388271554, 7672544560379181785324320290350, 22543424908109986923445574304695, 66255831375073245682145803993412, 194781951077045009504532440141396, 572783503254869268864047217594022, 1684792405001414647348856645732013, 4956937407787336387901741593452936, 14587765426674530465484714813300616, 42940753950790728299148874663609176, 126430973352845019344790249089690895, 372338273101134145496109926981954000, 1096780740183236388272441528847245066, 3231450663204362980554731118035642044, 9522885200305441053069920474784222079, 28069241670114708661852669209760908380, 82752629977973030745221962129345214936, 244016886093782321888697308342626742333, 719685797597200359808440118040501049515, 2122995392809778717115751666984513854737, 6263778537286879755046906861656257371389, 18484307155950499152062033476773319732083, 54556647215785861139168869263297189076433, 161052775284673849736973509097646627234736, 475513860462577865152939584703040055308473, 1404206597899854279669242235315364776065237, 4147345179697928605850235606984347636322434, 12251216958902729677442285550386020013605594, 36195679054955691421660475357757577042688188] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 150 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 3 2 2 3 2 1 + (x - x + x - x + 1) P(x) + (x - x + x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence 3 (n - 2) a(n - 1) (4 n - 13) a(n - 3) a(n) = ------------------ - 2 a(n - 2) + ------------------- n - 1 n - 1 5 (n - 4) a(n - 4) (4 n - 19) a(n - 5) 2 (n - 7) a(n - 6) - ------------------ + ------------------- - ------------------ n - 1 n - 1 n - 1 3 (n - 6) a(n - 7) (n - 7) a(n - 8) + ------------------ - ---------------- n - 1 n - 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 3, a(7) = 6, a(8) = 11] Just for fun, a(1000), equals 2526417588552551239087622713450782674607200349495548920098374171706326696905\ 987358984632091573940154973448477556026245607852146796696175220174654987\ 855874461174948050222440872108946014788406313744511562534901265968509284\ 696954151804670470433712742886374769364628648205940828707849254227729188\ 160424264587073078347824319516574225054899783553944301246415126657279029\ 7664919984405001150954576413689084208232811909097 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 3, 6, 11, 24, 53, 116, 261, 597, 1377, 3208, 7540, 17847, 42510, 101827, 245126, 592710, 1438905, 3505841, 8569961, 21012016, 51659463, 127329616, 314573101, 778844979, 1932193043, 4802416193, 11957066786, 29819210918, 74478329481, 186288350530, 466580654020, 1170090482218, 2937873493933, 7384811244576, 18582851192462, 46808845168667, 118022259461333, 297851482578911, 752343804020697, 1901936033410366, 4811958676118293, 12183690339863776, 30871118366907811, 78276051584747019, 198607586117122433, 504244586228095886, 1281014288157823436, 3256293260938671167, 8282090094784809942, 21076279669832244023, 53663215979973367078, 136703565278629257257, 348413920051606365109, 888415631303410400693, 2266390985297382939244, 5784218641490669913674, 14768602533175461412084, 37723576913059959189570, 96396065963852544869335, 246418456775384193136474, 630157907211050507794279, 1612065850169479898161270, 4125424301952818374373046, 10560934852242089082996167, 27044542703161579918191368, 69278151330674102881751756, 177520420186017784348428806, 455021484188559321940986022, 1166657521497332187638051722, 2992121704478148049224177563, 7676020787730604195115180260, 19697486679936546300310811022, 50559197165666329320968518355, 129807928435748370252434530892, 333358164897700274190418864591, 856302164478548731957014666276, 2200120148450921474862393099840, 5654140661698370047863287417184, 14534001765488032670590374660118, 37367997383741023386509197290397, 96096631510320455105379909994257, 247176980049779000805831259912359, 635912356770482453319652758048312, 1636341079706963296702745524874009, 4211489689376844826337823791129060, 10841292672959538903552987010730415, 27913092707747324523128458535889462, 71881083367693079248232340926059572, 185139591151355760086148154405710139, 476936254132963720949924420102440423, 1228842138771347352467775205360079295, 3166686001959774863542602151823218212, 8161792150076008893563071326754946445, 21039538144777814329082414357547689162, 54244484058143330790953988418597650491] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 151 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ HH HH + HH HH + H H + HH HH 4+ HH H H + H HH H H + HH HH HH HH 3+ H H H H + HH HH HH HH + HH H HH HH 2+ HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - x + x - x + 1) P(x) + (x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 4, 8, 18, 40, 91, 213, 506, 1218, 2966, 7294, 18090, 45194, 113632, 287327, 730186, 1863977, 4777509, 12289898, 31720176, 82117774, 213178108, 554823435, 1447397817, 3784127064, 9913343789, 26018984599, 68410290554, 180162323411, 475196423227, 1255186628492, 3319955888745, 8792478337054, 23313863133954, 61888878237389, 164467529111164, 437514750895099, 1165005085521436, 3105022797156006, 8282937176309562, 22114068273505488, 59088226972484393, 158003152307405371, 422813155834784436, 1132232400449265104, 3033990842200935686, 8135281602930567564, 21827264601898788794, 58598143522316010039, 157404179282151204174, 423045542218716765139, 1137594743452912452613, 3060618745844589852386, 8238424391588288975038, 22186297926687272423033, 59775628331378116451380, 161122179240285688242086, 434482162221497135042578, 1172108773956891793300694, 3163278407606545361590941, 8540335279906181472947846, 23066152471490359501137078, 62320812139929346207300967, 168439506971919225588258595, 455410615205746441651620508, 1231703823760505986269633728, 3332339148450456918147838167, 9018366051490575879544837616, 24413972116862043819738815721, 66111540013999155652084562422, 179077417530677201597463612360, 485205493746336496557728723796, 1315008887370019158511357609032, 3564894162691250059222924187153, 9666662859580349116822421353996, 26218968527010913350495426276500, 71131335168680718796826444485856, 193023408967072905414753719183441, 523914112753491222458732852061593, 1422357723794427722247327153795571, 3862369444640190578023693432554655, 10490416497562008857897222598208662, 28498590066657605402946101068227090, 77436123391842858541274538722111900, 210451173947131125254485079572740558, 572064013100223785974710999433555658, 1555326255238638414011723574401111725, 4229413184640997453434597390339399864, 11503199764674649726828795101413864073, 31292149913413969843695821586747884491, 85139020625938221899250912882028413646, 231684389965423872121630053321756963050, 630577096684214752553581697741642950844, 1716529914048561172329410428733040296684, 4673420308399441399562494693207807848635, 12725860269271441688238739378854893163811] ------------------------------------------------------------ Theorem Number, 152 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - 2 x + x - x + 1) P(x) + (2 x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 23, 46, 90, 178, 362, 744, 1535, 3194, 6704, 14152, 30019, 63993, 137027, 294528, 635254, 1374563, 2982983, 6490689, 14157809, 30951925, 67809639, 148847546, 327326203, 721035228, 1590820794, 3515055239, 7777682719, 17232166570, 38226855740, 84899762419, 188767040056, 420148088468, 936077556199, 2087530911028, 4659567865103, 10409537599216, 23274106872900, 52077999630392, 116616556556144, 261322452488208, 585990624975113, 1314890839669959, 2952311871586680, 6632807182343025, 14910247851249936, 33536291670621966, 75470769540757993, 169928986936543073, 382800909986262158, 862755122454412830, 1945377625304332347, 4388490288087053044, 9904091476130975332, 22361254774054848157, 50507256400673637226, 114125267802978496554, 257973369424416665508, 583348170337435269883, 1319580231355477362189, 2986029400547774283515, 6759245279299394441018, 15305372653077265127430, 34667865012474818872923, 78549578510974131598343, 178028874433432077107813, 403611252116127504636989, 915290323330934693322979, 2076222803712862095242661, 4710915866189376517310722, 10691779547594570193232569, 24271966712323362836221071, 55114699810857040680904084, 125179934207220209125653142, 284383356087736137374477620, 646209443923063259545590287, 1468722246098351522200998572, 3338881124332810099290521494, 7591977293065388173220549042, 17266301921829442269815327549, 39276443078065295093655855985, 89361682067950853306373719225, 203355033871187599510752721774, 462850820082071205035423702170, 1053677818530795805415739939561, 2399128776874249888355418021047, 5463569785243751528140091257717, 12444429345978153227988825135555, 28349632447886106488535958107141, 64594015240375900277795861639819, 147200074451998200230485801692760, 335500536871502998738039665095175] ------------------------------------------------------------ Theorem Number, 153 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H H H + H HH HHH HHH + HH H H HH H HH + H HH H H HH H 3+ H H H H + HH HH + H H + HH HH 2+ H H + H H + HH HH + H H 1+ H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-+*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 3 -1 + (x - 3 x + 6 x - 7 x + 7 x - 4 x + 1) P(x) 4 3 2 2 2 + (-3 x + 5 x - 9 x + 8 x - 3) P(x) + (2 x - 4 x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 5, 12, 29, 74, 199, 554, 1580, 4599, 13629, 41007, 124950, 384808, 1196010, 3747027, 11821256, 37523229, 119754660, 384047582, 1236976421, 3999780023, 12979240458, 42253474511, 137960509242, 451669185680, 1482402326444, 4876508223150, 16075907075992, 53100657695590, 175720809546355, 582495048581828, 1934011326455707, 6431021929933012, 21414877019549000, 71405175936331293, 238390696364328014, 796827183622750392, 2666406700941573674, 8932035228863731542, 29951087039643233174, 100528914233734415521, 337726663331342107359, 1135577613895114232222, 3821447399021126861689, 12870108500369910092496, 43377502675993578865587, 146305498040330514857658, 493806920018551613984013, 1667790945150358726863700, 5636405421129196150426245, 19060222673631623909309669, 64492388315933639743827121, 218340300664204189679262424, 739596870103919458535559424, 2506589512129250439904929232, 8499436166773644383255497931, 28834203095531408174101316777, 97865435244147141643724580885, 332312929827145883877564896974, 1128898818275654051949507939112, 3836598428445000055724268993497, 13044132578449189905222612800257, 44366572070761163512125173414595, 150960354073285919729699123970279, 513843778173627271234045629372531, 1749667435722230424981844473882405, 5959796016071331726800894082927865, 20307396272010013338048407694338611, 69218109534357021809344240032151246, 236006375173418561456532818720534102, 804937742592748279543019011219757331, 2746196546755413745801079650875838825, 9371909379169374468353399703869545521, 31992501744803496108426994087772295442, 109241748360914014259847741134534865867, 373117994182316404320539025462855575157, 1274728742041129800013073990663875472910, 4356126017486989008745040937927098462749, 14889884127520776439199122505070866766943, 50908193937206048172852892645702958973655, 174095247164727931352474741478183236910534, 595506430413343673847234697689138746032752, 2037435488027457279179217158357126956782541, 6972311652168179652233080636088514515646073, 23865083446147062045271868163117891384795476, 81703408011380205708376672572522315362235932, 279773335179699265305006937822281020199174023, 958207041437321309580759936460430793948328530, 3282443900489104842728265442779956311398980686, 11246523018497041223979591658610452594266227547, 38540771747913170357268601376105839730333735506, 132099727203511387502602637549167246847556748004, 452857008965978701094742813830497985574565158089, 1552731118955646347123917939554121904818246830295, 5324830582291895640932544203205749100761200265373, 18263673804706504903253280721851321222705981935308, 62652987313369405584985020744074808260772456463574, 214963759678046363792947428718327637564925230657702] ------------------------------------------------------------ Theorem Number, 154 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ HH HH + HH HH + H H + HH HH 4+ HH H H + H HH H H + HH HH HH HH 3+ H H H H + HH HH HH HH + HH H HH HH 2+ HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - x + x - x + 1) P(x) + (x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 4, 8, 18, 40, 91, 213, 506, 1218, 2966, 7294, 18090, 45194, 113632, 287327, 730186, 1863977, 4777509, 12289898, 31720176, 82117774, 213178108, 554823435, 1447397817, 3784127064, 9913343789, 26018984599, 68410290554, 180162323411, 475196423227, 1255186628492, 3319955888745, 8792478337054, 23313863133954, 61888878237389, 164467529111164, 437514750895099, 1165005085521436, 3105022797156006, 8282937176309562, 22114068273505488, 59088226972484393, 158003152307405371, 422813155834784436, 1132232400449265104, 3033990842200935686, 8135281602930567564, 21827264601898788794, 58598143522316010039, 157404179282151204174, 423045542218716765139, 1137594743452912452613, 3060618745844589852386, 8238424391588288975038, 22186297926687272423033, 59775628331378116451380, 161122179240285688242086, 434482162221497135042578, 1172108773956891793300694, 3163278407606545361590941, 8540335279906181472947846, 23066152471490359501137078, 62320812139929346207300967, 168439506971919225588258595, 455410615205746441651620508, 1231703823760505986269633728, 3332339148450456918147838167, 9018366051490575879544837616, 24413972116862043819738815721, 66111540013999155652084562422, 179077417530677201597463612360, 485205493746336496557728723796, 1315008887370019158511357609032, 3564894162691250059222924187153, 9666662859580349116822421353996, 26218968527010913350495426276500, 71131335168680718796826444485856, 193023408967072905414753719183441, 523914112753491222458732852061593, 1422357723794427722247327153795571, 3862369444640190578023693432554655, 10490416497562008857897222598208662, 28498590066657605402946101068227090, 77436123391842858541274538722111900, 210451173947131125254485079572740558, 572064013100223785974710999433555658, 1555326255238638414011723574401111725, 4229413184640997453434597390339399864, 11503199764674649726828795101413864073, 31292149913413969843695821586747884491, 85139020625938221899250912882028413646, 231684389965423872121630053321756963050, 630577096684214752553581697741642950844, 1716529914048561172329410428733040296684, 4673420308399441399562494693207807848635, 12725860269271441688238739378854893163811] ------------------------------------------------------------ Theorem Number, 155 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H + HH H + H HH + HH H 2+ H H + H HH + HH H + H HH 1+ H H H H H + HH H H HH H HH H HH + H H HH H HH H HH H +HH HHH HHH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-*+-+-+-+-*+-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 11 8 7 6 5 4 3 2 4 1 + (x + 5 x - 4 x + x + 8 x - 13 x + 11 x - 5 x + x) P(x) 10 8 7 6 5 4 3 2 3 + (x - 2 x + 7 x - 5 x - 5 x + 15 x - 17 x + 6 x + x - 1) P(x) 7 6 5 4 3 2 2 + (-x + 3 x + 2 x - 5 x + 10 x - 2 x - 5 x + 3) P(x) 4 3 2 + (x - 2 x + x + 3 x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 5, 11, 24, 56, 137, 342, 868, 2244, 5898, 15710, 42318, 115130, 315988, 873987, 2433850, 6818817, 19207583, 54368124, 154566571, 441170093, 1263743791, 3631928823, 10469366215, 30262294237, 87697552313, 254737832828, 741557710866, 2163091745386, 6321543350471, 18506961317790, 54270237686657, 159388976479292, 468796269858875, 1380709235668826, 4071726468668022, 12022149426495012, 35537281370217828, 105161912796014943, 311516186564212982, 923693984027211180, 2741448599955929354, 8143614844929624191, 24211486350973102526, 72040380979057674053, 214518725825060434720, 639255444595557293027, 1906294432131198569606, 5688516172292896015312, 16985934111754227612671, 50751626579012467559668, 151729466496455616456714, 453878163512655310718512, 1358465365027806620823730, 4068073895590287827596524, 12188538292628170746494937, 36536669130279271273984796, 109575443745411176634589183, 328773877188688943209402226, 986902694241744874001063914, 2963724710651864454152959060, 8903929895112963035115096535, 26760861777760479738154583626, 80461361687147615051359123250, 242012764225416332274809693066, 728195035609549205333788549825, 2191850210261955076448200735234, 6599683829476151407315998952988, 19878338456197719524514453209314, 59893198378853047065859564378161, 180514199286027803237968674182642, 544224147309109545341009069931750, 1641243874197093363128061723845171, 4951009406218658131790885463642588, 14939508254280379949450573966840561, 45091792880100598379264544493548085, 136136385534215301418904256079601606, 411115095603962266818532083813502917, 1241830232360723880955925599610699965, 3752043299066990230660285310808424077, 11339073735727996386620246050598846552, 34275899331747270617690918316024634201, 103633267292098387972530198824553304851, 313405184841647748330399551839462469697, 947998184578353904948223285745480323621, 2868143962814566734042495633455499430815, 8679292946943892987023681298595543891888, 26269731274270809187344129769013473494574, 79526668456480056173967770688485742475580, 240798560242717163434488091712960480611620, 729251021974818567525999460776053339798709, 2208922400001670223713295626979935584881835, 6692099232802723883171399688028284064369134, 20277811881705502440640261639290768794228070, 61454681299114118345047991421087817478357603, 186278368760806250961881329637233439138917486, 564731374967657902503340295115976495961162145, 1712347684359215469609616649122211481378906176] ------------------------------------------------------------ Theorem Number, 156 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 7+ HH + H HH + HH HH 6+ H HH + H HH + HH HH 5+ HH HH + H HH + HH HH 4+ H H + HH HH 3+ HH HH H + HH HH H HH + HH HH HH HH 2+ HH HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 10 6 5 4 3 2 3 1 + (x - 2 x + x - 2 x + 2 x - x + x - 1) P(x) 7 6 5 4 3 2 2 + (x + x - x + x - 3 x + 2 x - 2 x + 3) P(x) 4 3 2 + (x + x - x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 17, 34, 65, 133, 277, 572, 1199, 2548, 5435, 11661, 25202, 54731, 119348, 261396, 574697, 1267637, 2804766, 6223660, 13845748, 30876213, 69007994, 154549822, 346790404, 779545026, 1755254743, 3958377577, 8939866583, 20218263933, 45784748248, 103807846272, 235636648909, 535466068728, 1218071478804, 2773591163064, 6321494978078, 14420661400511, 32924586831606, 75232979162492, 172041452385538, 393713260872074, 901644278692778, 2066263403285064, 4738263632158243, 10872374445702404, 24962637101521442, 57346546860289679, 131815132773209257, 303148512760931479, 697540229550486854, 1605826655979897539, 3698588507211274869, 8522638804153427168, 19647441284817832815, 45313253580871653227, 104550383069514369365, 241324409472237417738, 557245419733284748247, 1287229693480919378873, 2974574026651753256696, 6876189561850918905880, 15900858902075293511482, 36782273871125357477166, 85113327142737810030492, 197012375003682813852835, 456165570455991041730121, 1056527571106193560850298, 2447738008468144928854371, 5672459058686943865573806, 13149125315278295774252093, 30488654546469742502130612, 70711902848157227103981151, 164042637933125322263068179, 380652001553129860482798995, 883494697499474927314001201, 2051075452619209350256289939, 4762759874731327435883409954, 11061975317641962863310400262, 25698117521092741572233415672, 59712096957248664358561521889, 138775742402184313315715377316, 322591509897645444074929806223, 750029645171417394902589537216, 1744167142214391071760524674061, 4056767007670417319587124446474, 9437403083691333062035763415892, 21958549988526521939432355828298, 51101285390525603579799232888122, 118942032240129949401856916912786, 276893431455538839876673911173260, 644706688082075915267897498545600, 1501351643055827208280357431912040, 3496809282282403489853994898965247] ------------------------------------------------------------ Theorem Number, 157 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 7+ HHH + H HH + HH HH 6+ H H + H H + HH HH 5+ HH HH + H H + HH HH 4+ H H + HH HH 3+ H HH HH + HH H HH HH + HH HH HH HH 2+ HH HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 3 4 3 2 2 -1 + (x + 2 x - x + x - x + 1) P(x) + (-2 x + x - 2 x + 2 x - 3) P(x) 2 + (x - x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 46, 94, 199, 439, 961, 2113, 4728, 10638, 24004, 54567, 124738, 286100, 658902, 1523561, 3533694, 8219807, 19175241, 44844787, 105117745, 246938893, 581271855, 1370801044, 3238379640, 7662884761, 18160172211, 43099390904, 102426035307, 243726755578, 580655580178, 1384932709103, 3306794777897, 7903693118041, 18909278995761, 45281554377074, 108530027712894, 260341203346223, 625005317744160, 1501608816996828, 3610336668014877, 8686460115399371, 20913632597618146, 50384354111881549, 121459160713524172, 292969666722436726, 707070780696631056, 1707424831884850449, 4125246497394886642, 9971935177734506502, 24116945055530422878, 58354041165562962555, 141259692927959905224, 342103485342872695337, 828862025018976189145, 2009029841925232601585, 4871515952686337473281, 11817073539321134031521, 28675998351756531656835, 69612131191902560435320, 169045763869085582882160, 410650213529812223368770, 997892674377742909288876, 2425690847473912236264926, 5898246126299524653958956, 14346379729918995608354591, 34905194678328946279956726, 84949845555910352980918162, 206802853409472003358290619, 503580166605875837883509918, 1226579626343965688411938881, 2988373791468938272294751313, 7282546274134082391128998885, 17751619375203540121720680747, 43280914162805330098090017988, 105549463086185917149368841094, 257462696364279602325677747884, 628158000160191822477935767084, 1532912888447769419915804392627, 3741604135787909331331389618256, 9134564337311934498324607449420, 22305163745068650394954160327031, 54476421551688686284866768832516, 133074664291962641673573493873036, 325135188262952797925515855558143, 794534125103762472134490936108882, 1941955965184663382273487480183164, 4747257141764020796262302049059161, 11607028692460250300264994515896655, 28383941376456748393597046710009226, 69421845442964953819449619451462120, 169820428896576356561050206615538112, 415482369434004891765163304919548259] ------------------------------------------------------------ Theorem Number, 158 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - 2 x + x - x + 1) P(x) + (2 x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 23, 46, 90, 178, 362, 744, 1535, 3194, 6704, 14152, 30019, 63993, 137027, 294528, 635254, 1374563, 2982983, 6490689, 14157809, 30951925, 67809639, 148847546, 327326203, 721035228, 1590820794, 3515055239, 7777682719, 17232166570, 38226855740, 84899762419, 188767040056, 420148088468, 936077556199, 2087530911028, 4659567865103, 10409537599216, 23274106872900, 52077999630392, 116616556556144, 261322452488208, 585990624975113, 1314890839669959, 2952311871586680, 6632807182343025, 14910247851249936, 33536291670621966, 75470769540757993, 169928986936543073, 382800909986262158, 862755122454412830, 1945377625304332347, 4388490288087053044, 9904091476130975332, 22361254774054848157, 50507256400673637226, 114125267802978496554, 257973369424416665508, 583348170337435269883, 1319580231355477362189, 2986029400547774283515, 6759245279299394441018, 15305372653077265127430, 34667865012474818872923, 78549578510974131598343, 178028874433432077107813, 403611252116127504636989, 915290323330934693322979, 2076222803712862095242661, 4710915866189376517310722, 10691779547594570193232569, 24271966712323362836221071, 55114699810857040680904084, 125179934207220209125653142, 284383356087736137374477620, 646209443923063259545590287, 1468722246098351522200998572, 3338881124332810099290521494, 7591977293065388173220549042, 17266301921829442269815327549, 39276443078065295093655855985, 89361682067950853306373719225, 203355033871187599510752721774, 462850820082071205035423702170, 1053677818530795805415739939561, 2399128776874249888355418021047, 5463569785243751528140091257717, 12444429345978153227988825135555, 28349632447886106488535958107141, 64594015240375900277795861639819, 147200074451998200230485801692760, 335500536871502998738039665095175] ------------------------------------------------------------ Theorem Number, 159 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 7+ HHH + H HH + HH HH 6+ H H + H H + HH HH 5+ HHH HH HH + H HH H H + HH HHH HH 4+ H H H + HH HH 3+ HH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 10 6 5 4 3 2 3 1 + (x - 2 x + x - 2 x + 2 x - x + x - 1) P(x) 7 6 5 4 3 2 2 + (x + x - x + x - 3 x + 2 x - 2 x + 3) P(x) 4 3 2 + (x + x - x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 17, 34, 65, 133, 277, 572, 1199, 2548, 5435, 11661, 25202, 54731, 119348, 261396, 574697, 1267637, 2804766, 6223660, 13845748, 30876213, 69007994, 154549822, 346790404, 779545026, 1755254743, 3958377577, 8939866583, 20218263933, 45784748248, 103807846272, 235636648909, 535466068728, 1218071478804, 2773591163064, 6321494978078, 14420661400511, 32924586831606, 75232979162492, 172041452385538, 393713260872074, 901644278692778, 2066263403285064, 4738263632158243, 10872374445702404, 24962637101521442, 57346546860289679, 131815132773209257, 303148512760931479, 697540229550486854, 1605826655979897539, 3698588507211274869, 8522638804153427168, 19647441284817832815, 45313253580871653227, 104550383069514369365, 241324409472237417738, 557245419733284748247, 1287229693480919378873, 2974574026651753256696, 6876189561850918905880, 15900858902075293511482, 36782273871125357477166, 85113327142737810030492, 197012375003682813852835, 456165570455991041730121, 1056527571106193560850298, 2447738008468144928854371, 5672459058686943865573806, 13149125315278295774252093, 30488654546469742502130612, 70711902848157227103981151, 164042637933125322263068179, 380652001553129860482798995, 883494697499474927314001201, 2051075452619209350256289939, 4762759874731327435883409954, 11061975317641962863310400262, 25698117521092741572233415672, 59712096957248664358561521889, 138775742402184313315715377316, 322591509897645444074929806223, 750029645171417394902589537216, 1744167142214391071760524674061, 4056767007670417319587124446474, 9437403083691333062035763415892, 21958549988526521939432355828298, 51101285390525603579799232888122, 118942032240129949401856916912786, 276893431455538839876673911173260, 644706688082075915267897498545600, 1501351643055827208280357431912040, 3496809282282403489853994898965247] ------------------------------------------------------------ Theorem Number, 160 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 7 6 3 2 2 4 3 2 1 + (x + x - 2 x + x - x + 1) P(x) + (x + 2 x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 5, 9, 15, 26, 48, 89, 165, 310, 591, 1138, 2205, 4297, 8427, 16622, 32938, 65530, 130863, 262249, 527197, 1062810, 2148146, 4352276, 8837592, 17982158, 36658592, 74865246, 153145667, 313763676, 643771865, 1322679887, 2721049245, 5604608198, 11557189002, 23857809325, 49300998950, 101977645247, 211133587197, 437515330510, 907391552370, 1883409945578, 3912262282749, 8132599519471, 16917470461058, 35215450619627, 73351807432625, 152882167526053, 318831083771137, 665291972000431, 1389000105612281, 2901494411312661, 6064042668767917, 12679900670970242, 26526198371525921, 55517844071636598, 116247016822421760, 243509692791621693, 510304958692509935, 1069834525683015957, 2243733025635694956, 4707477231914068428, 9880131493978309950, 20743874289698006871, 43567745659235343423, 91534327932366425904, 192372244159797445202, 404423446349987694296, 850475692006620639552, 1789021153134275019838, 3764380495324636198963, 7923054498549552785817, 16680521298024087725246, 35127021615881690891674, 73992001969543494910782, 155896761247337062814246, 328545486973552202438004, 692559921201329410518511, 1460226000605629209724777, 3079507053993080839704110, 6495885392920695093764064, 13705321751718305123493788, 28922220684172584949140422, 61046872393071002819789051, 128879108693661861817539403, 272136603848726764672198432, 574744475893032262155788238, 1214071321413282376714153974, 2565035305811338986244912025, 5420264847804173772189630486, 11455762461878381622441607611, 24215988133284034517908107932, 51198061405255014156095915233, 108262097724071227150484889909, 228965199188889890733294634337] ------------------------------------------------------------ Theorem Number, 161 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} To make it crystal clear here is such a path of semi-length 8 6+ H H + HHHH HHHH + H H H H 5+ HH HHH HH + HH H + H H + HH HH 4+ H H HH + H H HH H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 2 3 2 3 (x + 2 x - 3 x + 1) P(x) + (-2 x - x + 4 x - 2) P(x) + x - x + 1 = 0 The sequence a(n) satisfies the linear recurrence (7 n - 20) a(n - 1) 2 (7 n - 23) a(n - 2) (7 n - 26) a(n - 3) a(n) = ------------------- - --------------------- + ------------------- n - 2 n - 2 n - 2 2 (2 n - 7) a(n - 4) + -------------------- n - 2 subject to the initial conditions [a(1) = 1, a(2) = 1, a(3) = 2, a(4) = 4] Just for fun, a(1000), equals 8235746390933080467431311536856274089728305179579442375509746248407883283497\ 565781478754853559776862023051609338958345336727438155196363894430195515\ 444556423287922490411765317109186012765328426039643336604286127049210422\ 773408881212887835962798419317472538239101884774683617388054133207989729\ 866451232399693674215470078104215371378538223128948456372826121241951560\ 895707635690970474740004302573499588506007411419224026939664440417936239\ 506584737179318910257406721247670473326351168180750166295350885504291924\ 150317852140732684188770722817623569886156023912132267502637619068388630\ 1318521270485232 For the sake of the OEIS here are the first, 100, terms. [1, 1, 1, 2, 4, 9, 22, 58, 163, 483, 1494, 4783, 15740, 52956, 181391, 630533, 2218761, 7888266, 28291588, 102237141, 371884771, 1360527143, 5002837936, 18479695171, 68539149518, 255137783916, 952914971191, 3569834343547, 13410481705795, 50506214531252, 190661396754623, 721309231245930, 2734340902348660, 10384729349891801, 39508833052959062, 150556764433793122, 574604028257812551, 2196133765002623111, 8404921968965752371, 32207692734494651832, 123567904825171570007, 474617269824566355386, 1824933678818837074702, 7024125634258699052147, 27061810866951754785671, 104356677565423473137257, 402775764533193562448982, 1555852247194778679374801, 6014766389029773513342902, 23270147619699743008876442, 90093831591035950206998211, 349054331386847956567890747, 1353257428538389364335407949, 5249831449006596883287218598, 20378710829070480349853765305, 79152379106432483868793868282, 307607755667932328273940701518, 1196099515989815008181990112109, 4653345274135966755121774940981, 18112685962605211418277062361519, 70536220680328737640650868003150, 274818405130762127363775023157511, 1071215083196600681122389714471826, 4177323183244069093685741008136202, 16296848936793099441935749419523779, 63604423103740186283024835074100419, 248338098813162270659452650321497622, 969987771171029999449381574110315199, 3790101265933944939618704480857715284, 14814671741206925284291813349880162932, 57927549942538373437547202127871692979, 226582129078227610530624915866698146289, 886562118118615134989272323449529514677, 3470016654677134080958962113677550103074, 13585913523362049921786656972261958849929, 53208035532322666331381772934235887854662, 208445752260644142910821280376855464645554, 816832150505028367285449929808641793404649, 3201792565891852961532086177745940693859197, 12553676574768161918442923601638547491147259, 49233755168825636040190509798702664485568774, 193137363402581189781843747869732287019888207, 757840984470589034800991146439643751861772044, 2974374049385115360730728310785431473727584884, 11676592865799907305271600189113407813938977097, 45849774529775678994287493317245372378896223759, 180076204941122301719476603297392517178068338299, 707409923937410757893225686357741965383144870722, 2779581290322485827219473017204671948137505116329, 10923927891708038398667186903772046854896908583564, 42940526005207159580738533951311940846400161042252, 168827401527645830488158392515333238343950696104299, 663901382984815930428284363518445433570702229671429, 2611243420694303752657197022109899747540266225714237, 10272412328169638252044717519540824646909475201719354, 40418206582627380727519801114662572244091122051893759, 159059428206188977677275334863625816582883878433966332, 626062804643219538493657461465488203619840743484621724, 2464625148912081487251507784009490002286894109768942799, 9704133613649570792653803635613114442747623661282312917, 38215017094610905301981615724011907881762196971724186529] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 162 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H + HH + HH HH + H H + H H 3+ H H H H + H H H H H H + H HH H HH HH H + H H H H H H + H H H H H H 2+ H H H H H H + H H H H H H + H H H H H H + HH H HH HH H HH + H H H H H H 1+ H H H H + H H + H H + HH HH +H H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 2 2 (x + 3 x + 3 x - 2 x - x + 1) P(x) 5 4 3 2 4 3 2 + (-2 x - 4 x - x + 3 x + 2 x - 2) P(x) + x + x - x - x + 1 = 0 The sequence a(n) satisfies the linear recurrence 3 (n - 3) a(n - 1) 3 (n - 4) a(n - 2) (7 n - 29) a(n - 3) a(n) = ------------------ + ------------------ - ------------------- n - 2 n - 2 n - 2 9 (n - 4) a(n - 4) 3 (n - 5) a(n - 5) 2 (7 n - 32) a(n - 6) - ------------------ + ------------------ + --------------------- n - 2 n - 2 n - 2 (11 n - 52) a(n - 7) 3 (n - 5) a(n - 8) + -------------------- + ------------------ n - 2 n - 2 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 2, a(6) = 5, a(7) = 10, a(8) = 23] Just for fun, a(1000), equals 1535875550442896470095395825402152730540485059103186263221372801210323501244\ 111578111330181394715884418663118516141187615047640646477044894473357803\ 280274668210582403538317400444139846335917880226813081544286284810308729\ 072647413532602648950872610016959230151065821816599284663356610746800243\ 319291447553307283277134817529956699076645731800804659833853197106280275\ 021893508153099310089662316073536841861747025996168499724538716524639550\ 578422169119005948677208007890030693 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 2, 5, 10, 23, 54, 129, 317, 794, 2022, 5226, 13676, 36177, 96599, 260049, 705091, 1923871, 5278828, 14556831, 40321752, 112140498, 313018373, 876631744, 2462514328, 6936562032, 19589194824, 55451046515, 157306752376, 447158262791, 1273474767476, 3633119478220, 10381968098066, 29712930309197, 85160125438235, 244407751581020, 702340334430984, 2020705977007378, 5820402230049975, 16783053380443034, 48443202503099686, 139963962492011145, 404762620644449291, 1171564282548823780, 3393872490117955949, 9839475432357248051, 28548274571200159244, 82890502143463936992, 240842477782233628640, 700248347213646076273, 2037275920765430630412, 5930825566325687271777, 17275771005098358805489, 50350846660497998342893, 146829661190705224368913, 428400183540167551362836, 1250564138668682100196765, 3652370722363131893469317, 10672073370795706491503871, 31197566333949098811240969, 91239701690971305877812665, 266951165505227719446991440, 781373227089388454918837340, 2288011399507360177012440001, 6702321896661965308412015488, 19640591793504081872390683473, 57575940058609138924239645255, 168841774108237705791077565384, 495298043850689353798034091939, 1453438944634086434071361342474, 4266446216280524734268914482487, 12527691996807021824410833490622, 36796570730605031661551108337769, 108111389444692259051924439861599, 317731200083074237162198393694548, 934048053501838715982921569892333, 2746605497001170880027951059897851, 8078634368898013673128034690902148, 23767923398552104647316379507412729, 69944452421307104365515113219950301, 205883384600549446793428001836825007, 606167512898831453987377643603758496, 1785109209162200614319479019390848605, 5258177423204439347660160184559334513, 15491790657701878298307393633348809066, 45652193765306071921753371732920059627, 134559115778273182391365815932012805150, 396692424348014937680829010683476694375, 1169720069134376658629373194376390769530, 3449810510114653631841050285956378673623, 10176346501147241736346948992922602590389, 30024101604799559003910287303700174337197, 88598802405588036607244395624654392086364, 261495152070740315272955295758480227482047, 771926076709973082034432567707122551614768, 2279095045901399420133068593235002964846064, 6730111230265473608105224556506581870856631, 19877126509423817034725118069338167548089462] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 163 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 4+ H + HH + HH HH + H H + H H 3+ H H + H H + H HH + H H + H H 2+ H H + H H + H H + HH H + H H 1+ H H H H H H + H H H H H H H H H H + H H H H H H H H H H + HH HH HH HH HH HH HH HH HH HH +H HH HH HH HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-*+-+-+-+-*+-+-+-+-*+-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 5 4 3 2 3 (x - 2 x + 7 x - 9 x + 12 x - 14 x + 10 x - 7 x + 4 x - 1) P(x) 8 7 6 5 4 3 2 2 + (2 x - 7 x + 12 x - 16 x + 23 x - 18 x + 13 x - 9 x + 3) P(x) 7 6 5 4 3 2 6 4 + (3 x - 4 x + 7 x - 12 x + 10 x - 8 x + 6 x - 3) P(x) + x + 3 x 3 2 - x + 2 x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 4, 8, 18, 43, 107, 279, 756, 2110, 6037, 17640, 52456, 158288, 483568, 1492894, 4650704, 14601568, 46156995, 146782489, 469257258, 1507289667, 4862041172, 15743274434, 51152852674, 166727740156, 544993582602, 1786147562570, 5868064834144, 19321628076330, 63751777318042, 210754466774987, 697975087107177, 2315421894719860, 7693090494781241, 25598222027331448, 85294024580154610, 284572037791424554, 950601751638645939, 3179130706928813538, 10643746810073950944, 35672387552131100130, 119673593195733915195, 401857178134742726339, 1350618393155426577870, 4543209931421070301786, 15294846515110145969343, 51530425377117115234766, 173741587423603782151574, 586207574469557362541280, 1979218124244542799319645, 6686799535334889776000855, 22605489641467786941414103, 76466241842357597349351263, 258806785222998659644379449, 876439833546094218547775979, 2969614566519881761428112761, 10067023389154832346188732615, 34144207402697469332697607741, 115861817772408995037076400857, 393336058449361133198955361917, 1335920011416083390169608539566, 4539248043629602439869966785401, 15430072235316580650979902643615, 52471890167572910648457519213810, 178506719116668606641853615389704, 607499892275484997248806539885662, 2068219070372095059539330722214095, 7043697844838562923405403822540599, 23996843168988036140205693786358038, 81780978190379571611357495883258756, 278798896738905428977562298548354272, 950749960479359115869297043345124508, 3243203663431638386971922572769709059, 11066519580355582754503708128966076439, 37772285416504907067750012154875410798, 128960713439887320024276525321870170236, 440413147276017254199306291367776937065, 1504453129708127162206846786706422674943, 5140549753777339585054421186811635023123, 17569123083919020268315931672679734787381, 60061672910667023799590727140693425462132, 205375624405195940206188071345208621480266, 702428161825362847308113776450553501651394, 2403001197112171174304604878980487276107538, 8222477844715773539344006136719525737257985, 28141404876220548820431941449051025841531279, 96334299139933433311453500312549365692297382, 329842079999355892430366152512047465518715761, 1129585347059446104428707937597856978073528761, 3869170810510643588286699594961841108699713823, 13255639312157219311472436625646293112569889405, 45421921450094045893738653354575374038028375770, 155672019898697878192584926487102957689419501024, 533622460440399685530204075671873065586359432847, 1829508479059991656203558862329256709228769011741, 6273499901003238157409301436071442020268738721130, 21515870301276938946030622815245620845194845427609, 73804009021303313618741069038370335611748651676267] ------------------------------------------------------------ Theorem Number, 164 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 7 6 5 4 3 3 (x - 2 x - 5 x - 3 x - 2 x + 3 x + x - 1) P(x) 7 6 5 4 3 2 2 + (3 x + 8 x + 5 x + 6 x - 7 x + x - 3 x + 3) P(x) 7 6 5 4 3 2 6 4 3 + (-x - 4 x - 2 x - 6 x + 5 x - 2 x + 3 x - 3) P(x) + x + 2 x - x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 43, 88, 181, 389, 845, 1831, 4021, 8945, 19980, 44876, 101560, 231010, 527584, 1210453, 2788476, 6445177, 14945402, 34762125, 81073108, 189553176, 444229775, 1043335656, 2455327148, 5789089278, 13673287432, 32347923587, 76645650172, 181868152606, 432131947999, 1028094409010, 2448927381166, 5840049815756, 13942114199763, 33318698172213, 79702566442900, 190836157036246, 457334328621662, 1096917010956494, 2633078585793776, 6325404794268179, 15206619342001349, 36583326860847321, 88069812063175602, 212154504630779797, 511384369072275705, 1233394476770162448, 2976499939131468798, 7187032770060977636, 17362981676250949732, 41968309776585627857, 101492313014921186269, 245556679552748148628, 594388103914626682818, 1439399709664903706555, 3487218741318126496572, 8451957378207794626600, 20493205067305036075159, 49708582948760009479704, 120619210443382870051297, 292792615812251132529932, 710980136125717425588637, 1727046021860570120941582, 4196575501209940797812052, 10200621702455464613659962, 24802455440798284389096014, 60324716048583750707955995, 146765765674535755148007651, 357173774922625517418331537, 869473358467539692845081275, 2117150070857988843015817831, 5156588999541444162109803583, 12562783839142431859361805544, 30613913629937142427400197707, 74620580935239731515082935125, 181929247599493983737210197721, 443657637304923176383168028461, 1082162432233326872644453144656, 2640180374641018514672900497678, 6442717456740989437135783799985, 15725218378158726422687662180370, 38389658112166505912696121858576, 93738863949937324569728179979473, 228934372538591942169266458710404, 559224587509088449210338239175827, 1366292159965124659559109224182072, 3338728826968157284819565578655591, 8160132627279040287185529051465709, 19947574345781438581105186436718539, 48770603600956548080699610819303235, 119261355989728246212509483676650821, 291684525929087013741255678799558755] ------------------------------------------------------------ Theorem Number, 165 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H + HH + HH HH + H H + H H 3+ H H H H + H H H H H H + H HH H HH HH H + H H H H H H + H H H H H H 2+ H H H H H H + H H H H H H + H H H H H H + HH H HH H HH HH + H H H H H H 1+ H H H H + H H + H H + HH HH +H H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 2 2 (x + 3 x + 3 x - 2 x - x + 1) P(x) 5 4 3 2 4 3 2 + (-2 x - 4 x - x + 3 x + 2 x - 2) P(x) + x + x - x - x + 1 = 0 The sequence a(n) satisfies the linear recurrence 3 (n - 3) a(n - 1) 3 (n - 4) a(n - 2) (7 n - 29) a(n - 3) a(n) = ------------------ + ------------------ - ------------------- n - 2 n - 2 n - 2 9 (n - 4) a(n - 4) 3 (n - 5) a(n - 5) 2 (7 n - 32) a(n - 6) - ------------------ + ------------------ + --------------------- n - 2 n - 2 n - 2 (11 n - 52) a(n - 7) 3 (n - 5) a(n - 8) + -------------------- + ------------------ n - 2 n - 2 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 2, a(6) = 5, a(7) = 10, a(8) = 23] Just for fun, a(1000), equals 1535875550442896470095395825402152730540485059103186263221372801210323501244\ 111578111330181394715884418663118516141187615047640646477044894473357803\ 280274668210582403538317400444139846335917880226813081544286284810308729\ 072647413532602648950872610016959230151065821816599284663356610746800243\ 319291447553307283277134817529956699076645731800804659833853197106280275\ 021893508153099310089662316073536841861747025996168499724538716524639550\ 578422169119005948677208007890030693 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 2, 5, 10, 23, 54, 129, 317, 794, 2022, 5226, 13676, 36177, 96599, 260049, 705091, 1923871, 5278828, 14556831, 40321752, 112140498, 313018373, 876631744, 2462514328, 6936562032, 19589194824, 55451046515, 157306752376, 447158262791, 1273474767476, 3633119478220, 10381968098066, 29712930309197, 85160125438235, 244407751581020, 702340334430984, 2020705977007378, 5820402230049975, 16783053380443034, 48443202503099686, 139963962492011145, 404762620644449291, 1171564282548823780, 3393872490117955949, 9839475432357248051, 28548274571200159244, 82890502143463936992, 240842477782233628640, 700248347213646076273, 2037275920765430630412, 5930825566325687271777, 17275771005098358805489, 50350846660497998342893, 146829661190705224368913, 428400183540167551362836, 1250564138668682100196765, 3652370722363131893469317, 10672073370795706491503871, 31197566333949098811240969, 91239701690971305877812665, 266951165505227719446991440, 781373227089388454918837340, 2288011399507360177012440001, 6702321896661965308412015488, 19640591793504081872390683473, 57575940058609138924239645255, 168841774108237705791077565384, 495298043850689353798034091939, 1453438944634086434071361342474, 4266446216280524734268914482487, 12527691996807021824410833490622, 36796570730605031661551108337769, 108111389444692259051924439861599, 317731200083074237162198393694548, 934048053501838715982921569892333, 2746605497001170880027951059897851, 8078634368898013673128034690902148, 23767923398552104647316379507412729, 69944452421307104365515113219950301, 205883384600549446793428001836825007, 606167512898831453987377643603758496, 1785109209162200614319479019390848605, 5258177423204439347660160184559334513, 15491790657701878298307393633348809066, 45652193765306071921753371732920059627, 134559115778273182391365815932012805150, 396692424348014937680829010683476694375, 1169720069134376658629373194376390769530, 3449810510114653631841050285956378673623, 10176346501147241736346948992922602590389, 30024101604799559003910287303700174337197, 88598802405588036607244395624654392086364, 261495152070740315272955295758480227482047, 771926076709973082034432567707122551614768, 2279095045901399420133068593235002964846064, 6730111230265473608105224556506581870856631, 19877126509423817034725118069338167548089462] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 166 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 7 6 5 4 3 2 2 (x + 3 x + 4 x + 2 x - x - 2 x - x + 1) P(x) 6 5 4 3 2 5 4 2 + (-2 x - 4 x - 3 x + x + 3 x + 2 x - 2) P(x) + x + x - x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 4, 8, 16, 33, 70, 150, 327, 724, 1623, 3681, 8437, 19519, 45536, 107030, 253262, 602908, 1443070, 3470954, 8385578, 20340507, 49519730, 120960620, 296371546, 728194151, 1793822734, 4429426335, 10961605329, 27182576415, 67535928339, 168093551092, 419071987208, 1046409830685, 2616670011698, 6552275154848, 16428452398084, 41241152141282, 103649065442512, 260779618021894, 656796504105220, 1655823783167292, 4178326335276288, 10553000222691372, 26675752223358469, 67485051975526502, 170857040626370571, 432890244396224794, 1097560693166690293, 2784660137566683943, 7069629246725243597, 17959344777470420687, 45650238004771051545, 116103213416967047771, 295450394971008048336, 752237693339907678388, 1916228633922016792045, 4883748297887608403258, 12452752489393602069929, 31767014040001686325582, 81073622544524038530886, 206999092435241758311202, 528733513631754741849442, 1351073351501234035819687, 3453736074867883038676328, 8832063790026984856111437, 22593994962609018942548434, 57819831170065015160147258, 148016104724210103267209410, 379040037587504184490060968, 970959159026854695362349530, 2488012618157836350994116103, 6377288789306547119410546353, 16351130742851668466225675508, 41935725885251494891116908522, 107582545160831136156990545003, 276068900975039417577961191741, 708611271260935586778378177294, 1819325982174725461733301456785, 4672206168391097224635813991761, 12001615554273519602977258767978, 30836209618100401406574383084291, 79247078893538383198570456482935, 203706110082593778957038744931844, 523746317307284313148511444896921, 1346888803038138272803743432070913, 3464448567504291062470852345232066, 8913042328640841543950070604839795, 22935342432012856569972142351659937, 59029608373580246804602119121026026, 151956072880316631652959014065460778, 391244137913596746646621211123393927, 1007528749678183420473613625882526153, 2595046421486848446715296247659510435, 6685120019414906923717787261359040175, 17224556501071547042357604518932295461, 44387437195386122034755378429092171929] ------------------------------------------------------------ Theorem Number, 167 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 2 (x + 3 x + 2 x - x - x - x + 1) P(x) 5 4 3 2 4 + (-2 x - 3 x + x + x + 2 x - 2) P(x) + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 4, 7, 13, 31, 68, 151, 353, 830, 1974, 4770, 11640, 28655, 71137, 177846, 447385, 1131721, 2877047, 7346508, 18834532, 48462272, 125107961, 323947194, 841128592, 2189541500, 5712962262, 14938641202, 39141249503, 102747535602, 270188372857, 711657594649, 1877327468019, 4959442969022, 13119361391251, 34749264755755, 92151139526833, 244652427719736, 650228354697653, 1729918669812748, 4606876307510903, 12279701899270589, 32760484182475877, 87473433250584239, 233748476333341011, 625104546863703460, 1672912767097417426, 4480197378774438801, 12006364182059672476, 32196189195480675152, 86390268339099023784, 231943406867773673559, 623083386995175908592, 1674741886324264405491, 4503792002194003780483, 12117949758427063804038, 32620637607244253751313, 87853837000533097933303, 236715620047918300658849, 638093409766878741244939, 1720784029640200529466109, 4642449192227602970962471, 12529705204132005823084739, 33829995525108470689087028, 91374533506774924198871928, 246891162946274814345255866, 667326361919700464180466164, 1804341836783755969767902081, 4880258183200337657627349597, 13204016231004099310703911640, 35735888322210287302434695483, 96746346945051492129944136673, 261994616605299952299716358913, 709699417012216252826462746398, 1922991879447607695698409355563, 5211924795579765882550153279034, 14129715409831946127753654037827, 38316011030586066637914582753059, 103928796252920215410081803358278, 281966477169083994478108975309047, 765177829541777969692893114202601, 2076959557406565291855683538866509, 5638868945770438919899018896841685, 15312704020048106078050590219312734, 41591579374244913493655901742655268, 112992694770324482050201403155677648, 307032691453092694995591759198423486, 834461137862482346770494050695443513, 2268364636536495630406117530328420694, 6167411942613495429055053735583077660, 16771601256606925288684863153683394607, 45616902005536094888373955476881033944, 124095204548386658478210061054534756445, 337645125563796910137825400887087849425, 918841532591357251637815666765040689673, 2500885592781921396477331864964481145773, 6807984856027371010633861697801309493006] ------------------------------------------------------------ Theorem Number, 168 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 3 2 (x + 4 x + 4 x + 2 x - 3 x - x + 1) P(x) 7 6 5 4 3 2 6 4 3 + (-2 x - 4 x - x - x + 5 x - x + 2 x - 2) P(x) + x + x - 2 x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 16, 30, 56, 110, 216, 427, 860, 1746, 3569, 7364, 15307, 32009, 67334, 142416, 302663, 646031, 1384486, 2977909, 6426645, 13911978, 30200915, 65732898, 143413923, 313595163, 687144819, 1508571730, 3317922569, 7309678595, 16129313045, 35643218886, 78875659428, 174774870620, 387750760231, 861260255477, 1915124365594, 4262999134446, 9498739083878, 21184957996924, 47291188848596, 105658705445218, 236257445857091, 528694503932384, 1183990265577090, 2653392279620791, 5950474873926279, 13353228687666239, 29984296182635404, 67369519671951483, 151455416547760267, 340681282599710033, 766734008185798102, 1726495929499880140, 3889579288927147642, 8766943991330924627, 19769471877891215161, 44600139100704220528, 100661893945840143266, 227287461047749510237, 513406564122610083700, 1160158342765246864333, 2622633765230015042003, 5930856706035176271770, 13416885846141972833882, 30362383327524771174841, 68733011599111138672309, 155645297432095952323778, 352568632584310655173635, 798885376847742072720203, 1810734038572540097719042, 4105354542744134743448103, 9310415132809649662747530, 21120611967784825680218897, 47924750352934920938803331, 108774247977720255491636377, 246946153325595537553849633, 560771040004278719756311221, 1273718100463445848541909110, 2893762556614309851409015460, 6575848332156955986692361543, 14946434073696128773556355402, 33979577457896340549152610437, 77266402617848204240564334985, 175733104543382019162751390086, 399764798964070432229145750542, 909581419077380496710545668751, 2069963664743655297207114411687, 4711574821783439857678518992020, 10726298267792692107831401356403, 24423746955362712321076292407699, 55622647415401209237519460346954, 126696994280216636321751775842705, 288638769125621515697430788944262] ------------------------------------------------------------ Theorem Number, 169 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H H + HH HH 4+ H H H + H H HHH + HH HH H HH + H H HH H 3+ H H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 5 4 3 2 3 (x - 2 x + 7 x - 9 x + 12 x - 14 x + 10 x - 7 x + 4 x - 1) P(x) 8 7 6 5 4 3 2 2 + (2 x - 7 x + 12 x - 16 x + 23 x - 18 x + 13 x - 9 x + 3) P(x) 7 6 5 4 3 2 6 4 + (3 x - 4 x + 7 x - 12 x + 10 x - 8 x + 6 x - 3) P(x) + x + 3 x 3 2 - x + 2 x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 4, 8, 18, 43, 107, 279, 756, 2110, 6037, 17640, 52456, 158288, 483568, 1492894, 4650704, 14601568, 46156995, 146782489, 469257258, 1507289667, 4862041172, 15743274434, 51152852674, 166727740156, 544993582602, 1786147562570, 5868064834144, 19321628076330, 63751777318042, 210754466774987, 697975087107177, 2315421894719860, 7693090494781241, 25598222027331448, 85294024580154610, 284572037791424554, 950601751638645939, 3179130706928813538, 10643746810073950944, 35672387552131100130, 119673593195733915195, 401857178134742726339, 1350618393155426577870, 4543209931421070301786, 15294846515110145969343, 51530425377117115234766, 173741587423603782151574, 586207574469557362541280, 1979218124244542799319645, 6686799535334889776000855, 22605489641467786941414103, 76466241842357597349351263, 258806785222998659644379449, 876439833546094218547775979, 2969614566519881761428112761, 10067023389154832346188732615, 34144207402697469332697607741, 115861817772408995037076400857, 393336058449361133198955361917, 1335920011416083390169608539566, 4539248043629602439869966785401, 15430072235316580650979902643615, 52471890167572910648457519213810, 178506719116668606641853615389704, 607499892275484997248806539885662, 2068219070372095059539330722214095, 7043697844838562923405403822540599, 23996843168988036140205693786358038, 81780978190379571611357495883258756, 278798896738905428977562298548354272, 950749960479359115869297043345124508, 3243203663431638386971922572769709059, 11066519580355582754503708128966076439, 37772285416504907067750012154875410798, 128960713439887320024276525321870170236, 440413147276017254199306291367776937065, 1504453129708127162206846786706422674943, 5140549753777339585054421186811635023123, 17569123083919020268315931672679734787381, 60061672910667023799590727140693425462132, 205375624405195940206188071345208621480266, 702428161825362847308113776450553501651394, 2403001197112171174304604878980487276107538, 8222477844715773539344006136719525737257985, 28141404876220548820431941449051025841531279, 96334299139933433311453500312549365692297382, 329842079999355892430366152512047465518715761, 1129585347059446104428707937597856978073528761, 3869170810510643588286699594961841108699713823, 13255639312157219311472436625646293112569889405, 45421921450094045893738653354575374038028375770, 155672019898697878192584926487102957689419501024, 533622460440399685530204075671873065586359432847, 1829508479059991656203558862329256709228769011741, 6273499901003238157409301436071442020268738721130, 21515870301276938946030622815245620845194845427609, 73804009021303313618741069038370335611748651676267] ------------------------------------------------------------ Theorem Number, 170 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H H + HH HH 4+ H H H + H H HHH + HH HH H HH + H H HH H 3+ H H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 2 (x + 3 x + 2 x - x - x - x + 1) P(x) 5 4 3 2 4 + (-2 x - 3 x + x + x + 2 x - 2) P(x) + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 4, 7, 13, 31, 68, 151, 353, 830, 1974, 4770, 11640, 28655, 71137, 177846, 447385, 1131721, 2877047, 7346508, 18834532, 48462272, 125107961, 323947194, 841128592, 2189541500, 5712962262, 14938641202, 39141249503, 102747535602, 270188372857, 711657594649, 1877327468019, 4959442969022, 13119361391251, 34749264755755, 92151139526833, 244652427719736, 650228354697653, 1729918669812748, 4606876307510903, 12279701899270589, 32760484182475877, 87473433250584239, 233748476333341011, 625104546863703460, 1672912767097417426, 4480197378774438801, 12006364182059672476, 32196189195480675152, 86390268339099023784, 231943406867773673559, 623083386995175908592, 1674741886324264405491, 4503792002194003780483, 12117949758427063804038, 32620637607244253751313, 87853837000533097933303, 236715620047918300658849, 638093409766878741244939, 1720784029640200529466109, 4642449192227602970962471, 12529705204132005823084739, 33829995525108470689087028, 91374533506774924198871928, 246891162946274814345255866, 667326361919700464180466164, 1804341836783755969767902081, 4880258183200337657627349597, 13204016231004099310703911640, 35735888322210287302434695483, 96746346945051492129944136673, 261994616605299952299716358913, 709699417012216252826462746398, 1922991879447607695698409355563, 5211924795579765882550153279034, 14129715409831946127753654037827, 38316011030586066637914582753059, 103928796252920215410081803358278, 281966477169083994478108975309047, 765177829541777969692893114202601, 2076959557406565291855683538866509, 5638868945770438919899018896841685, 15312704020048106078050590219312734, 41591579374244913493655901742655268, 112992694770324482050201403155677648, 307032691453092694995591759198423486, 834461137862482346770494050695443513, 2268364636536495630406117530328420694, 6167411942613495429055053735583077660, 16771601256606925288684863153683394607, 45616902005536094888373955476881033944, 124095204548386658478210061054534756445, 337645125563796910137825400887087849425, 918841532591357251637815666765040689673, 2500885592781921396477331864964481145773, 6807984856027371010633861697801309493006] ------------------------------------------------------------ Theorem Number, 171 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 3+ H H + H H H H + HH HH HH HH 2.5+ H H H H + H H H H + H H H H + H H H H 2+ H H H H + H H H H + H H H H 1.5+ HH HH HH HH + H H H H + H H H H 1+ H H H H H H + HH H H H H H H + H H H H H H H H + H H H H H H H H 0.5+ H H H H H H H H + H H H H H H H H +H HH HH HH H -*-+-+-+-+*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The first, 100, terms of the sequence are [1, 1, 1, 2, 4, 8, 16, 34, 78, 185, 448, 1113, 2832, 7342, 19325, 51542, 139073, 379043, 1042119, 2887101, 8052601, 22594874, 63738278, 180661561, 514279275, 1469674381, 4214795642, 12126332537, 34991373446, 101243299167, 293665339342, 853764069369, 2487416034586, 7261371430246, 21236821462557, 62216820916336, 182568328505457, 536535845296402, 1579021413558462, 4653257135200309, 13730068676309043, 40560622789025057, 119956793170704530, 355146620868052672, 1052517656744170704, 3122252293977270120, 9270499597327761023, 27549639831297371824, 81938643209816600036, 243896340595328745956, 726524363301703783132, 2165754227538206399979, 6460540746352917364661, 19284870984011183128986, 57602456045641878422146, 172159342055409946368051, 514843344256571684022225, 1540510681480513851235715, 4612009270263275839268641, 13814740582537270579553281, 41401315155994550943056218, 124135660724602212421189503, 372376781791242427314383868, 1117545501021544639425929786, 3355349419576508977727908769, 10078453058856602355749440683, 30284999061821705257549321413, 91040175192217899003391306705, 273782056357565937165552736358, 823641180988210057938373877425, 2478719245133070022914347240318, 7462220956434794654696267636541, 22472729631733675388470762048427, 67699612540848006098876700193867, 204011682859497311016275456557591, 614976378606587857745897447184154, 1854353730220117824537071754296894, 5593116283922975889700880971173598, 16874804538600413409534222029778212, 50926513883094440046120224408611295, 153732691866727809599850431109206082, 464197204942422383544921774426946109, 1402005944842625633953888260420616917, 4235506833903839322997894556777501208, 12798715260970365782918660703943204144, 38683894507926184140048949049133909671, 116948400826031481549039184312407913732, 353635799631363075686471147436380875154, 1069580905258162953685276343278592539484, 3235670335121941333501269303451009913675, 9790523395780271402910437152921355490641, 29630327502284731313371489270188977001056, 89692029415502534374294896104750641633661, 271553955339781773944426254727375366565217, 822321019573473760044943574891663368432161, 2490621788720144544040885624836172039304681, 7544900644969655420426996846766590364369364, 22860035113771766116113005216447528226323839, 69274949576440306060447040743165723275435163, 209966440186969749370722586970726408619614169, 636496953270705679320987557370651440282853315] ------------------------------------------------------------ Theorem Number, 172 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 19 17 16 15 14 13 12 11 10 9 8 (x - 3 x - 3 x + x + 6 x + 5 x + 5 x - 4 x - 7 x - 12 x - x 7 6 5 4 3 3 16 14 13 + x + 10 x + 2 x + 2 x - 4 x - x + 1) P(x) + (3 x - 6 x - 6 x 12 11 10 9 8 6 5 4 3 2 - 3 x + 7 x + 7 x + 16 x - x - 18 x - 2 x - 6 x + 10 x - x 2 13 11 10 9 8 7 6 5 + 3 x - 3) P(x) + (3 x - 3 x - 3 x - 6 x + 2 x - x + 10 x - x 4 3 2 10 6 5 4 3 2 + 6 x - 8 x + 2 x - 3 x + 3) P(x) + x - 2 x + x - 2 x + 2 x - x + x - 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 5, 8, 13, 29, 58, 109, 224, 469, 962, 2004, 4252, 9040, 19301, 41563, 89980, 195525, 426833, 935706, 2058178, 4541767, 10053231, 22314028, 49652620, 110747551, 247557767, 554494496, 1244340818, 2797369080, 6299090767, 14206237723, 32085826176, 72567625371, 164336472104, 372610625607, 845820085338, 1922098715982, 4372439993387, 9956343355891, 22692518763045, 51766984724848, 118193127925529, 270074678950222, 617606728681446, 1413389735165284, 3236829392842055, 7417758864228774, 17010163891685398, 39031546594762198, 89615630050729955, 205874421628959923, 473218932824758023, 1088313251325592598, 2504201652272691910, 5765011816121712227, 13278193824888813318, 30596973864393382114, 70536151123238024372, 162679316350017450709, 375347930528973340133, 866385011098516035066, 2000587011422283874074, 4621343255190585868302, 10679186807475470377805, 24686666622984511599253, 57086890984494449675458, 132055217721375981546693, 305573531184986377077384, 707314945608209847639832, 1637732567993388063478190, 3793171352332367061541348, 8787952385570629968379636, 20365508673681019718977166, 47208682141769560467266124, 109462253636911869115750581, 253874894309395675440725721, 588959015626156096858744585, 1366650733136206654486516020, 3172009302049365401821933578, 7363990486154164923351386224, 17099812077001249363177056269, 39716082047248874128446242438, 92264807603841023580831955900, 214386848828249772877195858443, 498253569181261894314960926934, 1158219376625202957075370231454, 2692882159291120533526978907126, 6262215947461674813212714459568, 14565353079098544379221773377774, 33883983435461603108197061482155, 78839999338725373583734755178870, 183474523257584763789771107474471, 427051528522691727246479996839525, 994165194829588457736112300985513, 2314776392397718522798745297649492] ------------------------------------------------------------ Theorem Number, 173 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H H + H HH HHH + HH H H HH + H HH HH H 3+ H H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH H H 1+ H H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 7 6 5 4 3 3 (x - 2 x - 5 x - 3 x - 2 x + 3 x + x - 1) P(x) 7 6 5 4 3 2 2 + (3 x + 8 x + 5 x + 6 x - 7 x + x - 3 x + 3) P(x) 7 6 5 4 3 2 6 4 3 + (-x - 4 x - 2 x - 6 x + 5 x - 2 x + 3 x - 3) P(x) + x + 2 x - x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 43, 88, 181, 389, 845, 1831, 4021, 8945, 19980, 44876, 101560, 231010, 527584, 1210453, 2788476, 6445177, 14945402, 34762125, 81073108, 189553176, 444229775, 1043335656, 2455327148, 5789089278, 13673287432, 32347923587, 76645650172, 181868152606, 432131947999, 1028094409010, 2448927381166, 5840049815756, 13942114199763, 33318698172213, 79702566442900, 190836157036246, 457334328621662, 1096917010956494, 2633078585793776, 6325404794268179, 15206619342001349, 36583326860847321, 88069812063175602, 212154504630779797, 511384369072275705, 1233394476770162448, 2976499939131468798, 7187032770060977636, 17362981676250949732, 41968309776585627857, 101492313014921186269, 245556679552748148628, 594388103914626682818, 1439399709664903706555, 3487218741318126496572, 8451957378207794626600, 20493205067305036075159, 49708582948760009479704, 120619210443382870051297, 292792615812251132529932, 710980136125717425588637, 1727046021860570120941582, 4196575501209940797812052, 10200621702455464613659962, 24802455440798284389096014, 60324716048583750707955995, 146765765674535755148007651, 357173774922625517418331537, 869473358467539692845081275, 2117150070857988843015817831, 5156588999541444162109803583, 12562783839142431859361805544, 30613913629937142427400197707, 74620580935239731515082935125, 181929247599493983737210197721, 443657637304923176383168028461, 1082162432233326872644453144656, 2640180374641018514672900497678, 6442717456740989437135783799985, 15725218378158726422687662180370, 38389658112166505912696121858576, 93738863949937324569728179979473, 228934372538591942169266458710404, 559224587509088449210338239175827, 1366292159965124659559109224182072, 3338728826968157284819565578655591, 8160132627279040287185529051465709, 19947574345781438581105186436718539, 48770603600956548080699610819303235, 119261355989728246212509483676650821, 291684525929087013741255678799558755] ------------------------------------------------------------ Theorem Number, 174 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H H + H HH HHH + HH H H HH + H HH HH H 3+ H H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH H H 1+ H H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 3 2 (x + 4 x + 4 x + 2 x - 3 x - x + 1) P(x) 7 6 5 4 3 2 6 4 3 + (-2 x - 4 x - x - x + 5 x - x + 2 x - 2) P(x) + x + x - 2 x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 16, 30, 56, 110, 216, 427, 860, 1746, 3569, 7364, 15307, 32009, 67334, 142416, 302663, 646031, 1384486, 2977909, 6426645, 13911978, 30200915, 65732898, 143413923, 313595163, 687144819, 1508571730, 3317922569, 7309678595, 16129313045, 35643218886, 78875659428, 174774870620, 387750760231, 861260255477, 1915124365594, 4262999134446, 9498739083878, 21184957996924, 47291188848596, 105658705445218, 236257445857091, 528694503932384, 1183990265577090, 2653392279620791, 5950474873926279, 13353228687666239, 29984296182635404, 67369519671951483, 151455416547760267, 340681282599710033, 766734008185798102, 1726495929499880140, 3889579288927147642, 8766943991330924627, 19769471877891215161, 44600139100704220528, 100661893945840143266, 227287461047749510237, 513406564122610083700, 1160158342765246864333, 2622633765230015042003, 5930856706035176271770, 13416885846141972833882, 30362383327524771174841, 68733011599111138672309, 155645297432095952323778, 352568632584310655173635, 798885376847742072720203, 1810734038572540097719042, 4105354542744134743448103, 9310415132809649662747530, 21120611967784825680218897, 47924750352934920938803331, 108774247977720255491636377, 246946153325595537553849633, 560771040004278719756311221, 1273718100463445848541909110, 2893762556614309851409015460, 6575848332156955986692361543, 14946434073696128773556355402, 33979577457896340549152610437, 77266402617848204240564334985, 175733104543382019162751390086, 399764798964070432229145750542, 909581419077380496710545668751, 2069963664743655297207114411687, 4711574821783439857678518992020, 10726298267792692107831401356403, 24423746955362712321076292407699, 55622647415401209237519460346954, 126696994280216636321751775842705, 288638769125621515697430788944262] ------------------------------------------------------------ Theorem Number, 175 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 19 17 16 15 14 13 12 11 10 9 8 (x - 3 x - 3 x + x + 6 x + 5 x + 5 x - 4 x - 7 x - 12 x - x 7 6 5 4 3 3 16 14 13 + x + 10 x + 2 x + 2 x - 4 x - x + 1) P(x) + (3 x - 6 x - 6 x 12 11 10 9 8 6 5 4 3 2 - 3 x + 7 x + 7 x + 16 x - x - 18 x - 2 x - 6 x + 10 x - x 2 13 11 10 9 8 7 6 5 + 3 x - 3) P(x) + (3 x - 3 x - 3 x - 6 x + 2 x - x + 10 x - x 4 3 2 10 6 5 4 3 2 + 6 x - 8 x + 2 x - 3 x + 3) P(x) + x - 2 x + x - 2 x + 2 x - x + x - 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 5, 8, 13, 29, 58, 109, 224, 469, 962, 2004, 4252, 9040, 19301, 41563, 89980, 195525, 426833, 935706, 2058178, 4541767, 10053231, 22314028, 49652620, 110747551, 247557767, 554494496, 1244340818, 2797369080, 6299090767, 14206237723, 32085826176, 72567625371, 164336472104, 372610625607, 845820085338, 1922098715982, 4372439993387, 9956343355891, 22692518763045, 51766984724848, 118193127925529, 270074678950222, 617606728681446, 1413389735165284, 3236829392842055, 7417758864228774, 17010163891685398, 39031546594762198, 89615630050729955, 205874421628959923, 473218932824758023, 1088313251325592598, 2504201652272691910, 5765011816121712227, 13278193824888813318, 30596973864393382114, 70536151123238024372, 162679316350017450709, 375347930528973340133, 866385011098516035066, 2000587011422283874074, 4621343255190585868302, 10679186807475470377805, 24686666622984511599253, 57086890984494449675458, 132055217721375981546693, 305573531184986377077384, 707314945608209847639832, 1637732567993388063478190, 3793171352332367061541348, 8787952385570629968379636, 20365508673681019718977166, 47208682141769560467266124, 109462253636911869115750581, 253874894309395675440725721, 588959015626156096858744585, 1366650733136206654486516020, 3172009302049365401821933578, 7363990486154164923351386224, 17099812077001249363177056269, 39716082047248874128446242438, 92264807603841023580831955900, 214386848828249772877195858443, 498253569181261894314960926934, 1158219376625202957075370231454, 2692882159291120533526978907126, 6262215947461674813212714459568, 14565353079098544379221773377774, 33883983435461603108197061482155, 78839999338725373583734755178870, 183474523257584763789771107474471, 427051528522691727246479996839525, 994165194829588457736112300985513, 2314776392397718522798745297649492] ------------------------------------------------------------ Theorem Number, 176 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 13 12 11 10 9 8 7 6 3 2 (x + x - 2 x - 4 x - 3 x + 3 x + 4 x + 6 x - 3 x - x + 1) P(x) 10 9 8 7 6 5 3 2 7 + (2 x + 2 x - 2 x - 4 x - 6 x + x + 5 x - x + 2 x - 2) P(x) + x 6 3 2 + x - 2 x + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 6, 10, 19, 33, 58, 108, 200, 369, 693, 1317, 2514, 4829, 9344, 18189, 35592, 69989, 138246, 274200, 545940, 1090783, 2186343, 4395255, 8860180, 17906170, 36272959, 73639706, 149804622, 305325432, 623406188, 1274971376, 2611597375, 5357319501, 11004919758, 22635485192, 46614898206, 96108678206, 198370269901, 409866012138, 847687251598, 1754836188328, 3636009242250, 7540193770668, 15649198235420, 32504074898770, 67562356231747, 140533256797216, 292514736909839, 609254134880606, 1269754726390174, 2647896576486016, 5524994718069158, 11534607248785519, 24093773197898278, 50353464184534819, 105285448864725132, 220249118224532352, 460956131234061995, 965156511973390642, 2021725152698343735, 4236688329900954289, 8881883438230807832, 18627397182916201898, 39080715060247020314, 82022115622422152747, 172207762570687826247, 361678877010846007041, 759867396916920746124, 1596954324667818296589, 3357245924609224890483, 7060019872578142781287, 14851047729721395149780, 31248783492263029171750, 65770414878714109406951, 138466981593815657698333, 291592875976612309859386, 614213871210552443656780, 1294110799316931990023822, 2727279498663073507672105, 5748989295111391492048961, 12121446207030768286650281, 25563244831836776909695275, 53922960715096892884159306, 113769377614144303450467979, 240087037262751143517465894, 506759120361514589789712132, 1069847544922340934122333579, 2259059523710516236809422262, 4771083098316998636523021295, 10078315424868890475447526836, 21293093745583655561384647024, 44995356141694588627315053710, 95098362701380027278219259613, 201026434433379944826868248653] ------------------------------------------------------------ Theorem Number, 177 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ HH HH + HH HH + H H + HH HH 4+ H H + H H + HH HH 3+ H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HHH HHH HH HH + H H H H H H +HH HHHH HHHH HH -*-+-+-+-+*-+-+-+-+*-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 5 3 2 2 1 + (x + x - 2 x + 1) P(x) + (x + 2 x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence 6 (n - 3) a(n - 1) 4 (2 n - 7) a(n - 2) a(n) = ------------------ - -------------------- - a(n - 3) n - 2 n - 2 2 (2 n - 7) a(n - 4) 2 (2 n - 7) a(n - 6) + -------------------- - a(n - 5) + -------------------- n - 2 n - 2 subject to the initial conditions [a(1) = 1, a(2) = 1, a(3) = 2, a(4) = 4, a(5) = 8, a(6) = 18] Just for fun, a(1000), equals 6224665859314201016979561104165545762190845505511764978807996595737692852231\ 289808270735187141352703923458497516367684236403016906093293144377611727\ 352189003203467137255824578550520994706739911177644252059926720565504589\ 557234628983056200347172675628327750140629495857657185359325556935393336\ 796474135312176550273219428808989843818190011841471311555643072147493670\ 225956889373244976633590575865511904431275637530176464011411093243940205\ 749536686426091126573671437728262678039426623651804201751489134043211614\ 842024267213457418127172760977630634664368799028717124985975764277203103\ 6959149732655837 For the sake of the OEIS here are the first, 100, terms. [1, 1, 1, 2, 4, 8, 18, 45, 122, 354, 1084, 3458, 11379, 38348, 131670, 458889, 1618872, 5769135, 20735878, 75078884, 273574534, 1002433018, 3691281207, 13652372422, 50693499962, 188904707823, 706214223856, 2647947813867, 9955284948941, 37521015246216, 141738728886746, 536563910351055, 2035201074293395, 7733694636843407, 29437940387316452, 112233005848500961, 428531883562050570, 1638534667812222057, 6273388130046401008, 24048604822003888486, 92297246294968796318, 354626435363335842458, 1363991102966831374763, 5251538613785139661020, 20238320779022837725116, 78064783120688319238391, 301376880397313514634004, 1164451568505634053256169, 4502704669354042720866647, 17424122849064922479420494, 67474503103694569472775918, 261472822899807148188043085, 1013909316561660877898018079, 3934103083106433797226168049, 15274094890978311909015688675, 59335972908063894320803000865, 230634241365903981946745636476, 896940196552221462554187075622, 3490024933722091467982824858302, 13586598459032174139659546855765, 52917873605378914826689324547707, 206203548324792138236947397412716, 803868553121576866602324504261157, 3135180172300172769596749638155201, 12232698156401376877303973179258257, 47748347629121424798890391075980962, 186451162096389864477882470062774132, 728344927359073434690867685155575800, 2846224900067644940685738123840880787, 11126447524442811375010205510288874285, 43510526795853945353889205558403141758, 170207301528504141628800431186603737097, 666045313115422374004683582632821648809, 2607155982955205428204734398773628819673, 10208548141503569462824598354845430744314, 39984430832805262469219622489754277723761, 156654987746204193209813947781897591775252, 613932260348892715720334905145126065184621, 2406668588312403775372006175409686354616828, 9436882571100403699399760013077494147738610, 37012991177828993995821326777607653145818878, 145207902391131838482750985286671051584581276, 569815071126079280004665721684610250220631551, 2236570060433353434169770987420870617608441336, 8780785385892795108620632527048125872984217526, 34481327725098458777975918523549689478998218903, 135435318620152626991187367109060484431788621294, 532077487822681497622391899258556799161043673951, 2090790780782105019005674767049619496948584092183, 8217445531114689350426237362634037842021146659866, 32303647081347025541521251868248855338769841821818, 127014328649054707807542279088485212158840846191159, 499503292583172841290989369693919476763046874075101, 1964746748104571333950851063289082343052836412187001, 7729573955612977956161587182890394945878842511546777, 30414698794053325452618990707996964791669799435032235, 119698513046693440614942151654446010920097650519124310, 471161294363267735601347503379784430423834239140856342, 1854916941161805782268731035443222410550176600669397498, 7303848999727259531530228873428355074943883405686818209, 28764051362426773439023369099425961516385217723697502941] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 178 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 3 2 2 3 2 1 + (x + x - x - x + 1) P(x) + (x + x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 7) a(n - 1) (4 n - 17) a(n - 2) (n - 5) a(n - 3) a(n) = ------------------ + ------------------- - ---------------- n - 2 n - 2 n - 2 (5 n - 22) a(n - 4) (4 n - 17) a(n - 5) (n - 5) a(n - 6) - ------------------- - ------------------- + ---------------- n - 2 n - 2 n - 2 (5 n - 22) a(n - 7) 3 (n - 5) a(n - 8) + ------------------- + ------------------ n - 2 n - 2 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 3, a(7) = 6, a(8) = 12] Just for fun, a(1000), equals 8860862023292075968276277514714248593198929839769380856771491476834645411034\ 971894866292077857079552961537030062908551251261647144382496853687782821\ 054184668235733438566888858551955851111944135135787839723906153442696991\ 263785015057838776665886398587178566742546567522735087401082955419046964\ 387465387179652258465254660160763488653122285490983717702179160837248539\ 693902000749455565246146162113746561726289619446359962920995865101594027\ 98072823532463236599122791287290686 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 3, 6, 12, 28, 67, 163, 409, 1047, 2720, 7159, 19048, 51142, 138385, 376995, 1033115, 2845952, 7876302, 21888824, 61058893, 170902575, 479831873, 1350999629, 3813697119, 10791286286, 30602553756, 86962021759, 247586286749, 706142663596, 2017328796314, 5772096786694, 16539476231102, 47457342050583, 136346550591721, 392204416291603, 1129482421173659, 3256256113386519, 9397345776928842, 27146683687948823, 78493250837353534, 227160260729254220, 657959005719580783, 1907282045837204115, 5533054162491377407, 16063278227550962830, 46666904076383085772, 135667712558607666872, 394661414446836273133, 1148794020914769645815, 3345941551324715625346, 9750866064647774999374, 28431988455483862564098, 82947166290697018848452, 242112780921723361735501, 707046117556083065099702, 2065780034773677339223036, 6038367709611593213853662, 17658238896863824381246792, 51660799040420898435104415, 151200983620991436647323170, 442712766405318283383365368, 1296754766161140895530674505, 3799765063841219444556153664, 11138165760379116840837129933, 32660575398596894269004630395, 95803772582109903302659971705, 281116036367240287000597160307, 825141933187631607642170430582, 2422744979523191074317651806222, 7115720853948264721550464469138, 20905404259902466570682178134101, 61436033484557854416515787554557, 180596489848062297043580473736583, 531023478945026185132223045188242, 1561828335230849070577015587599862, 4594783181095102005982648845629943, 13520912751397068669158402972820535, 39797288428151734415529516833294183, 117166853197429318351440194184640001, 345030340472240118667348975322651831, 1016268735970876894277016612352986956, 2994030447810228732421196508522801508, 8822627125900727689992098802555040065, 26003480448270289308764833189852136625, 76657514628388286510106161064786323442, 226029756946856262771695312945809615724, 666595122574573979910207473435766415800, 1966266336978008980717249705195918553528, 5801020343999533654167963129230758635937, 17117741362043104820828161774821249467025, 50520403804121740187565594126636709015553, 149129567444169474699296647095777703816561, 440286794637589018357948150453173148034809, 1300112504445214435657315221574732403366232, 3839706689852153907055350498280218948390511, 11341892306305967810005135303941131373501596] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 179 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H H + HHHH HHHH + H H H H 5+ HH HHH HH + HH H + H H + HH HH 4+ H H HH + H H HH H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 7 6 4 2 3 1 + (x + 2 x - x - x - 2 x + 3 x - 1) P(x) 6 5 4 2 2 3 2 + (2 x + x + 2 x + x - 6 x + 3) P(x) + (x + x + 3 x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 4, 8, 17, 38, 90, 226, 595, 1626, 4583, 13255, 39152, 117632, 358379, 1104540, 3437534, 10787176, 34092504, 108416342, 346645635, 1113679878, 3593289935, 11638329754, 37826257535, 123327887440, 403249754800, 1321986604313, 4344384539207, 14308585907725, 47223762974482, 156154993535171, 517279512721917, 1716395204199984, 5704077870995768, 18983996816831575, 63268228255622253, 211127426884278728, 705395591765359037, 2359504087156624476, 7900995290800278546, 26484466908922828087, 88864073724588088699, 298445919260326762000, 1003205605602019452638, 3375055833370038475415, 11363752620369614839002, 38290991969078239124495, 129119188192510283827134, 435702970067638336401740, 1471237997718381515655961, 4971139384784656516703965, 16807309393261681339710744, 56858974801937271214176618, 192463501961806458549367052, 651833685891728036085694672, 2208794626314526388838255955, 7488512324566195972361000419, 25400920138005481936189410149, 86200438627398423962889600077, 292663432884995520011132877727, 994076051219370632401679112136, 3377975277695539232604789875543, 11483463522638435443420165861052, 39053778764590735260214744028411, 132868310291185248291927887818515, 452212595533463639447561125683626, 1539649070809086278831859924742111, 5243893547522691259065527773303234, 17866291918724072977961447260036381, 60891830322474637750552436362104287, 207598148314178819145969881814596126, 707984497135539785608629387494969248, 2415216119078895254512880464157359266, 8241695045438919774148317893449721900, 28132079874956214994051496481913538658, 96052464350363092991383456220764683736, 328044905624843906486485910858513572753, 1120657949699764053438714184735518723226, 3829348091994830090540765936502828433868, 13088374765183661773959550315278141251762, 44745867155596722406431032729662747065064, 153011402892271561815196042063134630398688, 523354250006880412472190047285558113292304, 1790467027162243322753693430925261434443773, 6126791872075190111368519055950746871602422, 20969779058966314248851114808976871975726427, 71787080271027796432625484374156603053117008, 245803630113711348682394578531982069883458376, 841817199668759568384197048764335720527595287, 2883585288228841543815032314714643011117463365, 9879418496568057882669009954460305569698680220, 33854131408980382617618344444952182795596663927, 116030423557513326707883799432379866683283721065, 397750052241093250199289662453849289266840594823, 1363719630603529068278211367514741724664144560802, 4676433740888708159615030564012359089021337713132, 16039016859440355938804763268391989690074811992256, 55018967565298271506684475923762005973565491298128] ------------------------------------------------------------ Theorem Number, 180 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 7+ HH + H HH + HH HH 6+ H HH + H HH + HH HH 5+ HH HH + H HH + HH HH 4+ H H HH + HH HH HH HH 3+ HH HHH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 6 5 3 2 3 1 + (x - 3 x - x + 2 x + x - 1) P(x) 6 5 3 2 2 3 2 + (3 x + x - 4 x - 2 x + 3) P(x) + (2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 24, 50, 100, 207, 447, 965, 2097, 4641, 10355, 23208, 52434, 119238, 272302, 624715, 1439586, 3329035, 7723353, 17974332, 41946862, 98136543, 230136785, 540859834, 1273645217, 3004841530, 7101517437, 16810654036, 39854525348, 94621732620, 224950428542, 535465514846, 1276128784737, 3044720948720, 7272175007841, 17386856915967, 41609801199087, 99670684111928, 238955727084254, 573360774386420, 1376837089630860, 3308764126191171, 7957256958283794, 19149694328506228, 46115679249370442, 111125023198040762, 267941826636175204, 646433889341639098, 1560460708781195478, 3768924211315976522, 9107707171651752850, 22020112055371883983, 53264882178765929245, 128903877018767159586, 312095509354848105514, 755960176581781422429, 1831866563301055461943, 4440851470520127715854, 10769869026085811825291, 26128886252691178913799, 63415069809483920442764, 153964356091782976724423, 373937747991975936017966, 908500784048788194813896, 2207973525844179375897501, 5367855788092394722780239, 13053960160887471570162186, 31755167086049496587497855, 77270472173378202155837143, 188077261542308295134223872, 457909065550704597273291014, 1115165194681535923031110438, 2716521249676194665104160483, 6619085378281378272215833397, 16132103128989793305097961362, 39326870592610533254429937069, 95893803896808211685222661778, 233879375049458391590317811366, 570546544663074707446614357597, 1392148793303975476592593878598, 3397608858830076013984401770568, 8293771449881197142382854212644, 20249746445904137513663328249672, 49450867782510794412659596235458, 120785019511666197733598404565551, 295076860334680412091808688807665, 721005064284596647207054443970788, 1762060340462307761691050403140135, 4307058934161988158319017861316542, 10529718506027909698423650877921477, 25747021917792169515323378704085982, 62966561145792275331841952840077732, 154015405894044114004769528686025159] ------------------------------------------------------------ Theorem Number, 181 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ H HH HH + HH H HH HH + H HH H H + HH HHH HH 4+ HH H H H + H HH H H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 3 2 2 3 2 1 + (x + x - x - x + 1) P(x) + (x + x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 7) a(n - 1) (4 n - 17) a(n - 2) (n - 5) a(n - 3) a(n) = ------------------ + ------------------- - ---------------- n - 2 n - 2 n - 2 (5 n - 22) a(n - 4) (4 n - 17) a(n - 5) (n - 5) a(n - 6) - ------------------- - ------------------- + ---------------- n - 2 n - 2 n - 2 (5 n - 22) a(n - 7) 3 (n - 5) a(n - 8) + ------------------- + ------------------ n - 2 n - 2 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 3, a(7) = 6, a(8) = 12] Just for fun, a(1000), equals 8860862023292075968276277514714248593198929839769380856771491476834645411034\ 971894866292077857079552961537030062908551251261647144382496853687782821\ 054184668235733438566888858551955851111944135135787839723906153442696991\ 263785015057838776665886398587178566742546567522735087401082955419046964\ 387465387179652258465254660160763488653122285490983717702179160837248539\ 693902000749455565246146162113746561726289619446359962920995865101594027\ 98072823532463236599122791287290686 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 3, 6, 12, 28, 67, 163, 409, 1047, 2720, 7159, 19048, 51142, 138385, 376995, 1033115, 2845952, 7876302, 21888824, 61058893, 170902575, 479831873, 1350999629, 3813697119, 10791286286, 30602553756, 86962021759, 247586286749, 706142663596, 2017328796314, 5772096786694, 16539476231102, 47457342050583, 136346550591721, 392204416291603, 1129482421173659, 3256256113386519, 9397345776928842, 27146683687948823, 78493250837353534, 227160260729254220, 657959005719580783, 1907282045837204115, 5533054162491377407, 16063278227550962830, 46666904076383085772, 135667712558607666872, 394661414446836273133, 1148794020914769645815, 3345941551324715625346, 9750866064647774999374, 28431988455483862564098, 82947166290697018848452, 242112780921723361735501, 707046117556083065099702, 2065780034773677339223036, 6038367709611593213853662, 17658238896863824381246792, 51660799040420898435104415, 151200983620991436647323170, 442712766405318283383365368, 1296754766161140895530674505, 3799765063841219444556153664, 11138165760379116840837129933, 32660575398596894269004630395, 95803772582109903302659971705, 281116036367240287000597160307, 825141933187631607642170430582, 2422744979523191074317651806222, 7115720853948264721550464469138, 20905404259902466570682178134101, 61436033484557854416515787554557, 180596489848062297043580473736583, 531023478945026185132223045188242, 1561828335230849070577015587599862, 4594783181095102005982648845629943, 13520912751397068669158402972820535, 39797288428151734415529516833294183, 117166853197429318351440194184640001, 345030340472240118667348975322651831, 1016268735970876894277016612352986956, 2994030447810228732421196508522801508, 8822627125900727689992098802555040065, 26003480448270289308764833189852136625, 76657514628388286510106161064786323442, 226029756946856262771695312945809615724, 666595122574573979910207473435766415800, 1966266336978008980717249705195918553528, 5801020343999533654167963129230758635937, 17117741362043104820828161774821249467025, 50520403804121740187565594126636709015553, 149129567444169474699296647095777703816561, 440286794637589018357948150453173148034809, 1300112504445214435657315221574732403366232, 3839706689852153907055350498280218948390511, 11341892306305967810005135303941131373501596] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 182 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 7 6 5 4 3 2 2 4 3 2 1 + (x + x + x - x - x - x + 1) P(x) + (x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 13, 27, 57, 122, 269, 605, 1379, 3182, 7420, 17453, 41357, 98634, 236566, 570211, 1380506, 3355528, 8185280, 20031389, 49166375, 121002963, 298537692, 738233460, 1829385845, 4542215715, 11298528028, 28152340251, 70258317672, 175601508371, 439507839770, 1101481096871, 2763928226505, 6943617188432, 17463308610473, 43966681055061, 110803591314883, 279508718029179, 705713141395523, 1783338434049627, 4510189067496665, 11415502088207975, 28914756126315351, 73291605054009739, 185902808224083404, 471848246294267687, 1198372494643665002, 3045394236766909042, 7743675807786471786, 19701218942356810935, 50150147311769770266, 127724997572390171097, 325458808262111252105, 829707261507611876254, 2116192588906284015186, 5399828958003920513740, 13784555460579860793317, 35203612522652740364321, 89940949867922712140206, 229878222115467330486132, 587763931062371858642318, 1503376118181573715492116, 3846689074287382715336194, 9845926103531341883449308, 25209930548939165325681225, 64569593220925809983353741, 165432828463614467696091458, 423983228232161138542127417, 1086939209804757271772782924, 2787326502503975028357540702, 7149785357374383356755014728, 18344988147682731284390067244, 47082332510245985594850527490, 120868063607826687877377441737, 310366824810174497951080379711, 797161550726939334098105323361, 2047962538520677057918137011882, 5262591774393682820710492185268, 13526232782451771125033589219572, 34773716938692040062180157785472, 89417006035041206300195848318547, 229975535400889609582190376440923, 591607252505314635177965446303011, 1522206540138937137410957478162416, 3917417870346613394306707106257430, 10083481058857910286087790951922170, 25959926405844391913481266197640325, 66846240398637627465766894999925009, 172158833514831085495533759750852249, 443464362514862318990873457216395456, 1142519379979763280935871522685254468, 2944030220631010667545133900797281562, 7587404552052090177381410902098091913, 19557575074829519830141191357108835960] ------------------------------------------------------------ Theorem Number, 183 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 7+ HHH + H HH + HH HH 6+ HH H H + H HH H H + HH HH HH HH 5+ HH H HH + H H + HH HH 4+ H H + HH HH 3+ HH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 4 3 2 1 + (x + x + x + x - x - x - x + 1) P(x) + (x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 9, 20, 43, 94, 216, 505, 1198, 2888, 7046, 17359, 43137, 107986, 272046, 689198, 1754652, 4486913, 11519069, 29677863, 76709558, 198858270, 516900481, 1346928260, 3517812058, 9206982999, 24144187305, 63430675750, 166926209959, 439987494362, 1161457799001, 3070255463861, 8126765806794, 21537753472145, 57146877005927, 151798160546375, 403642671170599, 1074387128983240, 2862437766174048, 7633132414976498, 20372410447459630, 54417377540703801, 145469599771072044, 389162759164454635, 1041839048797994230, 2791046978146480666, 7482013275489230337, 20069799089461878199, 53867882389413538721, 144666793314662499979, 388731932228847260560, 1045116905170070363120, 2811280699994517254169, 7565891179843013798830, 20371584272556101353559, 54877087351154823046446, 147894461371360500600404, 398750098993789333077497, 1075552191594669068647971, 2902271444012572374471826, 7834562869658274177153008, 21157109326212823185371492, 57155440767212455161258473, 154459142393974795917404387, 417560856723240437282759617, 1129201890148987098172382917, 3054673524396563025177593345, 8265998275897394297578700464, 22374802833318826872122652538, 60583279272141959914982521230, 164086330649285086653717627671, 444543898955742904386559063447, 1204692560613252897219505954158, 3265532089955788106865668972679, 8854107516472847109120249657859, 24012966115006271427327717275224, 65140965498791000199417379317401, 176753178774204821202519803270294, 479713926425134540506383824683065, 1302257781851035311627912843109992, 3535971195076524443101782142887928, 9603184357286987364921458068688114, 26086412423311750697378497706485168, 70876753808269982532179552022657879, 192611209804903641890607349672943792, 523534801206376171916739816450584025, 1423291500980846384191200772841684210, 3870121247506880066064768564504438295, 10525333776394036585049608115707480862, 28630307203934076060493508135753907692, 77892072069382575356484169897529778150, 211951244648000788109732632639813482287, 576836257741440102366311224036503009129, 1570151092397116337614826283811806160336, 4274655842190291606027844557863134575971] ------------------------------------------------------------ Theorem Number, 184 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 1 + (x + 2 x + x + x - 2 x - x - x + 1) P(x) 4 3 2 + (2 x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 5, 9, 15, 27, 52, 99, 191, 380, 767, 1561, 3212, 6676, 13980, 29462, 62457, 133095, 284904, 612325, 1320835, 2858528, 6204794, 13504896, 29467046, 64443065, 141232079, 310127824, 682241562, 1503390100, 3318125585, 7334286838, 16234066006, 35980324422, 79843282903, 177385141221, 394524626312, 878382600262, 1957594535824, 4366863711975, 9749996633366, 21787566703548, 48726502251068, 109058113252455, 244271064473613, 547511721413636, 1228030388231890, 2756175501803298, 6189779845643445, 13909237626615638, 31273831159967158, 70355639977282178, 158360839466453771, 356631087517226215, 803535502075715446, 1811330859313674816, 4084981786161499396, 9216695405000005915, 20803989056321610682, 46978386063151976380, 106126487292393541621, 239838068767767185118, 542220278033115896654, 1226285592006812766558, 2774347696927967271630, 6278832855959591570828, 14214818277513329553301, 32191694467833655881358, 72926011196323997616532, 165254507682930495490441, 374586998426594973124882, 849331174613651352346800, 1926295937143221477067260, 4370058634352064414375042, 9916689051600915661994716, 22509108319009393216650851, 51104501330215476319608970, 116055707226039146541380078, 263619574451311192836750119, 598949259458766324199929106, 1361134842968359225485601500, 3093916712876970178692847268, 7034125719967269865497972628, 15995707321136515987498854278, 36381989171910295666778485227, 82766964402142036623410315006, 188327268860574074593683978712, 428600872006951760458663393481, 975606498851851793659981066972, 2221141899698881971264987257058, 5057736078329066998909302286110, 11518941426334046103018614433700, 26238795102693120770431186038224, 59778984596015548358887856377059, 136215011845586736829636505040690] ------------------------------------------------------------ Theorem Number, 185 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 4+ H + HH + HH HH + H H + H H 3+ H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H H HH + H H H H 1+ H H H H H + H H H H H H + H H H H H H + HH HH HH HH HH HH +H HH HH H -*-+-+-+-+*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 7 6 4 2 3 1 + (x + 2 x - x - x - 2 x + 3 x - 1) P(x) 6 5 4 2 2 3 2 + (2 x + x + 2 x + x - 6 x + 3) P(x) + (x + x + 3 x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 4, 8, 17, 38, 90, 226, 595, 1626, 4583, 13255, 39152, 117632, 358379, 1104540, 3437534, 10787176, 34092504, 108416342, 346645635, 1113679878, 3593289935, 11638329754, 37826257535, 123327887440, 403249754800, 1321986604313, 4344384539207, 14308585907725, 47223762974482, 156154993535171, 517279512721917, 1716395204199984, 5704077870995768, 18983996816831575, 63268228255622253, 211127426884278728, 705395591765359037, 2359504087156624476, 7900995290800278546, 26484466908922828087, 88864073724588088699, 298445919260326762000, 1003205605602019452638, 3375055833370038475415, 11363752620369614839002, 38290991969078239124495, 129119188192510283827134, 435702970067638336401740, 1471237997718381515655961, 4971139384784656516703965, 16807309393261681339710744, 56858974801937271214176618, 192463501961806458549367052, 651833685891728036085694672, 2208794626314526388838255955, 7488512324566195972361000419, 25400920138005481936189410149, 86200438627398423962889600077, 292663432884995520011132877727, 994076051219370632401679112136, 3377975277695539232604789875543, 11483463522638435443420165861052, 39053778764590735260214744028411, 132868310291185248291927887818515, 452212595533463639447561125683626, 1539649070809086278831859924742111, 5243893547522691259065527773303234, 17866291918724072977961447260036381, 60891830322474637750552436362104287, 207598148314178819145969881814596126, 707984497135539785608629387494969248, 2415216119078895254512880464157359266, 8241695045438919774148317893449721900, 28132079874956214994051496481913538658, 96052464350363092991383456220764683736, 328044905624843906486485910858513572753, 1120657949699764053438714184735518723226, 3829348091994830090540765936502828433868, 13088374765183661773959550315278141251762, 44745867155596722406431032729662747065064, 153011402892271561815196042063134630398688, 523354250006880412472190047285558113292304, 1790467027162243322753693430925261434443773, 6126791872075190111368519055950746871602422, 20969779058966314248851114808976871975726427, 71787080271027796432625484374156603053117008, 245803630113711348682394578531982069883458376, 841817199668759568384197048764335720527595287, 2883585288228841543815032314714643011117463365, 9879418496568057882669009954460305569698680220, 33854131408980382617618344444952182795596663927, 116030423557513326707883799432379866683283721065, 397750052241093250199289662453849289266840594823, 1363719630603529068278211367514741724664144560802, 4676433740888708159615030564012359089021337713132, 16039016859440355938804763268391989690074811992256, 55018967565298271506684475923762005973565491298128] ------------------------------------------------------------ Theorem Number, 186 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 7+ HH + H HH + HH HH 6+ H HH + H HH + HH HH 5+ HH HH + H HH + HH HH 4+ H H HH + HH HH HH HH 3+ HH HHH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 4 3 2 1 + (x + x + x + x - x - x - x + 1) P(x) + (x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 9, 20, 43, 94, 216, 505, 1198, 2888, 7046, 17359, 43137, 107986, 272046, 689198, 1754652, 4486913, 11519069, 29677863, 76709558, 198858270, 516900481, 1346928260, 3517812058, 9206982999, 24144187305, 63430675750, 166926209959, 439987494362, 1161457799001, 3070255463861, 8126765806794, 21537753472145, 57146877005927, 151798160546375, 403642671170599, 1074387128983240, 2862437766174048, 7633132414976498, 20372410447459630, 54417377540703801, 145469599771072044, 389162759164454635, 1041839048797994230, 2791046978146480666, 7482013275489230337, 20069799089461878199, 53867882389413538721, 144666793314662499979, 388731932228847260560, 1045116905170070363120, 2811280699994517254169, 7565891179843013798830, 20371584272556101353559, 54877087351154823046446, 147894461371360500600404, 398750098993789333077497, 1075552191594669068647971, 2902271444012572374471826, 7834562869658274177153008, 21157109326212823185371492, 57155440767212455161258473, 154459142393974795917404387, 417560856723240437282759617, 1129201890148987098172382917, 3054673524396563025177593345, 8265998275897394297578700464, 22374802833318826872122652538, 60583279272141959914982521230, 164086330649285086653717627671, 444543898955742904386559063447, 1204692560613252897219505954158, 3265532089955788106865668972679, 8854107516472847109120249657859, 24012966115006271427327717275224, 65140965498791000199417379317401, 176753178774204821202519803270294, 479713926425134540506383824683065, 1302257781851035311627912843109992, 3535971195076524443101782142887928, 9603184357286987364921458068688114, 26086412423311750697378497706485168, 70876753808269982532179552022657879, 192611209804903641890607349672943792, 523534801206376171916739816450584025, 1423291500980846384191200772841684210, 3870121247506880066064768564504438295, 10525333776394036585049608115707480862, 28630307203934076060493508135753907692, 77892072069382575356484169897529778150, 211951244648000788109732632639813482287, 576836257741440102366311224036503009129, 1570151092397116337614826283811806160336, 4274655842190291606027844557863134575971] ------------------------------------------------------------ Theorem Number, 187 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 4+ H + HH + HH HH + H H + H H 3+ H H + H H + HH H + H H + H H 2+ H H + H H + H H + HH HH + H H 1+ H H H H H H + H H H H H H H H H H + H H H H H H H H H H + HH HH HH HH HH HH HH HH HH HH +H HH HH HH HH H -*-+-+-+-+*-+-+-+-+*-+-+-+-+*-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 16 15 14 13 12 11 10 9 8 1 + (x - x + 6 x - 6 x + 18 x - 20 x + 34 x - 38 x + 40 x 7 6 5 4 3 2 4 13 - 40 x + 30 x - 26 x + 21 x - 14 x + 10 x - 5 x + 1) P(x) + (x 12 11 10 9 8 7 6 5 4 - 2 x + 9 x - 11 x + 29 x - 30 x + 45 x - 40 x + 36 x - 39 x 3 2 3 + 26 x - 21 x + 15 x - 4) P(x) + 9 8 7 6 5 4 3 2 (-x + 6 x - 7 x + 16 x - 11 x + 22 x - 17 x + 12 x - 15 x + 6) 2 5 4 3 2 P(x) + (x - 2 x + 5 x - x + 5 x - 4) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 1, 1, 2, 4, 8, 16, 33, 72, 164, 385, 929, 2302, 5840, 15111, 39757, 106109, 286706, 782895, 2157299, 5991418, 16754254, 47133780, 133304680, 378797699, 1080937675, 3096280119, 8899537396, 25659333522, 74191700152, 215076839680, 624984670580, 1820129112192, 5311545870919, 15529666967498, 45485087616947, 133441401606583, 392085514552888, 1153713520188876, 3399415949251933, 10029177048139525, 29624392272099090, 87604798820875723, 259342755462266401, 768535092178287956, 2279681122434373908, 6768375941367910082, 20112936791682062882, 59817630236510978087, 178044647096438862509, 530345732633818958522, 1580902724946170334773, 4715780087511170459341, 14076398258518775876396, 42044309205950898883276, 125657775418684525282240, 375774621674018331570617, 1124375927719242353602914, 3366140922258788228051364, 10082793760566366735918753, 30216834851528764843156805, 90600097910177860813128100, 271776877001635112478414645, 815630397943282664266638306, 2448863906442666183560448424, 7355629884158953643930957993, 22103090309753427783590493944, 66444383239904652378075241174, 199815976489513500075595905999, 601123318248188990641792230597, 1809061723036316227985596979165, 5446216242789368898725261988498, 16401498208300928395050396115929, 49410012350418902703302101326748, 148896693672221242827909559044314, 448838211413611377469327964824974, 1353397796957554208981282171507347, 4082144142443893016917450931257201, 12316149177447690389881162822938496, 37169089927293809170079493663299174, 112203416185148453398803931080693932, 338800706501319369696382907969208572, 1023278009928211254986459641428529088, 3091371705830182285136346003095432953, 9341450589312301474826731216085100489, 28234511565375956705606901847882309569, 85358457499283402993308375153738965078, 258113493055080447379653073472541104222, 780675170598065191844926634793792438671, 2361691994037213726820461707344108161220, 7146068639249103743135894366785630040238, 21627186493192152310573023594624255353631, 65466592489038222003770498496859393128085, 198209443092575491290699990137175971768889, 600222027601206427086019256822220933481158, 1817944647598719163400906295383418984699691, 5507173198404019898922370364442446872032905, 16686084596223294364458606091843282550184879, 50565709242439808017941737571515784259619335, 153261158470844421358596187137846703234962646, 464601788900210564675892966077306594592266035] ------------------------------------------------------------ Theorem Number, 188 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 13 12 11 10 9 8 7 6 4 3 2 1 + (x + x + 2 x + x + x - 3 x - x - 4 x + x + 2 x + x - 1) 3 9 8 7 6 5 4 3 2 2 P(x) + (x + 3 x + x + 4 x - x - 2 x - 4 x - 2 x + 3) P(x) 5 4 3 2 + (x + x + 2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 6, 10, 20, 38, 71, 141, 287, 583, 1202, 2521, 5325, 11323, 24282, 52406, 113664, 247757, 542480, 1192313, 2629716, 5818748, 12912436, 28729813, 64079490, 143246101, 320883803, 720195496, 1619321024, 3647055270, 8226810087, 18584803194, 42041964361, 95229445722, 215968599608, 490356105984, 1114569328616, 2536017260869, 5775957109341, 13167432219753, 30044425752651, 68611060035416, 156810360349206, 358665774457819, 820965478557937, 1880467407759439, 4310223033012889, 9885862452845120, 22688139458714478, 52100512493277179, 119710934944822697, 275210416296780580, 633032733279400641, 1456831279717572075, 3354332676055975953, 7726969805373657763, 17807843238326490566, 41058742911104314735, 94707794486141835055, 218547278951873016885, 504520534310020995281, 1165146383227358880327, 2691813944679198933254, 6221103655060471930962, 14382781044804077557878, 33263407167250290261739, 76954607918587159600611, 178091158645065231338585, 412273954575974881103440, 954687857089931041054750, 2211389126498755280117222, 5123817785575377937814225, 11875270986534273727976053, 27530333413891900687626772, 63840220572849222278104788, 148077560084497124425798169, 343552566984827422999062338, 797266374037095566105098803, 1850620324705013667862682500, 4296672645787599932254592314, 9978052601349627898050764057, 23176908557720826553542054224, 53846703494421872456717997242, 125127973383987991144787652593, 290830021503801381930721105937, 676100921983193825969329450941, 1572060634527419283928666538120, 3656037156311472113594424613186, 8504202409975272430583173001536, 19785018164064980155520108020417, 46038106439011476416261131849967, 107145737836783124653205894881028, 249406152369906834599933074660180, 580647632830318254723094356140953, 1352040998078175397319979202051475] ------------------------------------------------------------ Theorem Number, 189 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 7+ HHH + H HH + HH HH 6+ H H + H H + HH HH 5+ HHH HH HH + H HH H H + HH HHH HH 4+ H H H + HH HH 3+ HH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 6 5 3 2 3 1 + (x - 3 x - x + 2 x + x - 1) P(x) 6 5 3 2 2 3 2 + (3 x + x - 4 x - 2 x + 3) P(x) + (2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 24, 50, 100, 207, 447, 965, 2097, 4641, 10355, 23208, 52434, 119238, 272302, 624715, 1439586, 3329035, 7723353, 17974332, 41946862, 98136543, 230136785, 540859834, 1273645217, 3004841530, 7101517437, 16810654036, 39854525348, 94621732620, 224950428542, 535465514846, 1276128784737, 3044720948720, 7272175007841, 17386856915967, 41609801199087, 99670684111928, 238955727084254, 573360774386420, 1376837089630860, 3308764126191171, 7957256958283794, 19149694328506228, 46115679249370442, 111125023198040762, 267941826636175204, 646433889341639098, 1560460708781195478, 3768924211315976522, 9107707171651752850, 22020112055371883983, 53264882178765929245, 128903877018767159586, 312095509354848105514, 755960176581781422429, 1831866563301055461943, 4440851470520127715854, 10769869026085811825291, 26128886252691178913799, 63415069809483920442764, 153964356091782976724423, 373937747991975936017966, 908500784048788194813896, 2207973525844179375897501, 5367855788092394722780239, 13053960160887471570162186, 31755167086049496587497855, 77270472173378202155837143, 188077261542308295134223872, 457909065550704597273291014, 1115165194681535923031110438, 2716521249676194665104160483, 6619085378281378272215833397, 16132103128989793305097961362, 39326870592610533254429937069, 95893803896808211685222661778, 233879375049458391590317811366, 570546544663074707446614357597, 1392148793303975476592593878598, 3397608858830076013984401770568, 8293771449881197142382854212644, 20249746445904137513663328249672, 49450867782510794412659596235458, 120785019511666197733598404565551, 295076860334680412091808688807665, 721005064284596647207054443970788, 1762060340462307761691050403140135, 4307058934161988158319017861316542, 10529718506027909698423650877921477, 25747021917792169515323378704085982, 62966561145792275331841952840077732, 154015405894044114004769528686025159] ------------------------------------------------------------ Theorem Number, 190 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 1 + (x + 2 x + x + x - 2 x - x - x + 1) P(x) 4 3 2 + (2 x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 5, 9, 15, 27, 52, 99, 191, 380, 767, 1561, 3212, 6676, 13980, 29462, 62457, 133095, 284904, 612325, 1320835, 2858528, 6204794, 13504896, 29467046, 64443065, 141232079, 310127824, 682241562, 1503390100, 3318125585, 7334286838, 16234066006, 35980324422, 79843282903, 177385141221, 394524626312, 878382600262, 1957594535824, 4366863711975, 9749996633366, 21787566703548, 48726502251068, 109058113252455, 244271064473613, 547511721413636, 1228030388231890, 2756175501803298, 6189779845643445, 13909237626615638, 31273831159967158, 70355639977282178, 158360839466453771, 356631087517226215, 803535502075715446, 1811330859313674816, 4084981786161499396, 9216695405000005915, 20803989056321610682, 46978386063151976380, 106126487292393541621, 239838068767767185118, 542220278033115896654, 1226285592006812766558, 2774347696927967271630, 6278832855959591570828, 14214818277513329553301, 32191694467833655881358, 72926011196323997616532, 165254507682930495490441, 374586998426594973124882, 849331174613651352346800, 1926295937143221477067260, 4370058634352064414375042, 9916689051600915661994716, 22509108319009393216650851, 51104501330215476319608970, 116055707226039146541380078, 263619574451311192836750119, 598949259458766324199929106, 1361134842968359225485601500, 3093916712876970178692847268, 7034125719967269865497972628, 15995707321136515987498854278, 36381989171910295666778485227, 82766964402142036623410315006, 188327268860574074593683978712, 428600872006951760458663393481, 975606498851851793659981066972, 2221141899698881971264987257058, 5057736078329066998909302286110, 11518941426334046103018614433700, 26238795102693120770431186038224, 59778984596015548358887856377059, 136215011845586736829636505040690] ------------------------------------------------------------ Theorem Number, 191 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 13 12 11 10 9 8 7 6 4 3 2 1 + (x + x + 2 x + x + x - 3 x - x - 4 x + x + 2 x + x - 1) 3 9 8 7 6 5 4 3 2 2 P(x) + (x + 3 x + x + 4 x - x - 2 x - 4 x - 2 x + 3) P(x) 5 4 3 2 + (x + x + 2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 6, 10, 20, 38, 71, 141, 287, 583, 1202, 2521, 5325, 11323, 24282, 52406, 113664, 247757, 542480, 1192313, 2629716, 5818748, 12912436, 28729813, 64079490, 143246101, 320883803, 720195496, 1619321024, 3647055270, 8226810087, 18584803194, 42041964361, 95229445722, 215968599608, 490356105984, 1114569328616, 2536017260869, 5775957109341, 13167432219753, 30044425752651, 68611060035416, 156810360349206, 358665774457819, 820965478557937, 1880467407759439, 4310223033012889, 9885862452845120, 22688139458714478, 52100512493277179, 119710934944822697, 275210416296780580, 633032733279400641, 1456831279717572075, 3354332676055975953, 7726969805373657763, 17807843238326490566, 41058742911104314735, 94707794486141835055, 218547278951873016885, 504520534310020995281, 1165146383227358880327, 2691813944679198933254, 6221103655060471930962, 14382781044804077557878, 33263407167250290261739, 76954607918587159600611, 178091158645065231338585, 412273954575974881103440, 954687857089931041054750, 2211389126498755280117222, 5123817785575377937814225, 11875270986534273727976053, 27530333413891900687626772, 63840220572849222278104788, 148077560084497124425798169, 343552566984827422999062338, 797266374037095566105098803, 1850620324705013667862682500, 4296672645787599932254592314, 9978052601349627898050764057, 23176908557720826553542054224, 53846703494421872456717997242, 125127973383987991144787652593, 290830021503801381930721105937, 676100921983193825969329450941, 1572060634527419283928666538120, 3656037156311472113594424613186, 8504202409975272430583173001536, 19785018164064980155520108020417, 46038106439011476416261131849967, 107145737836783124653205894881028, 249406152369906834599933074660180, 580647632830318254723094356140953, 1352040998078175397319979202051475] ------------------------------------------------------------ Theorem Number, 192 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 4 3 2 2 1 + (x + 2 x + 2 x + x - 2 x - x - x + 1) P(x) 5 4 3 2 + (x + 2 x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 4, 7, 12, 19, 32, 57, 102, 184, 338, 631, 1193, 2278, 4386, 8512, 16640, 32727, 64698, 128497, 256288, 513091, 1030652, 2076551, 4195389, 8497633, 17251501, 35097685, 71545640, 146109780, 298889934, 612391857, 1256577012, 2581964630, 5312226040, 10942958613, 22568085609, 46593874760, 96296775298, 199214136675, 412508168198, 854927060291, 1773336983575, 3681307349391, 7647959447370, 15900359496772, 33080520927256, 68869800315583, 143471049030427, 299066191297844, 623774018719246, 1301769563196978, 2718180873152235, 5678728121019830, 11869812600023930, 24822684507644365, 51934768256973512, 108708894175429681, 227647361551092841, 476918760095371805, 999549561565926862, 2095736135278881599, 4395777411189686495, 9223512701958825628, 19360365984653297116, 40652059395072526234, 85388375869205205243, 179414611502079962724, 377099148730245402023, 792844459482558248785, 1667444254566230073271, 3507856998483416898418, 7381696118116215545446, 15537845884622259716287, 32714673959131873020032, 68898287436832289381615, 145139401140260625493636, 305823205702351606859661, 644556519941560063235675, 1358796405023640572966082, 2865154456516923094879448, 6042817833843456908071545, 12747540906403185592012971, 26897164309201694344507690, 56764595715167086292090332, 119822245028973861534159847, 252978817559414921672077464, 534214521684417321803286560, 1128314411252867525432280150, 2383557588085094296518911183, 5036170748773190613622743871, 10642723994868396804864305247, 22494743116136658770993174434, 47553615795218186780095160580, 100544600599431729835469608976] ------------------------------------------------------------ Theorem Number, 193 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H H + HH H H HH + H H HH H + HH HHH HH 4+ H H H H + HHH H H + HH H HH HH + H HH H H 3+ H H H H + HH H HH HH + H HH H H + HH HHH HH 2+ H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 2 2 (x + 2 x + 3 x + 1) P(x) + (-2 x - 4 x - 3) P(x) + x + 2 = 0 The sequence a(n) satisfies the linear recurrence (n - 6) a(n - 1) 2 (5 n - 9) a(n - 2) (7 n - 12) a(n - 3) a(n) = ---------------- + -------------------- + ------------------- n n n 2 (2 n - 3) a(n - 4) + -------------------- n subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 2] Just for fun, a(1000), equals 2707402735309983135132830930008358711802938857520701248531880121950828753517\ 140752980183535403777375986917306444296419260986060477589548556037762608\ 021391966650114470915341831522545407557152112906015028533538118042316594\ 340362095907181262581463733389867106522904048397840452453634970960991796\ 328749896297368545963349355352685770120780078298280998772571430722901273\ 660635726521459623968336222250887848827628418225763502189310375750913801\ 821705239424552610367168661296086653401250057857184414981984454071957672\ 076561465920959215339701706896029839097374603799877083172542408439659336\ 99045560577715111 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 2, 6, 19, 61, 202, 683, 2348, 8184, 28855, 102731, 368813, 1333684, 4853436, 17761181, 65320691, 241300829, 894958140, 3331323651, 12441078958, 46601721324, 175040968111, 659136721385, 2487852579751, 9410480922018, 35667266269063, 135437338519384, 515185187633072, 1962888995885257, 7490161337785610, 28622617684287782, 109525065687923287, 419633448379599729, 1609712400708367719, 6181869221288227590, 23766138410960724427, 91461996710839705220, 352325235554247008286, 1358461203781110949095, 5242422512475231564943, 20247914731362280357935, 78266428782266936507914, 302762581270996091283485, 1172049337968363646029502, 4540389106303823332882930, 17600745181071853504438211, 68272835148377794652881905, 264991860266036223754203505, 1029139661480224539550961920, 3999110130393192524416641481, 15548565371366563074036365724, 60484815419416135440060503826, 235409030896578746454938625017, 916669327708883797673506627545, 3571146067505648476179200827817, 13918766967943560133905666549106, 54273197922571442881749410881583, 211716721356469888341404676796334, 826235637713623931765154260337622, 3225714404808472635627059301729219, 12598406531836491166958567443614077, 49222966515931937084040409751741834, 192387742859539393257343619269364075, 752211005008601123765419649488675380, 2942050117299498640098929109092648976, 11510764917181279902273582368532288187, 45050318007808847615401615690430270607, 176370968535127374265932852537610587529, 690697801882101172559066805828004952612, 2705684263921738445594472455745399412849, 10602071025644854806761514736824822294400, 41555209958417235357175681704555998222990, 162921422120640167006891744475746095356621, 638918897125978985286972610434787025980777, 2506254184785669386096509206434455194145957, 9833620956080278279340996694438859463493994, 38592953476594305864187170748027053434580927, 151497724648535755916943831190097432839515504, 594846876092460114173086045243882191816411580, 2336161018937274322245908328646264688936808065, 9176910481971497883734299610298999753113471421, 36056540772898692298020250501235236313954764127, 141697846737818771044867204236218593283643237152, 556970170681048640358660244722051206491150693389, 2189717327264842579050960435031081416074057995722, 8610524664770983008154075961989320888879471783844, 33865281766239231547573828485394004757403853224635, 133217546348476293285872842963649909620502810789285, 524140873557676455968478988712814805097265479349831, 2062589238447783499393097092212211270483825767171102, 8118090813684709703726204866752429900747420056833347, 31957284743833043185529275807435121843662551723245924, 125822720420760672705482442375454286403314506437654032, 495472976413795393661373786846954064039214715418616759, 1951421549579213815983255020280040772941570986524290035, 7686896034143582529038882769747548028406498054051997445, 30284353891598387169443168471393176638878880774133421836, 119330555501567809279429014421734187209650473184512496907] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 194 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 7+ HH + H HH + HH HH 6+ H HH + H HH + HH HH 5+ HH HH HHH + H HH HH H + HH HHH HH 4+ H H H + HH HH 3+ HH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 2 2 (x + 2 x + 2 x) P(x) + (-2 x - 3 x - 1) P(x) + x + 1 = 0 The sequence a(n) satisfies the linear recurrence (n - 2) a(n - 1) (3 n - 5) a(n - 2) (8 n - 13) a(n - 3) a(n) = ---------------- + 3/2 ------------------ + 1/2 ------------------- n + 1 n + 1 n + 1 (n - 2) a(n - 4) + 3/2 ---------------- n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1] Just for fun, a(1000), equals 7343404681614865434684763507210347427020178928606879284591986851267868435212\ 249556011530055635260692642873173384899274398440091540022367641511462811\ 008588715366229430034654248374874429801785393016846152183226987120054638\ 377243219486082547546109186858130940779431142925904831163236883434548069\ 717001456259229640670551010840014168855155974450769022681600038685269854\ 610212079351914130074608602819585260369730841413939433497804288630925688\ 421813161058031288831930158762023521 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 3, 7, 17, 43, 110, 286, 753, 2003, 5376, 14540, 39589, 108427, 298512, 825664, 2293271, 6393539, 17885835, 50191175, 141247519, 398537101, 1127203038, 3195229662, 9076078057, 25830193513, 73643406563, 210312889095, 601558143229, 1723166250419, 4942839351320, 14196809911512, 40826123674963, 117541006104195, 338779909115505, 977457947592893, 2822979976721987, 8160665572497041, 23611944396102714, 68376681868569674, 198169750797798271, 574782976348372753, 1668371611006438146, 4846075456079115622, 14085849428228221307, 40969362561239301701, 119235949321876092360, 347229239641557994136, 1011757164052160972473, 2949698621922111594973, 8604216545991491941581, 25111278043912438342801, 73323304601568480731897, 214201905567546134575859, 626044889720284349690094, 1830549990322901795312942, 5354823514457078459010313, 15670757112005494774639051, 45878599087262878735384752, 134369299666916034362964172, 393689966411509441770005261, 1153899343965725112108795443, 3383264305703491359375922560, 9923244697941619216569794624, 29115000903164573472390932419, 85451758118279984454158890195, 250877793948601314014941971921, 736777877691289303907872561885, 2164414038110594458093425771443, 6360187870629936233099648197313, 18694848840332640943329103122570, 54965868576235609763116372140890, 161651671060236413178711527166019, 475532533517495969396124008055577, 1399233799660317951564059213015592, 4118201121063760930956059935817164, 12123536696685281838476884788352799, 35698751359647955775282558186295793, 105141975845442998724664146494190416, 309739251405543919680997857296802976, 912664235708260313995444264873081893, 2689789263907547466230267373821274617, 7928951321295704744188805383739325609, 23377681170924021285051888390957923933, 68940317392992307735795082187711452173, 203343055362393936480696319767811457959, 599884647429347603580336902386533043922, 1770054610915592642421723575150356117650, 5223773514153323773797895768878478839271, 15419101377748600955972333826656656984531, 45520727173998870390619930777957739468395, 134410477602615207286484828173887369300119, 396944068749335739220874303030506094827475, 1172455237187575683372434195700210272843517, 3463638304632421047516261496691878570034312, 10233794644880525218408203904987488653900680, 30241783348700004437279186446509997511733421, 89380608824471922293587992331001273395222029] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 195 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ H HH HH + HH H HH H + H HH H H + HH HHH HH 4+ H H H + H H + HH HH 3+ H H H + HH HH HH HH + HH HH H HH 2+ HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 3 2 3 3 2 2 (x + x + 2 x + 2 x + 1) P(x) + (-x - 2 x - 4 x - 4) P(x) + (x + 5) P(x) + x - 2 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 4, 11, 32, 92, 268, 798, 2417, 7409, 22931, 71578, 225118, 712686, 2269259, 7262408, 23348234, 75371198, 244209752, 793922885, 2588944642, 8466114815, 27756449911, 91216824823, 300427664158, 991493522072, 3278408704177, 10859355762318, 36029871072603, 119727441254516, 398433520053897, 1327735921331329, 4430236913543148, 14800290107786456, 49500857672038125, 165740223550798760, 555509330233154118, 1863718969612078606, 6258556568274075767, 21035517099873132816, 70761884332297082831, 238229723402968015453, 802651770776295634942, 2706315280847069804470, 9131368928663148753789, 30830943127004895626333, 104164344729652022372536, 352144882787617743382427, 1191195467930829139371432, 4031753696860260535873179, 13653525013507045702446578, 46262221293727887222695060, 156830531738403548802204614, 531923519587394486140914326, 1804989353314779366565664573, 6127732370256599427590772425, 20812191613032286308588468250, 70716743818038013656607563236, 240384775526146338138014715838, 817459182249753014198547008431, 2780955496918837237493060283095, 9464235518382892911813311314354, 32220740443783794237463184568288, 109733477209625400990531366044899, 373845193764580440887137160551830, 1274057680137629184323415138515915, 4343369647109098310744030885522394, 14811562449213920489492625724619388, 50525134016892416697757791490754471, 172402235236229664416152856563053560, 588441888978895199681690449432647588, 2009028683724481037829010313005954953, 6860998043629993597783808224242316984, 23437102464843862375040696880957159526, 80081646422341743520187352214736746920, 273697998868827525743390492229576164130, 935657747917930688808279271290726527487, 3199384967684844509260377112657893906714, 10942523432757272457692151978400404581662, 37434107846219747936050803202237082526945, 128089670611089198479202378078493541289683, 438384347626859838482204945655608043874262, 1500679845818741594816355330826884005371688, 5138199822194022758666722428274036108948099, 17596317209266075384479571684091183910114292, 60272391338699587154458877593205805356653629, 206489975699880005353440032861612462027288756, 707557210953922690143418121324842138988153154, 2424958869126915747945014715952827946054380045, 8312385344477138064417392518336035933606252824, 28498614099560963291759771398502393392750152072, 97723036759639719118573772264179641008200374123, 335153470480484634702632317052439643497669875866, 1149641683003293282818706915888024378617536908271, 3944135317413206096688826552058903952174456391494, 13533501959434215913619309932320102488558849813204, 46444709604471049971163662121618872580306645497669, 159414797827033471632169097078014363215217008372911] ------------------------------------------------------------ Theorem Number, 196 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H H + HH H H HH + H HH HH H + HH H H HH 4+ H H H H + H HH HH H + HH H H HH + H HH H H 3+ H H H H + HH H HH HH + H HH H H + HH HHH HH 2+ H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 2 1 + P(x) x + P(x) x + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence 2 (518 n - 1031 n + 780) a(n - 1) a(n) = -1/195 -------------------------------- n + 2 3 2 (1036 n - 4335 n + 4367 n - 1170) a(n - 2) - 1/195 -------------------------------------------- (n + 2) (n + 1) 3 2 (2590 n - 11115 n + 11939 n - 1812) a(n - 3) + 1/195 ---------------------------------------------- (n + 2) (n + 1) 2 (n - 3) (1036 n - 1865 n + 1074) a(n - 4) + 1/65 ------------------------------------------ (n + 2) (n + 1) 161 (74 n - 55) (n - 3) (n - 4) a(n - 5) + --- ------------------------------------ 195 (n + 2) (n + 1) subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 1456286415952438964652364581682787386720721652723427223868652268704217615734\ 013661511014559584083968554107289545538396109682528052843351355027899510\ 336875839145124807012715538480932028100745003225030603205050905923754789\ 120913436417646081847745149534148691270052401108733311860953962982834165\ 142352139116820596104418788220221329115699403613078628786955744338071295\ 785210326056171164845618097 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 4, 8, 13, 31, 71, 144, 318, 729, 1611, 3604, 8249, 18803, 42907, 98858, 228474, 528735, 1228800, 2865180, 6693712, 15676941, 36807239, 86584783, 204060509, 481823778, 1139565120, 2699329341, 6403500057, 15211830451, 36183117255, 86171536894, 205459894230, 490417795075, 1171809537516, 2802705573178, 6709741472212, 16077633792535, 38557470978465, 92543928713413, 222292252187223, 534346494991678, 1285375448785399, 3094086155360823, 7452774657572624, 17962853486458385, 43320517724198861, 104535253974750259, 252390609561764423, 609699521485999914, 1473608895007574938, 3563397472304118410, 8620933920694737411, 20866285270840567272, 50527739882573522706, 122405798503806040965, 296657476936886314309, 719256672330233630085, 1744545018773625965860, 4232966162296687137816, 10274640808649742568633, 24948409910226791411907, 60599469176019760272535, 147244830551438483529790, 357892285312932720768838, 870164978202711215226483, 2116332778149031726542808, 5148677470688468794716894, 12529484860432553111063488, 30499528076281485551927755, 74262935424150986850220725, 180870207357836188170341069, 440630810522439231975963839, 1073724026979876073732588658, 2617084008072054452326466527, 6380388743696091681799439061, 15558886191823487315141504508, 37949779254079097680017924283, 92584221613943667202286169191, 225922433438019608173638548805, 551409374869368431174631180753, 1346105758407373798214652723522, 3286793196606678831626181940722, 8026971507652658919500628768874, 19607185640982275440523729181033, 47902823795394726062462438796736, 117054324948067108888357885549374, 286083328086281443692788683942327, 699317934466646636669713105276839, 1709748244857899463255482766619759, 4180839106559551650627420507840427, 10225086357877339275440446600065760, 25011587559313816213376886296261184, 61190615947546730766871192612064833, 149725673741988363612845353038008205, 366415846156516055604065564079552535, 896845046441850691487935128438196159] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 197 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 3+ H H H + H H H H H H + HH HH HH HH HH HH 2.5+ H H H H H H + H H H H H H + H H H H H H + H H H H H H 2+ H H H H H H + H H H H H H + H H H H H H 1.5+ HH HH HH HH HH HH + H H H H H H + H H H H H H 1+ H H H H H + H H H H + H H H H + H H H H 0.5+ H H H H + H H H H +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 2 2 (x + 2 x + 2 x) P(x) + (-2 x - 3 x - 1) P(x) + x + 1 = 0 The sequence a(n) satisfies the linear recurrence (n - 2) a(n - 1) (3 n - 5) a(n - 2) (8 n - 13) a(n - 3) a(n) = ---------------- + 3/2 ------------------ + 1/2 ------------------- n + 1 n + 1 n + 1 (n - 2) a(n - 4) + 3/2 ---------------- n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1] Just for fun, a(1000), equals 7343404681614865434684763507210347427020178928606879284591986851267868435212\ 249556011530055635260692642873173384899274398440091540022367641511462811\ 008588715366229430034654248374874429801785393016846152183226987120054638\ 377243219486082547546109186858130940779431142925904831163236883434548069\ 717001456259229640670551010840014168855155974450769022681600038685269854\ 610212079351914130074608602819585260369730841413939433497804288630925688\ 421813161058031288831930158762023521 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 3, 7, 17, 43, 110, 286, 753, 2003, 5376, 14540, 39589, 108427, 298512, 825664, 2293271, 6393539, 17885835, 50191175, 141247519, 398537101, 1127203038, 3195229662, 9076078057, 25830193513, 73643406563, 210312889095, 601558143229, 1723166250419, 4942839351320, 14196809911512, 40826123674963, 117541006104195, 338779909115505, 977457947592893, 2822979976721987, 8160665572497041, 23611944396102714, 68376681868569674, 198169750797798271, 574782976348372753, 1668371611006438146, 4846075456079115622, 14085849428228221307, 40969362561239301701, 119235949321876092360, 347229239641557994136, 1011757164052160972473, 2949698621922111594973, 8604216545991491941581, 25111278043912438342801, 73323304601568480731897, 214201905567546134575859, 626044889720284349690094, 1830549990322901795312942, 5354823514457078459010313, 15670757112005494774639051, 45878599087262878735384752, 134369299666916034362964172, 393689966411509441770005261, 1153899343965725112108795443, 3383264305703491359375922560, 9923244697941619216569794624, 29115000903164573472390932419, 85451758118279984454158890195, 250877793948601314014941971921, 736777877691289303907872561885, 2164414038110594458093425771443, 6360187870629936233099648197313, 18694848840332640943329103122570, 54965868576235609763116372140890, 161651671060236413178711527166019, 475532533517495969396124008055577, 1399233799660317951564059213015592, 4118201121063760930956059935817164, 12123536696685281838476884788352799, 35698751359647955775282558186295793, 105141975845442998724664146494190416, 309739251405543919680997857296802976, 912664235708260313995444264873081893, 2689789263907547466230267373821274617, 7928951321295704744188805383739325609, 23377681170924021285051888390957923933, 68940317392992307735795082187711452173, 203343055362393936480696319767811457959, 599884647429347603580336902386533043922, 1770054610915592642421723575150356117650, 5223773514153323773797895768878478839271, 15419101377748600955972333826656656984531, 45520727173998870390619930777957739468395, 134410477602615207286484828173887369300119, 396944068749335739220874303030506094827475, 1172455237187575683372434195700210272843517, 3463638304632421047516261496691878570034312, 10233794644880525218408203904987488653900680, 30241783348700004437279186446509997511733421, 89380608824471922293587992331001273395222029] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 198 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H + HH + HH HH + H H + H H 3+ H H H H + H H H H H H + H HH H HH HH H + H H H H H H + H H H H H H 2+ H H H H H H + H H H H H H + H H H H H H + HH H HH H HH HH + H H H H H H 1+ H H H H + H H + H H + HH HH +H H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 2 2 1 + (x + x + x) P(x) + (-x - x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence 2 (n - 1) a(n - 1) (n - 1) a(n - 2) 2 (n - 1) a(n - 3) a(n) = ------------------ + ---------------- + ------------------ n + 1 n + 1 n + 1 (n - 3) a(n - 4) - ---------------- n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1] Just for fun, a(1000), equals 8234659334401657747875389407339732571279656787453168797588442711849374055534\ 399731682304762377542535497852520568576497017328122037389731286460846010\ 443044461555595775102715778648717413957811227465415071230974603488309732\ 107343673557325707026620173863825091561733935419104095484779003697596919\ 255210685186353505572823322729205796609579316204499517633378703540535080\ 3409820335280448356407307981663834920780317426749 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 2, 5, 10, 22, 50, 113, 260, 605, 1418, 3350, 7967, 19055, 45810, 110637, 268301, 653066, 1594980, 3907395, 9599326, 23643751, 58374972, 144442170, 358136905, 889671937, 2214015802, 5518884019, 13778312440, 34448765740, 86247503194, 216212219905, 542679337066, 1363662087715, 3430394504590, 8638347021316, 21774280683160, 54936700030009, 138729047625190, 350622839271715, 886878410779009, 2245035973560322, 5687260991374732, 14417497508837605, 36573842414782720, 92839581311917417, 235812199941976300, 599320573575117058, 1524059250959474845, 3877794297358396855, 9871859585738505998, 25144093549048182413, 64074878999855307016, 163360408628999551420, 416683052591437311350, 1063305463526486150711, 2714544416605796805210, 6932923285598087446149, 17713749965177310901250, 45276590676999361770020, 115771243190730143287029, 296132708151904856692605, 757749959340385997844066, 1939611300690566481202245, 4966475689992131813080275, 12721037219231181524918736, 32593672118747960232345768, 83536591670236848918980793, 214165343693311340487787210, 549220913503729937675166115, 1408857141116990223897481488, 3614977486126526634363849600, 9278116932131800430461411537, 23819172672221244289361500615, 61165029262413086366062908710, 157103765800777960498332811967, 403621422467675998133976904567, 1037202229331537017532021569084, 2665948789464832385353926176270, 6853886233863366236936936795735, 17624483272650789398585868641894, 45330245872305860180937387609005, 116613755674453256884710203714666, 300054011066011799837634994176260, 772209302119215355747708116143270, 1987717304715678281668936564065731, 5117482079734060283987442566130410, 13177661616778411036873823419266269, 33938991026722940778160525149248795, 87425144864325959468794760227423610, 225241810423464354245477849720958404, 580410529842003323164622447324796887, 1495869269272260873776735256040173482, 3855871824791709224683110411230106875, 9940785017959226569084248730088041700, 25632237357502162050592459300023620282, 66102628816394659298417014543745019551, 170496711589999877722570711772593704641] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 199 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H H + HH H H HH + H HH HH H + HH H H HH 4+ H H H H + H HH HH H + HH H H HH + H HH H H 3+ H H H H + HH H HH HH + H HH H H + HH HHH HH 2+ H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 2 2 (x + x + 2 x) P(x) + (-x - 3 x - 1) P(x) + x + 1 = 0 The sequence a(n) satisfies the linear recurrence (n - 1) a(n - 1) (n - 2) a(n - 2) (7 n - 17) a(n - 3) a(n) = 3/2 ---------------- + 3/2 ---------------- + 1/2 ------------------- n + 1 n + 1 n + 1 (3 n - 13) a(n - 4) (5 n - 22) a(n - 5) (n - 5) a(n - 6) + 3/2 ------------------- + 1/2 ------------------- + 3/2 ---------------- n + 1 n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 2, a(6) = 5] Just for fun, a(1000), equals 3110044087209480005117630829262690785178962570065949042859259790317737040379\ 759701100163072347133061628900994344134390606794155733990395791805633590\ 537024931819516117069440621896922710132357543951542671730020154374140252\ 481152668325136984199423840335486951160089481484427605735584117701225360\ 185443273546843074814776347936817145348029484175062959758647141754841797\ 753245545859806563436028413641584301856998781416920512719059909064340976\ 0 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 2, 5, 11, 25, 59, 141, 341, 834, 2059, 5124, 12840, 32371, 82050, 208966, 534479, 1372334, 3535944, 9139675, 23692836, 61582898, 160460533, 419044485, 1096638614, 2875503146, 7553584753, 19876005353, 52383517030, 138263312497, 365449260129, 967207210893, 2563018745281, 6799767407380, 18060050175493, 48017685199887, 127796083238768, 340444518042078, 907752844356051, 2422494371813441, 6470152610896746, 17294449355027437, 46261996007121913, 123837829563019809, 331726717724960709, 889187531069931542, 2384956488565065577, 6400749401946081761, 17188349671696188130, 46182861841598432798, 124154272173533996975, 333939679211499088158, 898652419534228226604, 2419497096058108744166, 6517195853819678326088, 17562721011645909868766, 47349085933302136830686, 127706762787997722778519, 344582115297767116357518, 930127395634033290690560, 2511641225135101661616031, 6784737997508785947833423, 18334284135645956344807060, 49561625230123678806569948, 134021151295179820442198667, 362529278870271414806465126, 980958575089081220186575686, 2655168958539467728055683765, 7188922641889272855844422284, 19469823265240457710850266212, 52745243719287993339338899111, 142930323501759518804118082079, 387420007592312848246718803936, 1050396463254153780076312101790, 2848623504656927168323648787217, 7727244004300524919097211948918, 20966177067251074388309237930728, 56900529025354369049987241325696, 154459018041216523663673725397056, 419380000231229978551289480086232, 1138930702193997869259515535996900, 3093711169515805375308283798382397, 8405294433076864048756502355472832, 22840977954137951310899630769331300, 62081617647281364849542856240229601, 168770275549107517139128210128571621, 458893179504305904371260921975082624, 1247981426695721573015941182507884778, 3394561989992914077350719681100010261, 9234996898972555075219612487910040538, 25128437916389102357822582745659397424, 68386180935302267072806640040633043691, 186141742299768525598859015802344525816, 506745913283408122311836317282617805046, 1379768832938553665690210141896566743301, 3757427285946611381207676416183901480105, 10233911186736146182371403793350552546986, 27877779223509090151328083619029560684242] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 200 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH + HH H + H HH + HH H 4+ H HH + H HH + HH HH 3+ H H H + HH H HH HH + HH HH H HH 2+ HH HH H HH + H H HH H + HH HH H HH + H H HH H 1+ HH H HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 1 + (x + x) P(x) + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 1) a(n - 1) (2 n - 1) a(n - 2) 3 (2 n - 5) a(n - 3) a(n) = ------------------ - ------------------ + -------------------- n + 1 n + 1 n + 1 (n - 2) a(n - 4) 2 (2 n - 7) a(n - 5) - ---------------- + -------------------- n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 3134627918704846396589045582068293464668313386749891925759492579301366817419\ 266879466277019895299966556024437351904982229786354769359885950645415854\ 487762568754124509622513897418551096745213564746196714896494879980931898\ 708502918937008296833152741264657083164783324054454272242916859440693717\ 64365001116217342900388877265443604205502181219138765202835544779530 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 42, 84, 170, 354, 740, 1549, 3269, 6945, 14811, 31711, 68177, 147091, 318313, 690837, 1503351, 3279445, 7169907, 15708485, 34482475, 75830981, 167042763, 368548926, 814341362, 1801867812, 3992172298, 8855912464, 19668236110, 43729885814, 97330067640, 216844435346, 483571453768, 1079354856828, 2411240775366, 5391038526802, 12062679137620, 27010935099196, 60526554883902, 135721815227704, 304535849636366, 683755276072670, 1536121565708096, 3453050076522008, 7766456476976834, 17477390474177854, 39350957376284872, 88644320911742304, 199781711595784890, 450466438264400446, 1016164744759318568, 2293264805607373804, 5177575503207130558, 11694317639185843274, 26423715558659122260, 59728010995453379421, 135058602016778295285, 305507289981338348257, 691307640696528081979, 1564829833276593856647, 3543272754560789200681, 8025639916478510155339, 18183959683820504446497, 41212333247384488368873, 93431281455612601484243, 211875294974960472423745, 480604617462452614426815, 1090465639820281676643073, 2474853003558648387670983, 5618201029263006344110809, 12757123617698163352573711, 28974313940315478259062979, 65822724585319538752564981, 149567876677718197286698671, 339936780414026450535924453, 772774892529317395366017603, 1757116730487566683312107729, 3996122690354078859729209647, 9090033908542877936842438557, 20681339112467629598326301035, 47062642985297219558085765929, 107116551464339266855499450015, 243847122183568901454753070765, 555210537090801909689698475727, 1264372572489361483075032316117, 2879838126594606020044275873923, 6560472630207720552639437125041, 14947710059210582396858604974621, 34063179534729044501719950750975, 77636370512116064921945474111629, 176975591352231844422902101233579, 403485801251068480867972873270661, 920043891167408188870554973636539] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 201 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H H H + H HH HHH HHH + HH H H HH H HH + H HH H H HH H 3+ H H H H H + HH H HH HH + H HH H H + HH HHH HH 2+ H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 3 2 3 3 2 2 (x + x + 2 x + 2 x + 1) P(x) + (-x - 2 x - 4 x - 4) P(x) + (x + 5) P(x) + x - 2 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 4, 11, 32, 92, 268, 798, 2417, 7409, 22931, 71578, 225118, 712686, 2269259, 7262408, 23348234, 75371198, 244209752, 793922885, 2588944642, 8466114815, 27756449911, 91216824823, 300427664158, 991493522072, 3278408704177, 10859355762318, 36029871072603, 119727441254516, 398433520053897, 1327735921331329, 4430236913543148, 14800290107786456, 49500857672038125, 165740223550798760, 555509330233154118, 1863718969612078606, 6258556568274075767, 21035517099873132816, 70761884332297082831, 238229723402968015453, 802651770776295634942, 2706315280847069804470, 9131368928663148753789, 30830943127004895626333, 104164344729652022372536, 352144882787617743382427, 1191195467930829139371432, 4031753696860260535873179, 13653525013507045702446578, 46262221293727887222695060, 156830531738403548802204614, 531923519587394486140914326, 1804989353314779366565664573, 6127732370256599427590772425, 20812191613032286308588468250, 70716743818038013656607563236, 240384775526146338138014715838, 817459182249753014198547008431, 2780955496918837237493060283095, 9464235518382892911813311314354, 32220740443783794237463184568288, 109733477209625400990531366044899, 373845193764580440887137160551830, 1274057680137629184323415138515915, 4343369647109098310744030885522394, 14811562449213920489492625724619388, 50525134016892416697757791490754471, 172402235236229664416152856563053560, 588441888978895199681690449432647588, 2009028683724481037829010313005954953, 6860998043629993597783808224242316984, 23437102464843862375040696880957159526, 80081646422341743520187352214736746920, 273697998868827525743390492229576164130, 935657747917930688808279271290726527487, 3199384967684844509260377112657893906714, 10942523432757272457692151978400404581662, 37434107846219747936050803202237082526945, 128089670611089198479202378078493541289683, 438384347626859838482204945655608043874262, 1500679845818741594816355330826884005371688, 5138199822194022758666722428274036108948099, 17596317209266075384479571684091183910114292, 60272391338699587154458877593205805356653629, 206489975699880005353440032861612462027288756, 707557210953922690143418121324842138988153154, 2424958869126915747945014715952827946054380045, 8312385344477138064417392518336035933606252824, 28498614099560963291759771398502393392750152072, 97723036759639719118573772264179641008200374123, 335153470480484634702632317052439643497669875866, 1149641683003293282818706915888024378617536908271, 3944135317413206096688826552058903952174456391494, 13533501959434215913619309932320102488558849813204, 46444709604471049971163662121618872580306645497669, 159414797827033471632169097078014363215217008372911] ------------------------------------------------------------ Theorem Number, 202 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 2 2 (x + x + 2 x) P(x) + (-x - 3 x - 1) P(x) + x + 1 = 0 The sequence a(n) satisfies the linear recurrence (n - 1) a(n - 1) (n - 2) a(n - 2) (7 n - 17) a(n - 3) a(n) = 3/2 ---------------- + 3/2 ---------------- + 1/2 ------------------- n + 1 n + 1 n + 1 (3 n - 13) a(n - 4) (5 n - 22) a(n - 5) (n - 5) a(n - 6) + 3/2 ------------------- + 1/2 ------------------- + 3/2 ---------------- n + 1 n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 2, a(6) = 5] Just for fun, a(1000), equals 3110044087209480005117630829262690785178962570065949042859259790317737040379\ 759701100163072347133061628900994344134390606794155733990395791805633590\ 537024931819516117069440621896922710132357543951542671730020154374140252\ 481152668325136984199423840335486951160089481484427605735584117701225360\ 185443273546843074814776347936817145348029484175062959758647141754841797\ 753245545859806563436028413641584301856998781416920512719059909064340976\ 0 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 2, 5, 11, 25, 59, 141, 341, 834, 2059, 5124, 12840, 32371, 82050, 208966, 534479, 1372334, 3535944, 9139675, 23692836, 61582898, 160460533, 419044485, 1096638614, 2875503146, 7553584753, 19876005353, 52383517030, 138263312497, 365449260129, 967207210893, 2563018745281, 6799767407380, 18060050175493, 48017685199887, 127796083238768, 340444518042078, 907752844356051, 2422494371813441, 6470152610896746, 17294449355027437, 46261996007121913, 123837829563019809, 331726717724960709, 889187531069931542, 2384956488565065577, 6400749401946081761, 17188349671696188130, 46182861841598432798, 124154272173533996975, 333939679211499088158, 898652419534228226604, 2419497096058108744166, 6517195853819678326088, 17562721011645909868766, 47349085933302136830686, 127706762787997722778519, 344582115297767116357518, 930127395634033290690560, 2511641225135101661616031, 6784737997508785947833423, 18334284135645956344807060, 49561625230123678806569948, 134021151295179820442198667, 362529278870271414806465126, 980958575089081220186575686, 2655168958539467728055683765, 7188922641889272855844422284, 19469823265240457710850266212, 52745243719287993339338899111, 142930323501759518804118082079, 387420007592312848246718803936, 1050396463254153780076312101790, 2848623504656927168323648787217, 7727244004300524919097211948918, 20966177067251074388309237930728, 56900529025354369049987241325696, 154459018041216523663673725397056, 419380000231229978551289480086232, 1138930702193997869259515535996900, 3093711169515805375308283798382397, 8405294433076864048756502355472832, 22840977954137951310899630769331300, 62081617647281364849542856240229601, 168770275549107517139128210128571621, 458893179504305904371260921975082624, 1247981426695721573015941182507884778, 3394561989992914077350719681100010261, 9234996898972555075219612487910040538, 25128437916389102357822582745659397424, 68386180935302267072806640040633043691, 186141742299768525598859015802344525816, 506745913283408122311836317282617805046, 1379768832938553665690210141896566743301, 3757427285946611381207676416183901480105, 10233911186736146182371403793350552546986, 27877779223509090151328083619029560684242] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 203 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ HH HH + HH HH + H H + HH HH 4+ HH HH H H + H HH H HH H H + HH HH HH HH HH HH 3+ H H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 2 6 3 5 4 (2/3 x - 2/3 x - 1/3 x + x - 7/3 x - 2 x ) P(x) 2 3 4 5 6 3 + (3 x + 5/3 x + 16/3 x - x + 8/3 x - 2 x + 1/3) P(x) 2 3 4 5 6 2 + (-13/3 x - 7/3 x - 13/3 x + 1/3 x - x + 4/3 x - 4/3) P(x) 2 3 2 3 6 x x + (7/3 x + 4/3 x + 5/3 x - 1/3 x + 5/3) P(x) - 2/3 - ---- - ---- - x/3 3 3 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 3, 7, 19, 48, 121, 317, 849, 2295, 6260, 17238, 47854, 133739, 375956, 1062401, 3016277, 8599474, 24609932, 70669849, 203568713, 588065209, 1703241119, 4945083315, 14389269638, 41956671794, 122573721675, 358732833944, 1051650021319, 3087816637484, 9079668947948, 26735518466880, 78826538835286, 232696402517695, 687718936535023, 2034742616034626, 6026433186023432, 17866547852261327, 53018709639292465, 157473025966248023, 468115910382008156, 1392687902962686815, 4146593736062635360, 12355252182420621497, 36840057739895400360, 109921981134962228626, 328195143196523201849, 980509151053480576710, 2931111056099916428518, 8767259623829233653529, 26238370715167170480195, 78567372729745697245616, 235381016350865721960474, 705531486493133865568695, 2115775025331736766683348, 6347803107567504498169503, 19053352882058645083609387, 57214597951772507551374941, 171879295088324112115065383, 516553908179985049784429602, 1553021606315984842645727000, 4670934225785637901889762088, 14053655774696028074514025940, 42298923075695460971060081332, 127355908859469296003652021430, 383578483407227242749937621890, 1155661056021261035032999901158, 3482922080329031429341930164353, 10500021486009174863793116503591, 31664019125455362828186476027855, 95514155518702200550450182653772, 288198598360766351188179770936806, 869831483270155216594836640508450, 2625997799047454637431985440636704, 7929878539369504357620908259520421, 23952383644286012279224306377795492, 72366599256033165279640212095424965, 218691604789707075464678503322225571, 661040369867576438102443513497241597, 1998587998897188701021777125058009669, 6043876443538976269861876583647698410, 18281110099369700523079519993375302915, 55307240904778956059405392476852229467, 167360013242688139883001717054017845828, 506535164142841409350513499921534631078, 1533393689247309355471033764839930986817, 4642821299523705002131206131030855365586, 14060235875545685061853701333191717834124, 42587658796005704113951209436349900988837, 129019004168836179306537256487662770413742, 390931495897254832321098655083383022057085, 1184739980168234080513806618328901897524787, 3591031691545390089794462930538742784318970, 10886485419793160134651388334281597323809962, 33008585800436114963525019054587044570171903, 100100317276562279735610578588213615758404467, 303607080044986725110911174494083798485502180, 920989916642033540827031744168974681449917909] ------------------------------------------------------------ Theorem Number, 204 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 7+ HH + H HH + HH HH 6+ H HH + H HH + HH HH 5+ HH HH + H HH + HH HH 4+ H H + HH HH 3+ HH HH H + HH HH H HH + HH HH HH HH 2+ HH HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 3 2 3 1 + x P(x) + (-x + x) P(x) + (x - x - 1) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 45, 90, 187, 401, 851, 1815, 3925, 8517, 18528, 40542, 89084, 196294, 433960, 962393, 2139725, 4768821, 10653090, 23847852, 53488512, 120189110, 270525714, 609872556, 1376944183, 3113165351, 7047902436, 15975594178, 36254691983, 82367121385, 187327128656, 426463015182, 971794420430, 2216459060360, 5059629805741, 11559371030889, 26429562274813, 60474319746292, 138472393707020, 317287830492994, 727493771194568, 1669087821473347, 3831706461817534, 8801529240735440, 20228664934673007, 46516855554565603, 107023436481333016, 246356874740823900, 567361687949009896, 1307246629320351536, 3013355124126816199, 6949151353923786831, 16032292599544117909, 37002915021440054034, 85437203538695002412, 197343801364604072373, 455995108480664878088, 1054028117510157798185, 2437220586009809344922, 5637462008726047404420, 13044104958896714457266, 30191362314092522717088, 69901256387316040143189, 161889000866928364911673, 375038792667449550811129, 869076638050116790409663, 2014464785302280486552292, 4670655362261462738855867, 10832019860018976585731355, 25127635081038417757157497, 58304423258510204427627370, 135318255823699614530347478, 314133072780421992235087867, 729408461607772465334189902, 1694046655330387389384834123, 3935274500031711929656356229, 9143606614506172522351127413, 21249596815155851886618287769, 49393797017310482802102109124, 114836688005465996914314854101, 267038248650994935249780214935, 621082017773923851102098581060, 1444791837130910209523805410400, 3361558483284669762056309772047, 7822641317082942993733966553444, 18207145325605714380902195077507, 42384238894815170678689367097517, 98682343013046975787558512672747, 229797628412303496855469928034966, 535206276701024419566301617271889, 1246708738543803536060775647603284, 2904528122457933390105864083112466, 6767863837948236105079296085122986] ------------------------------------------------------------ Theorem Number, 205 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 2 1 + P(x) x + P(x) x + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence 2 (518 n - 1031 n + 780) a(n - 1) a(n) = -1/195 -------------------------------- n + 2 3 2 (1036 n - 4335 n + 4367 n - 1170) a(n - 2) - 1/195 -------------------------------------------- (n + 2) (n + 1) 3 2 (2590 n - 11115 n + 11939 n - 1812) a(n - 3) + 1/195 ---------------------------------------------- (n + 2) (n + 1) 2 (n - 3) (1036 n - 1865 n + 1074) a(n - 4) + 1/65 ------------------------------------------ (n + 2) (n + 1) 161 (74 n - 55) (n - 3) (n - 4) a(n - 5) + --- ------------------------------------ 195 (n + 2) (n + 1) subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 1456286415952438964652364581682787386720721652723427223868652268704217615734\ 013661511014559584083968554107289545538396109682528052843351355027899510\ 336875839145124807012715538480932028100745003225030603205050905923754789\ 120913436417646081847745149534148691270052401108733311860953962982834165\ 142352139116820596104418788220221329115699403613078628786955744338071295\ 785210326056171164845618097 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 4, 8, 13, 31, 71, 144, 318, 729, 1611, 3604, 8249, 18803, 42907, 98858, 228474, 528735, 1228800, 2865180, 6693712, 15676941, 36807239, 86584783, 204060509, 481823778, 1139565120, 2699329341, 6403500057, 15211830451, 36183117255, 86171536894, 205459894230, 490417795075, 1171809537516, 2802705573178, 6709741472212, 16077633792535, 38557470978465, 92543928713413, 222292252187223, 534346494991678, 1285375448785399, 3094086155360823, 7452774657572624, 17962853486458385, 43320517724198861, 104535253974750259, 252390609561764423, 609699521485999914, 1473608895007574938, 3563397472304118410, 8620933920694737411, 20866285270840567272, 50527739882573522706, 122405798503806040965, 296657476936886314309, 719256672330233630085, 1744545018773625965860, 4232966162296687137816, 10274640808649742568633, 24948409910226791411907, 60599469176019760272535, 147244830551438483529790, 357892285312932720768838, 870164978202711215226483, 2116332778149031726542808, 5148677470688468794716894, 12529484860432553111063488, 30499528076281485551927755, 74262935424150986850220725, 180870207357836188170341069, 440630810522439231975963839, 1073724026979876073732588658, 2617084008072054452326466527, 6380388743696091681799439061, 15558886191823487315141504508, 37949779254079097680017924283, 92584221613943667202286169191, 225922433438019608173638548805, 551409374869368431174631180753, 1346105758407373798214652723522, 3286793196606678831626181940722, 8026971507652658919500628768874, 19607185640982275440523729181033, 47902823795394726062462438796736, 117054324948067108888357885549374, 286083328086281443692788683942327, 699317934466646636669713105276839, 1709748244857899463255482766619759, 4180839106559551650627420507840427, 10225086357877339275440446600065760, 25011587559313816213376886296261184, 61190615947546730766871192612064833, 149725673741988363612845353038008205, 366415846156516055604065564079552535, 896845046441850691487935128438196159] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 206 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ H HH HH + HH H HH HH + H HH H H + HH H HH HH 4+ H HH H H + H HH H H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 2 1 + (x + x) P(x) + (-x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 1) a(n - 1) (2 n - 1) a(n - 2) 3 (2 n - 5) a(n - 3) a(n) = ------------------ - ------------------ + -------------------- n + 1 n + 1 n + 1 (n - 2) a(n - 4) 2 (2 n - 7) a(n - 5) - ---------------- + -------------------- n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1] Just for fun, a(1000), equals 3134627918704846396589045582068293464668313386749891925759492579301366817419\ 266879466277019895299966556024437351904982229786354769359885950645415854\ 487762568754124509622513897418551096745213564746196714896494879980931898\ 708502918937008296833152741264657083164783324054454272242916859440693717\ 64365001116217342900388877265443604205502181219138765202835544779530 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 42, 84, 170, 354, 740, 1549, 3269, 6945, 14811, 31711, 68177, 147091, 318313, 690837, 1503351, 3279445, 7169907, 15708485, 34482475, 75830981, 167042763, 368548926, 814341362, 1801867812, 3992172298, 8855912464, 19668236110, 43729885814, 97330067640, 216844435346, 483571453768, 1079354856828, 2411240775366, 5391038526802, 12062679137620, 27010935099196, 60526554883902, 135721815227704, 304535849636366, 683755276072670, 1536121565708096, 3453050076522008, 7766456476976834, 17477390474177854, 39350957376284872, 88644320911742304, 199781711595784890, 450466438264400446, 1016164744759318568, 2293264805607373804, 5177575503207130558, 11694317639185843274, 26423715558659122260, 59728010995453379421, 135058602016778295285, 305507289981338348257, 691307640696528081979, 1564829833276593856647, 3543272754560789200681, 8025639916478510155339, 18183959683820504446497, 41212333247384488368873, 93431281455612601484243, 211875294974960472423745, 480604617462452614426815, 1090465639820281676643073, 2474853003558648387670983, 5618201029263006344110809, 12757123617698163352573711, 28974313940315478259062979, 65822724585319538752564981, 149567876677718197286698671, 339936780414026450535924453, 772774892529317395366017603, 1757116730487566683312107729, 3996122690354078859729209647, 9090033908542877936842438557, 20681339112467629598326301035, 47062642985297219558085765929, 107116551464339266855499450015, 243847122183568901454753070765, 555210537090801909689698475727, 1264372572489361483075032316117, 2879838126594606020044275873923, 6560472630207720552639437125041, 14947710059210582396858604974621, 34063179534729044501719950750975, 77636370512116064921945474111629, 176975591352231844422902101233579, 403485801251068480867972873270661, 920043891167408188870554973636539] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 207 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H H + HHH HH H + HH H H HH + H HH H H 3+ H H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 3 3 3 2 3 1 + P(x) x + (-x + x) P(x) + (x - x - 1) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 45, 90, 187, 401, 851, 1815, 3925, 8517, 18528, 40542, 89084, 196294, 433960, 962393, 2139725, 4768821, 10653090, 23847852, 53488512, 120189110, 270525714, 609872556, 1376944183, 3113165351, 7047902436, 15975594178, 36254691983, 82367121385, 187327128656, 426463015182, 971794420430, 2216459060360, 5059629805741, 11559371030889, 26429562274813, 60474319746292, 138472393707020, 317287830492994, 727493771194568, 1669087821473347, 3831706461817534, 8801529240735440, 20228664934673007, 46516855554565603, 107023436481333016, 246356874740823900, 567361687949009896, 1307246629320351536, 3013355124126816199, 6949151353923786831, 16032292599544117909, 37002915021440054034, 85437203538695002412, 197343801364604072373, 455995108480664878088, 1054028117510157798185, 2437220586009809344922, 5637462008726047404420, 13044104958896714457266, 30191362314092522717088, 69901256387316040143189, 161889000866928364911673, 375038792667449550811129, 869076638050116790409663, 2014464785302280486552292, 4670655362261462738855867, 10832019860018976585731355, 25127635081038417757157497, 58304423258510204427627370, 135318255823699614530347478, 314133072780421992235087867, 729408461607772465334189902, 1694046655330387389384834123, 3935274500031711929656356229, 9143606614506172522351127413, 21249596815155851886618287769, 49393797017310482802102109124, 114836688005465996914314854101, 267038248650994935249780214935, 621082017773923851102098581060, 1444791837130910209523805410400, 3361558483284669762056309772047, 7822641317082942993733966553444, 18207145325605714380902195077507, 42384238894815170678689367097517, 98682343013046975787558512672747, 229797628412303496855469928034966, 535206276701024419566301617271889, 1246708738543803536060775647603284, 2904528122457933390105864083112466, 6767863837948236105079296085122986] ------------------------------------------------------------ Theorem Number, 208 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 2 3 1 + P(x) x + (x - x - 1) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 1) a(n - 1) (n - 2) a(n - 2) (2 n - 7) a(n - 3) a(n) = ------------------ - ---------------- + ------------------ n + 1 n + 1 n + 1 2 (n - 5) a(n - 4) (n - 8) a(n - 6) + ------------------ - ---------------- n + 1 n + 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 2] Just for fun, a(1000), equals 1176950714989472039879867097164787467731168548390670636801992136416332972086\ 898277348698087993919571204091110820742448542481636855988358314899690938\ 672843649138648682133884841687298042373689270639308856986357452480963423\ 383401026848945091071625435678738959251923453512607097256262920947479103\ 356615910327226686348752585420237437 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 22, 42, 80, 152, 292, 568, 1112, 2185, 4313, 8557, 17050, 34089, 68370, 137542, 277475, 561185, 1137595, 2311014, 4704235, 9593662, 19598920, 40103635, 82185653, 168666493, 346613232, 713200114, 1469254621, 3030218948, 6256281188, 12930039374, 26748697772, 55386529370, 114785051382, 238083048103, 494216315763, 1026681547651, 2134372036796, 4440242721757, 9243424565624, 19254704030249, 40133535117994, 83701671288887, 174665494666782, 364684302692317, 761824952311410, 1592257031239222, 3329531677118927, 6965586177249102, 14579064797995464, 30527584089316653, 63949861857983311, 134018617814709631, 280972131660117384, 589289169477022354, 1236390172104441711, 2595012857019532078, 5448483097227962922, 11443510685976418890, 24042863051171641274, 50530333059247995344, 106231480858589059892, 223401249061635751536, 469943442677589917028, 988848424941723500999, 2081299761445939780379, 4381845845937552096915, 9227711560622784969822, 19437589516267668693817, 40954291792943394506568, 86310235142767979004036, 181940086112454163969993, 383614626529901470057969, 809021619679751516526841, 1706557543555582414238756, 3600603520716009828550165, 7598380174796313007709308, 16038212465609702505254728, 33859308082950214419570804, 71496599403133792320146486, 150999659234389445825596123, 318968628398739129595019281, 673906439126292155812743214, 1424062795029526232414783635, 3009781004697296226116225432, 6362316258145714947705909656, 13451435620129537321656044592, 28444187117982772743078140610, 60157315325649798906283809598, 127248293230730230066048509796, 269204737445574334829247116243, 569612221084905567519367331211] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 209 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} To make it crystal clear here is such a path of semi-length 8 5+ H H H + HH H HH H HH H + H HH H HH H H + HH HHH HHH HH 4+ H H H H + H H + HH HH + H H 3+ H H H + HH HH H HH + H H HH H + HH HHH HH 2+ H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 2 2 2 1 + (x + 2 x - x + 1) P(x) + (-2 x + x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence (5 n - 11) a(n - 1) 6 (n - 2) a(n - 2) 4 (2 n - 5) a(n - 3) a(n) = ------------------- - ------------------ + -------------------- n - 1 n - 1 n - 1 2 (2 n - 5) a(n - 5) - a(n - 4) + -------------------- n - 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 2, a(5) = 5] Just for fun, a(1000), equals 1459341104055037878590661433818194709926483197855691409550477860693417350767\ 855210086176057831769184334103823262892996511760055728667691876054701698\ 415640252220316692992809510161002761290881497449019645307421748762437970\ 405369359132921846771387275336684598136656312309520095533790705704605794\ 139507926994477686139780531759766001128230358820788339963914501449862328\ 049824353748023042827286105953806315043814621933170652321093298810459088\ 225975148962886179869207476753551941234958257299974272725863670553837016\ 313871566103225359234089001106220929249590551580493606221963063175419332\ 89894907704741021 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 2, 5, 15, 46, 146, 478, 1601, 5461, 18909, 66295, 234888, 839737, 3025492, 10974568, 40046366, 146902283, 541422759, 2003906815, 7445135351, 27756686458, 103807556637, 389349890556, 1464193789255, 5519683723137, 20854841660142, 78959675239316, 299533749880137, 1138336932329736, 4333401093001624, 16522403592329745, 63090132533494558, 241243118828748442, 923673494818488466, 3540950348475963099, 13590311612165945101, 52217972121007012825, 200847538188863335557, 773296015328349147198, 2980137669614948639803, 11495246258475141466602, 44378553292960824810679, 171468243965767544390977, 663031119303227639217856, 2565717239190465482223020, 9935605158824222769022949, 38501476193668571146615422, 149295132998886263447343568, 579278708551975793726498276, 2249014495151539697686562647, 8736741272368235924317766829, 33958584094838896215639715479, 132063694297332882823831529425, 513857229640457022142475045244, 2000407716382241891562974095537, 7791191600851362448612042753882, 30359265330564397686907483946047, 118351485188343519801039242205119, 461577541938409468266304743584600, 1800934344556991108902386995459376, 7029547953934463425169950871560465, 27449065522677764513275596184709906, 107224242631642407982604774644384501, 419004347117546547326160237993052263, 1637944887051246936808270632076588146, 6405168711853788812636590265754918814, 25055779717076169994395996841014307971, 98045361945211331547982296735212526580, 383781360134223720179165825876165947192, 1502707575952229538760506819330215519818, 5885659000612811487826719364731935033813, 23059088876472540941686721997590932493297, 90367478880190072087062355074600165362979, 354243471862396081802464695778897218883807, 1389018509531781937969273988272927083915056, 5447881407293821144751044083112406174678665, 21372619636159951771936795030579497034735614, 83867856961242371860322394023223116705830077, 329183680354655585832763376003281962816836023, 1292359662972213292899000420878390779898111144, 5074909732615062651924778171347380540729358224, 19932907565326170529001595453483031500623540139, 78308345552521584347207327769453635599333193768, 307707621798292148024947599730777748238859282786, 1209369723882907752651644824063999555222566941546, 4754101646277522020587443857319868485672976150155, 18692368687624611728006091842185301882996099468235, 73509718359034189909302894013960968626092823146139, 289139769563902855224592506284368012391425429294723, 1137500831999911735314798524935668071016027896803090, 4475841157768486098555460993288096254124212618769209, 17614687597901517845144888013181142375479995129852490, 69334742048141058544271937101150230746809873370457349, 272961211538415116287583139173329000931114682702178879, 1074789684261792720953089674432680040020601481577399022, 4232695777993015356096649049452087555492398818421883074, 16671711023550864292663075610142610006417340932796382781, 65676716325627776246369894980107962228522160238007048754] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 210 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H H + HH HH 4+ H H H + HHH H H + HH H HH HH + H HH H H 3+ H H H H H + HH H HH HH H HH + H HH H H HH H + HH HHH HHH HH 2+ H H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 2 2 2 1 + (x + x - x + 1) P(x) + (-x + x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence 3 (n - 2) a(n - 1) 6 a(n - 2) (n - 7) a(n - 3) (2 n - 11) a(n - 4) a(n) = ------------------ - ---------- - ---------------- + ------------------- n - 1 n - 1 n - 1 n - 1 (2 n - 5) a(n - 5) 3 (n - 4) a(n - 6) + ------------------ + ------------------ n - 1 n - 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 2, a(6) = 5] Just for fun, a(1000), equals 2878856782144216993304909072623096791238839518518732785727996575633444455066\ 956567407176766906801478386293066728857038629471924993382928181488681958\ 710041927681409642009723491606649261960519836795925263537967363891891877\ 002397732039295842231821680571756238560688066459575319018806752729651146\ 307687782271404813967049306258778771027776415891416409801808828208302403\ 754056299604571215754189083212255788144789217206287683307324816030984715\ 912400078786930743617163678096273514 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 2, 5, 11, 26, 64, 160, 408, 1057, 2773, 7354, 19684, 53108, 144285, 394395, 1083893, 2993169, 8301373, 23113093, 64579846, 181021143, 508903147, 1434529326, 4053774130, 11481626162, 32588767361, 92680384675, 264061121393, 753641895385, 2154377477842, 6167816874173, 17682947621645, 50763953628170, 145915335052953, 419914634020438, 1209784305460203, 3489122437332586, 10073078554564695, 29108790529242620, 84194127583139219, 243734132718405830, 706171736309540520, 2047610949222803792, 5941724010837544122, 17254056039415814789, 50138381671889842151, 145793214118265385208, 424209581629997008364, 1235062051960423283060, 3597922845367850458123, 10487208390879264918883, 30584663463760785497977, 89243080822651069459771, 260533876350475763594648, 760965285519087098174307, 2223663640882079998345485, 6500846940194260255293338, 19013441322241949351689809, 55633340339776650260833834, 162849771289646446389856459, 476882138630038856845066186, 1397016057262264292104843536, 4094048498713833927119509096, 12002199935911865573364250368, 35198186649823939053399904897, 103258726164565685925228187639, 303023253756858648014378213498, 889536432693308266926378432142, 2612078093684686900158388271554, 7672544560379181785324320290350, 22543424908109986923445574304695, 66255831375073245682145803993412, 194781951077045009504532440141396, 572783503254869268864047217594022, 1684792405001414647348856645732013, 4956937407787336387901741593452936, 14587765426674530465484714813300616, 42940753950790728299148874663609176, 126430973352845019344790249089690895, 372338273101134145496109926981954000, 1096780740183236388272441528847245066, 3231450663204362980554731118035642044, 9522885200305441053069920474784222079, 28069241670114708661852669209760908380, 82752629977973030745221962129345214936, 244016886093782321888697308342626742333, 719685797597200359808440118040501049515, 2122995392809778717115751666984513854737, 6263778537286879755046906861656257371389, 18484307155950499152062033476773319732083, 54556647215785861139168869263297189076433, 161052775284673849736973509097646627234736, 475513860462577865152939584703040055308473, 1404206597899854279669242235315364776065237, 4147345179697928605850235606984347636322434, 12251216958902729677442285550386020013605594, 36195679054955691421660475357757577042688188] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 211 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H H + H HH HHH + HH H H HH + H HH HH H 3+ H H H H H + HH H HH H H HH + H HH H H HH H + HH HHH HHH HH 2+ H H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 3 -1 + (x + 3 x - x + 2 x - x + 1) P(x) 4 3 2 2 2 + (-3 x + x - 4 x + 2 x - 3) P(x) + (2 x - x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 4, 10, 27, 75, 213, 618, 1829, 5502, 16766, 51638, 160510, 502936, 1586915, 5037898, 16080235, 51573656, 166126983, 537210234, 1743335620, 5675610217, 18531855864, 60672908113, 199134937139, 655080466555, 2159549717786, 7133254708699, 23605300357009, 78248448348415, 259799644349161, 863882438059948, 2876633874448522, 9591630118557203, 32021763420976533, 107031813937668680, 358152189955471182, 1199730957398468158, 4022889016490524078, 13502318085820052452, 45360149060387967073, 152516792901902446460, 513239903623380228634, 1728488199311870556341, 5825597234987627520020, 19648449862710960974468, 66315737830868245888713, 223971572659045980013396, 756912089375355151840065, 2559547558143509099472648, 8660357142068863862107857, 29319290281835060734836478, 99313188455823088233137906, 336579373400955588030931881, 1141266121031792548722158167, 3871661712452036848055847470, 13140489346376262760361654857, 44619255232323372515516410045, 151573431350978167631715794500, 515118930493289031968941039048, 1751336452045707226974825075965, 5956670663491505612092714479448, 20267681246079204003255012865920, 68986771314773128486278704491480, 234900475364654332600465402316920, 800117117051503591337045132089775, 2726278992841291666370074042805552, 9292440348616198047813541633462860, 31683115635946347629623785374232487, 108058924915649277676057259022053450, 368658517648600298195076388866196657, 1258099938148112126731753336113447860, 4294669672171115651292913118800043505, 14664417073874898414157875861063587347, 50086080693227785224603418797689014647, 171113162673527886251575588257860242944, 584737505909401426725277499566850600057, 1998696255538134167700268645424394567403, 6833421694220982386351699617760291091198, 23368594017576346412900144749396392537494, 79933226555544808570154485203703170303082, 273476527083818725459119768102976825183174, 935854562232197114417776416047740913475791, 3203243746003208985602790946785892576204084, 10966366023767169445457040146336010139424470, 37551258376630702470631501349315514985767247, 128609549273471674748920099576880547705705392, 440561888663121764943313503687497097424833621, 1509467316392012568455815312423254958860615388, 5172752952822098618520373830943462076045468471, 17729611172602185872624136898905502208011067108, 60779120265889375477319148720241010268357965932, 208394216696886394027227843877823994272813709965, 714646633388317059244644903816395674073736347859, 2451150134945316687182000995138448683956077170923, 8408525152767898640307450568476492573560279508421, 28849588699589082878771616007427495302335252329135, 98998333177530072769467116809527730538555485758775] ------------------------------------------------------------ Theorem Number, 212 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 3 4 3 2 2 -1 + (x + 2 x - x + x - x + 1) P(x) + (-2 x + x - 2 x + 2 x - 3) P(x) 2 + (x - x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 46, 94, 199, 439, 961, 2113, 4728, 10638, 24004, 54567, 124738, 286100, 658902, 1523561, 3533694, 8219807, 19175241, 44844787, 105117745, 246938893, 581271855, 1370801044, 3238379640, 7662884761, 18160172211, 43099390904, 102426035307, 243726755578, 580655580178, 1384932709103, 3306794777897, 7903693118041, 18909278995761, 45281554377074, 108530027712894, 260341203346223, 625005317744160, 1501608816996828, 3610336668014877, 8686460115399371, 20913632597618146, 50384354111881549, 121459160713524172, 292969666722436726, 707070780696631056, 1707424831884850449, 4125246497394886642, 9971935177734506502, 24116945055530422878, 58354041165562962555, 141259692927959905224, 342103485342872695337, 828862025018976189145, 2009029841925232601585, 4871515952686337473281, 11817073539321134031521, 28675998351756531656835, 69612131191902560435320, 169045763869085582882160, 410650213529812223368770, 997892674377742909288876, 2425690847473912236264926, 5898246126299524653958956, 14346379729918995608354591, 34905194678328946279956726, 84949845555910352980918162, 206802853409472003358290619, 503580166605875837883509918, 1226579626343965688411938881, 2988373791468938272294751313, 7282546274134082391128998885, 17751619375203540121720680747, 43280914162805330098090017988, 105549463086185917149368841094, 257462696364279602325677747884, 628158000160191822477935767084, 1532912888447769419915804392627, 3741604135787909331331389618256, 9134564337311934498324607449420, 22305163745068650394954160327031, 54476421551688686284866768832516, 133074664291962641673573493873036, 325135188262952797925515855558143, 794534125103762472134490936108882, 1941955965184663382273487480183164, 4747257141764020796262302049059161, 11607028692460250300264994515896655, 28383941376456748393597046710009226, 69421845442964953819449619451462120, 169820428896576356561050206615538112, 415482369434004891765163304919548259] ------------------------------------------------------------ Theorem Number, 213 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 7+ HHH + H HH + HH HH 6+ HH H H + H HH H H + HH HH HH HH 5+ HH H HH + H H + HH HH 4+ H H + HH HH 3+ HH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 2 2 2 1 + (x + x - x + 1) P(x) + (-x + x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence 3 (n - 2) a(n - 1) 6 a(n - 2) (n - 7) a(n - 3) (2 n - 11) a(n - 4) a(n) = ------------------ - ---------- - ---------------- + ------------------- n - 1 n - 1 n - 1 n - 1 (2 n - 5) a(n - 5) 3 (n - 4) a(n - 6) + ------------------ + ------------------ n - 1 n - 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 2, a(6) = 5] Just for fun, a(1000), equals 2878856782144216993304909072623096791238839518518732785727996575633444455066\ 956567407176766906801478386293066728857038629471924993382928181488681958\ 710041927681409642009723491606649261960519836795925263537967363891891877\ 002397732039295842231821680571756238560688066459575319018806752729651146\ 307687782271404813967049306258778771027776415891416409801808828208302403\ 754056299604571215754189083212255788144789217206287683307324816030984715\ 912400078786930743617163678096273514 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 2, 5, 11, 26, 64, 160, 408, 1057, 2773, 7354, 19684, 53108, 144285, 394395, 1083893, 2993169, 8301373, 23113093, 64579846, 181021143, 508903147, 1434529326, 4053774130, 11481626162, 32588767361, 92680384675, 264061121393, 753641895385, 2154377477842, 6167816874173, 17682947621645, 50763953628170, 145915335052953, 419914634020438, 1209784305460203, 3489122437332586, 10073078554564695, 29108790529242620, 84194127583139219, 243734132718405830, 706171736309540520, 2047610949222803792, 5941724010837544122, 17254056039415814789, 50138381671889842151, 145793214118265385208, 424209581629997008364, 1235062051960423283060, 3597922845367850458123, 10487208390879264918883, 30584663463760785497977, 89243080822651069459771, 260533876350475763594648, 760965285519087098174307, 2223663640882079998345485, 6500846940194260255293338, 19013441322241949351689809, 55633340339776650260833834, 162849771289646446389856459, 476882138630038856845066186, 1397016057262264292104843536, 4094048498713833927119509096, 12002199935911865573364250368, 35198186649823939053399904897, 103258726164565685925228187639, 303023253756858648014378213498, 889536432693308266926378432142, 2612078093684686900158388271554, 7672544560379181785324320290350, 22543424908109986923445574304695, 66255831375073245682145803993412, 194781951077045009504532440141396, 572783503254869268864047217594022, 1684792405001414647348856645732013, 4956937407787336387901741593452936, 14587765426674530465484714813300616, 42940753950790728299148874663609176, 126430973352845019344790249089690895, 372338273101134145496109926981954000, 1096780740183236388272441528847245066, 3231450663204362980554731118035642044, 9522885200305441053069920474784222079, 28069241670114708661852669209760908380, 82752629977973030745221962129345214936, 244016886093782321888697308342626742333, 719685797597200359808440118040501049515, 2122995392809778717115751666984513854737, 6263778537286879755046906861656257371389, 18484307155950499152062033476773319732083, 54556647215785861139168869263297189076433, 161052775284673849736973509097646627234736, 475513860462577865152939584703040055308473, 1404206597899854279669242235315364776065237, 4147345179697928605850235606984347636322434, 12251216958902729677442285550386020013605594, 36195679054955691421660475357757577042688188] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 214 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 4 3 2 2 3 2 1 + (x - x + x - x + 1) P(x) + (x - x + x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence 3 (n - 2) a(n - 1) (4 n - 13) a(n - 3) a(n) = ------------------ - 2 a(n - 2) + ------------------- n - 1 n - 1 5 (n - 4) a(n - 4) (4 n - 19) a(n - 5) 2 (n - 7) a(n - 6) - ------------------ + ------------------- - ------------------ n - 1 n - 1 n - 1 3 (n - 6) a(n - 7) (n - 7) a(n - 8) + ------------------ - ---------------- n - 1 n - 1 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 3, a(7) = 6, a(8) = 11] Just for fun, a(1000), equals 2526417588552551239087622713450782674607200349495548920098374171706326696905\ 987358984632091573940154973448477556026245607852146796696175220174654987\ 855874461174948050222440872108946014788406313744511562534901265968509284\ 696954151804670470433712742886374769364628648205940828707849254227729188\ 160424264587073078347824319516574225054899783553944301246415126657279029\ 7664919984405001150954576413689084208232811909097 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 3, 6, 11, 24, 53, 116, 261, 597, 1377, 3208, 7540, 17847, 42510, 101827, 245126, 592710, 1438905, 3505841, 8569961, 21012016, 51659463, 127329616, 314573101, 778844979, 1932193043, 4802416193, 11957066786, 29819210918, 74478329481, 186288350530, 466580654020, 1170090482218, 2937873493933, 7384811244576, 18582851192462, 46808845168667, 118022259461333, 297851482578911, 752343804020697, 1901936033410366, 4811958676118293, 12183690339863776, 30871118366907811, 78276051584747019, 198607586117122433, 504244586228095886, 1281014288157823436, 3256293260938671167, 8282090094784809942, 21076279669832244023, 53663215979973367078, 136703565278629257257, 348413920051606365109, 888415631303410400693, 2266390985297382939244, 5784218641490669913674, 14768602533175461412084, 37723576913059959189570, 96396065963852544869335, 246418456775384193136474, 630157907211050507794279, 1612065850169479898161270, 4125424301952818374373046, 10560934852242089082996167, 27044542703161579918191368, 69278151330674102881751756, 177520420186017784348428806, 455021484188559321940986022, 1166657521497332187638051722, 2992121704478148049224177563, 7676020787730604195115180260, 19697486679936546300310811022, 50559197165666329320968518355, 129807928435748370252434530892, 333358164897700274190418864591, 856302164478548731957014666276, 2200120148450921474862393099840, 5654140661698370047863287417184, 14534001765488032670590374660118, 37367997383741023386509197290397, 96096631510320455105379909994257, 247176980049779000805831259912359, 635912356770482453319652758048312, 1636341079706963296702745524874009, 4211489689376844826337823791129060, 10841292672959538903552987010730415, 27913092707747324523128458535889462, 71881083367693079248232340926059572, 185139591151355760086148154405710139, 476936254132963720949924420102440423, 1228842138771347352467775205360079295, 3166686001959774863542602151823218212, 8161792150076008893563071326754946445, 21039538144777814329082414357547689162, 54244484058143330790953988418597650491] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 215 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - x + x - x + 1) P(x) + (x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 4, 8, 18, 40, 91, 213, 506, 1218, 2966, 7294, 18090, 45194, 113632, 287327, 730186, 1863977, 4777509, 12289898, 31720176, 82117774, 213178108, 554823435, 1447397817, 3784127064, 9913343789, 26018984599, 68410290554, 180162323411, 475196423227, 1255186628492, 3319955888745, 8792478337054, 23313863133954, 61888878237389, 164467529111164, 437514750895099, 1165005085521436, 3105022797156006, 8282937176309562, 22114068273505488, 59088226972484393, 158003152307405371, 422813155834784436, 1132232400449265104, 3033990842200935686, 8135281602930567564, 21827264601898788794, 58598143522316010039, 157404179282151204174, 423045542218716765139, 1137594743452912452613, 3060618745844589852386, 8238424391588288975038, 22186297926687272423033, 59775628331378116451380, 161122179240285688242086, 434482162221497135042578, 1172108773956891793300694, 3163278407606545361590941, 8540335279906181472947846, 23066152471490359501137078, 62320812139929346207300967, 168439506971919225588258595, 455410615205746441651620508, 1231703823760505986269633728, 3332339148450456918147838167, 9018366051490575879544837616, 24413972116862043819738815721, 66111540013999155652084562422, 179077417530677201597463612360, 485205493746336496557728723796, 1315008887370019158511357609032, 3564894162691250059222924187153, 9666662859580349116822421353996, 26218968527010913350495426276500, 71131335168680718796826444485856, 193023408967072905414753719183441, 523914112753491222458732852061593, 1422357723794427722247327153795571, 3862369444640190578023693432554655, 10490416497562008857897222598208662, 28498590066657605402946101068227090, 77436123391842858541274538722111900, 210451173947131125254485079572740558, 572064013100223785974710999433555658, 1555326255238638414011723574401111725, 4229413184640997453434597390339399864, 11503199764674649726828795101413864073, 31292149913413969843695821586747884491, 85139020625938221899250912882028413646, 231684389965423872121630053321756963050, 630577096684214752553581697741642950844, 1716529914048561172329410428733040296684, 4673420308399441399562494693207807848635, 12725860269271441688238739378854893163811] ------------------------------------------------------------ Theorem Number, 216 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - 2 x + x - x + 1) P(x) + (2 x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 23, 46, 90, 178, 362, 744, 1535, 3194, 6704, 14152, 30019, 63993, 137027, 294528, 635254, 1374563, 2982983, 6490689, 14157809, 30951925, 67809639, 148847546, 327326203, 721035228, 1590820794, 3515055239, 7777682719, 17232166570, 38226855740, 84899762419, 188767040056, 420148088468, 936077556199, 2087530911028, 4659567865103, 10409537599216, 23274106872900, 52077999630392, 116616556556144, 261322452488208, 585990624975113, 1314890839669959, 2952311871586680, 6632807182343025, 14910247851249936, 33536291670621966, 75470769540757993, 169928986936543073, 382800909986262158, 862755122454412830, 1945377625304332347, 4388490288087053044, 9904091476130975332, 22361254774054848157, 50507256400673637226, 114125267802978496554, 257973369424416665508, 583348170337435269883, 1319580231355477362189, 2986029400547774283515, 6759245279299394441018, 15305372653077265127430, 34667865012474818872923, 78549578510974131598343, 178028874433432077107813, 403611252116127504636989, 915290323330934693322979, 2076222803712862095242661, 4710915866189376517310722, 10691779547594570193232569, 24271966712323362836221071, 55114699810857040680904084, 125179934207220209125653142, 284383356087736137374477620, 646209443923063259545590287, 1468722246098351522200998572, 3338881124332810099290521494, 7591977293065388173220549042, 17266301921829442269815327549, 39276443078065295093655855985, 89361682067950853306373719225, 203355033871187599510752721774, 462850820082071205035423702170, 1053677818530795805415739939561, 2399128776874249888355418021047, 5463569785243751528140091257717, 12444429345978153227988825135555, 28349632447886106488535958107141, 64594015240375900277795861639819, 147200074451998200230485801692760, 335500536871502998738039665095175] ------------------------------------------------------------ Theorem Number, 217 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH + HH H + H H + HH HH 4+ H H + H H + HH HH 3+ H H H H + HH HH HH HH HH HH + HH H HH HH H HH 2+ HH HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 3 -1 + (x + 3 x - x + 2 x - x + 1) P(x) 4 3 2 2 2 + (-3 x + x - 4 x + 2 x - 3) P(x) + (2 x - x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 4, 10, 27, 75, 213, 618, 1829, 5502, 16766, 51638, 160510, 502936, 1586915, 5037898, 16080235, 51573656, 166126983, 537210234, 1743335620, 5675610217, 18531855864, 60672908113, 199134937139, 655080466555, 2159549717786, 7133254708699, 23605300357009, 78248448348415, 259799644349161, 863882438059948, 2876633874448522, 9591630118557203, 32021763420976533, 107031813937668680, 358152189955471182, 1199730957398468158, 4022889016490524078, 13502318085820052452, 45360149060387967073, 152516792901902446460, 513239903623380228634, 1728488199311870556341, 5825597234987627520020, 19648449862710960974468, 66315737830868245888713, 223971572659045980013396, 756912089375355151840065, 2559547558143509099472648, 8660357142068863862107857, 29319290281835060734836478, 99313188455823088233137906, 336579373400955588030931881, 1141266121031792548722158167, 3871661712452036848055847470, 13140489346376262760361654857, 44619255232323372515516410045, 151573431350978167631715794500, 515118930493289031968941039048, 1751336452045707226974825075965, 5956670663491505612092714479448, 20267681246079204003255012865920, 68986771314773128486278704491480, 234900475364654332600465402316920, 800117117051503591337045132089775, 2726278992841291666370074042805552, 9292440348616198047813541633462860, 31683115635946347629623785374232487, 108058924915649277676057259022053450, 368658517648600298195076388866196657, 1258099938148112126731753336113447860, 4294669672171115651292913118800043505, 14664417073874898414157875861063587347, 50086080693227785224603418797689014647, 171113162673527886251575588257860242944, 584737505909401426725277499566850600057, 1998696255538134167700268645424394567403, 6833421694220982386351699617760291091198, 23368594017576346412900144749396392537494, 79933226555544808570154485203703170303082, 273476527083818725459119768102976825183174, 935854562232197114417776416047740913475791, 3203243746003208985602790946785892576204084, 10966366023767169445457040146336010139424470, 37551258376630702470631501349315514985767247, 128609549273471674748920099576880547705705392, 440561888663121764943313503687497097424833621, 1509467316392012568455815312423254958860615388, 5172752952822098618520373830943462076045468471, 17729611172602185872624136898905502208011067108, 60779120265889375477319148720241010268357965932, 208394216696886394027227843877823994272813709965, 714646633388317059244644903816395674073736347859, 2451150134945316687182000995138448683956077170923, 8408525152767898640307450568476492573560279508421, 28849588699589082878771616007427495302335252329135, 98998333177530072769467116809527730538555485758775] ------------------------------------------------------------ Theorem Number, 218 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH H + HH H HH H + H HH H H + HH HHH HH 4+ H H H + H H + HH HH 3+ H H H + HH HH HH HH + HH HH H HH 2+ HH HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - x + x - x + 1) P(x) + (x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 4, 8, 18, 40, 91, 213, 506, 1218, 2966, 7294, 18090, 45194, 113632, 287327, 730186, 1863977, 4777509, 12289898, 31720176, 82117774, 213178108, 554823435, 1447397817, 3784127064, 9913343789, 26018984599, 68410290554, 180162323411, 475196423227, 1255186628492, 3319955888745, 8792478337054, 23313863133954, 61888878237389, 164467529111164, 437514750895099, 1165005085521436, 3105022797156006, 8282937176309562, 22114068273505488, 59088226972484393, 158003152307405371, 422813155834784436, 1132232400449265104, 3033990842200935686, 8135281602930567564, 21827264601898788794, 58598143522316010039, 157404179282151204174, 423045542218716765139, 1137594743452912452613, 3060618745844589852386, 8238424391588288975038, 22186297926687272423033, 59775628331378116451380, 161122179240285688242086, 434482162221497135042578, 1172108773956891793300694, 3163278407606545361590941, 8540335279906181472947846, 23066152471490359501137078, 62320812139929346207300967, 168439506971919225588258595, 455410615205746441651620508, 1231703823760505986269633728, 3332339148450456918147838167, 9018366051490575879544837616, 24413972116862043819738815721, 66111540013999155652084562422, 179077417530677201597463612360, 485205493746336496557728723796, 1315008887370019158511357609032, 3564894162691250059222924187153, 9666662859580349116822421353996, 26218968527010913350495426276500, 71131335168680718796826444485856, 193023408967072905414753719183441, 523914112753491222458732852061593, 1422357723794427722247327153795571, 3862369444640190578023693432554655, 10490416497562008857897222598208662, 28498590066657605402946101068227090, 77436123391842858541274538722111900, 210451173947131125254485079572740558, 572064013100223785974710999433555658, 1555326255238638414011723574401111725, 4229413184640997453434597390339399864, 11503199764674649726828795101413864073, 31292149913413969843695821586747884491, 85139020625938221899250912882028413646, 231684389965423872121630053321756963050, 630577096684214752553581697741642950844, 1716529914048561172329410428733040296684, 4673420308399441399562494693207807848635, 12725860269271441688238739378854893163811] ------------------------------------------------------------ Theorem Number, 219 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 5 3 2 4 1 + (x + x + 2 x + 3 x + 2 x + 3 x - 3 x + 2 x) P(x) 10 7 6 5 4 3 2 3 + (x + x - 6 x - x - 2 x - 8 x + 7 x - 5 x - 1) P(x) 7 6 5 4 3 2 2 + (-x + 3 x - x + x + 7 x - 5 x + 4 x + 3) P(x) 4 3 2 + (x - 2 x + x - x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 2, 3, 7, 18, 44, 109, 280, 735, 1954, 5255, 14288, 39217, 108505, 302302, 847429, 2388592, 6765497, 19246740, 54970312, 157562416, 453097016, 1306839319, 3779538739, 10958392378, 31846583680, 92749692974, 270663243009, 791322671236, 2317564004952, 6798531375987, 19973753306397, 58765902899481, 173131371863628, 510713506115097, 1508340867919083, 4459786618285185, 13200644997642960, 39112787528848850, 116001279632237698, 344355206748910149, 1023131191333303333, 3042419221532643555, 9054267496265684000, 26966134344132354244, 80371256036254553851, 239709920294859563755, 715421183374306007562, 2136565638960719056486, 6384665872837409917126, 19090490663261938862990, 57114067020897547781467, 170964846855150763189694, 512034794172729323663479, 1534308541126455320397434, 4599796479480732910229762, 13796525827540580085304946, 41399870101856911693167035, 124285255191656772092061739, 373271847488557793060517405, 1121527133521793199398529651, 3371067683449113643208929271, 10136611244295805178335506076, 30491619569690255288608488922, 91754119429669930918439578524, 276199730126408645572551619700, 831704332336391375603521245725, 2505291996504647791163520786378, 7548962005207718512325143407531, 22753679484774428262141551847782, 68603728073198831689741859679122, 206905395935030304046916480848346, 624195196971857633889350855456976, 1883606423186986155277619004361039, 5685619623395209730334405327840435, 17166439619614140305204033759142919, 51843517807962177918812419431253021, 156609328191448317164163442286830882, 473202422733253589702545651921133223, 1430144112363376670826848484744426602, 4323282306043807828111513035709202120, 13072115360399069790669916059912438838, 39534318291653934703315545185353834326, 119590436552338302559220773081934909782, 361834731326808756208297833835888676769, 1094998489140762391359812725618698478528, 3314394486204804522148317021872597821536, 10034143201570208161313294880666515660127, 30383646042639021286863897593477035894106, 92019764881724708060825041166437152084740, 278741858403344620559751350060783939727561, 844503328063769443224654633785899215174158, 2559039323627586379562839652619725161977135, 7755813282661770390894741126403860743539833, 23509908834096397085071851401290457807032194, 71276477994521894144434592480478231691440190, 216128343340502595627075199672200914399781898, 655459690737824635537416520631391620380551798] ------------------------------------------------------------ Theorem Number, 220 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H H + HH H H HH + H HH HH H + HH H H HH 4+ H H H H + H HH HH H + HH H H HH + H HH H H 3+ H H H H + HH H HH HH + H HH H H + HH HHH HH 2+ H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 10 6 5 4 3 2 3 1 + (x - 2 x + x - 2 x + 2 x - x + x - 1) P(x) 7 6 5 4 3 2 2 + (x + x - x + x - 3 x + 2 x - 2 x + 3) P(x) 4 3 2 + (x + x - x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 17, 34, 65, 133, 277, 572, 1199, 2548, 5435, 11661, 25202, 54731, 119348, 261396, 574697, 1267637, 2804766, 6223660, 13845748, 30876213, 69007994, 154549822, 346790404, 779545026, 1755254743, 3958377577, 8939866583, 20218263933, 45784748248, 103807846272, 235636648909, 535466068728, 1218071478804, 2773591163064, 6321494978078, 14420661400511, 32924586831606, 75232979162492, 172041452385538, 393713260872074, 901644278692778, 2066263403285064, 4738263632158243, 10872374445702404, 24962637101521442, 57346546860289679, 131815132773209257, 303148512760931479, 697540229550486854, 1605826655979897539, 3698588507211274869, 8522638804153427168, 19647441284817832815, 45313253580871653227, 104550383069514369365, 241324409472237417738, 557245419733284748247, 1287229693480919378873, 2974574026651753256696, 6876189561850918905880, 15900858902075293511482, 36782273871125357477166, 85113327142737810030492, 197012375003682813852835, 456165570455991041730121, 1056527571106193560850298, 2447738008468144928854371, 5672459058686943865573806, 13149125315278295774252093, 30488654546469742502130612, 70711902848157227103981151, 164042637933125322263068179, 380652001553129860482798995, 883494697499474927314001201, 2051075452619209350256289939, 4762759874731327435883409954, 11061975317641962863310400262, 25698117521092741572233415672, 59712096957248664358561521889, 138775742402184313315715377316, 322591509897645444074929806223, 750029645171417394902589537216, 1744167142214391071760524674061, 4056767007670417319587124446474, 9437403083691333062035763415892, 21958549988526521939432355828298, 51101285390525603579799232888122, 118942032240129949401856916912786, 276893431455538839876673911173260, 644706688082075915267897498545600, 1501351643055827208280357431912040, 3496809282282403489853994898965247] ------------------------------------------------------------ Theorem Number, 221 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 7+ HHH + H HH + HH HH 6+ H H + H H + HH HH 5+ HH HH + H H + HH HH 4+ HH H H + HH HH HH HH 3+ HH HHH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 3 4 3 2 2 -1 + (x + 2 x - x + x - x + 1) P(x) + (-2 x + x - 2 x + 2 x - 3) P(x) 2 + (x - x + 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 21, 46, 94, 199, 439, 961, 2113, 4728, 10638, 24004, 54567, 124738, 286100, 658902, 1523561, 3533694, 8219807, 19175241, 44844787, 105117745, 246938893, 581271855, 1370801044, 3238379640, 7662884761, 18160172211, 43099390904, 102426035307, 243726755578, 580655580178, 1384932709103, 3306794777897, 7903693118041, 18909278995761, 45281554377074, 108530027712894, 260341203346223, 625005317744160, 1501608816996828, 3610336668014877, 8686460115399371, 20913632597618146, 50384354111881549, 121459160713524172, 292969666722436726, 707070780696631056, 1707424831884850449, 4125246497394886642, 9971935177734506502, 24116945055530422878, 58354041165562962555, 141259692927959905224, 342103485342872695337, 828862025018976189145, 2009029841925232601585, 4871515952686337473281, 11817073539321134031521, 28675998351756531656835, 69612131191902560435320, 169045763869085582882160, 410650213529812223368770, 997892674377742909288876, 2425690847473912236264926, 5898246126299524653958956, 14346379729918995608354591, 34905194678328946279956726, 84949845555910352980918162, 206802853409472003358290619, 503580166605875837883509918, 1226579626343965688411938881, 2988373791468938272294751313, 7282546274134082391128998885, 17751619375203540121720680747, 43280914162805330098090017988, 105549463086185917149368841094, 257462696364279602325677747884, 628158000160191822477935767084, 1532912888447769419915804392627, 3741604135787909331331389618256, 9134564337311934498324607449420, 22305163745068650394954160327031, 54476421551688686284866768832516, 133074664291962641673573493873036, 325135188262952797925515855558143, 794534125103762472134490936108882, 1941955965184663382273487480183164, 4747257141764020796262302049059161, 11607028692460250300264994515896655, 28383941376456748393597046710009226, 69421845442964953819449619451462120, 169820428896576356561050206615538112, 415482369434004891765163304919548259] ------------------------------------------------------------ Theorem Number, 222 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 4 3 2 2 3 2 1 + (x + x - 2 x + x - x + 1) P(x) + (2 x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 23, 46, 90, 178, 362, 744, 1535, 3194, 6704, 14152, 30019, 63993, 137027, 294528, 635254, 1374563, 2982983, 6490689, 14157809, 30951925, 67809639, 148847546, 327326203, 721035228, 1590820794, 3515055239, 7777682719, 17232166570, 38226855740, 84899762419, 188767040056, 420148088468, 936077556199, 2087530911028, 4659567865103, 10409537599216, 23274106872900, 52077999630392, 116616556556144, 261322452488208, 585990624975113, 1314890839669959, 2952311871586680, 6632807182343025, 14910247851249936, 33536291670621966, 75470769540757993, 169928986936543073, 382800909986262158, 862755122454412830, 1945377625304332347, 4388490288087053044, 9904091476130975332, 22361254774054848157, 50507256400673637226, 114125267802978496554, 257973369424416665508, 583348170337435269883, 1319580231355477362189, 2986029400547774283515, 6759245279299394441018, 15305372653077265127430, 34667865012474818872923, 78549578510974131598343, 178028874433432077107813, 403611252116127504636989, 915290323330934693322979, 2076222803712862095242661, 4710915866189376517310722, 10691779547594570193232569, 24271966712323362836221071, 55114699810857040680904084, 125179934207220209125653142, 284383356087736137374477620, 646209443923063259545590287, 1468722246098351522200998572, 3338881124332810099290521494, 7591977293065388173220549042, 17266301921829442269815327549, 39276443078065295093655855985, 89361682067950853306373719225, 203355033871187599510752721774, 462850820082071205035423702170, 1053677818530795805415739939561, 2399128776874249888355418021047, 5463569785243751528140091257717, 12444429345978153227988825135555, 28349632447886106488535958107141, 64594015240375900277795861639819, 147200074451998200230485801692760, 335500536871502998738039665095175] ------------------------------------------------------------ Theorem Number, 223 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 7+ HHH + H HH + HH HH 6+ H H + H H + HH HH 5+ HHH HH HH + H HH H H + HH HHH HH 4+ H H H + HH HH 3+ HH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 10 6 5 4 3 2 3 1 + (x - 2 x + x - 2 x + 2 x - x + x - 1) P(x) 7 6 5 4 3 2 2 + (x + x - x + x - 3 x + 2 x - 2 x + 3) P(x) 4 3 2 + (x + x - x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 17, 34, 65, 133, 277, 572, 1199, 2548, 5435, 11661, 25202, 54731, 119348, 261396, 574697, 1267637, 2804766, 6223660, 13845748, 30876213, 69007994, 154549822, 346790404, 779545026, 1755254743, 3958377577, 8939866583, 20218263933, 45784748248, 103807846272, 235636648909, 535466068728, 1218071478804, 2773591163064, 6321494978078, 14420661400511, 32924586831606, 75232979162492, 172041452385538, 393713260872074, 901644278692778, 2066263403285064, 4738263632158243, 10872374445702404, 24962637101521442, 57346546860289679, 131815132773209257, 303148512760931479, 697540229550486854, 1605826655979897539, 3698588507211274869, 8522638804153427168, 19647441284817832815, 45313253580871653227, 104550383069514369365, 241324409472237417738, 557245419733284748247, 1287229693480919378873, 2974574026651753256696, 6876189561850918905880, 15900858902075293511482, 36782273871125357477166, 85113327142737810030492, 197012375003682813852835, 456165570455991041730121, 1056527571106193560850298, 2447738008468144928854371, 5672459058686943865573806, 13149125315278295774252093, 30488654546469742502130612, 70711902848157227103981151, 164042637933125322263068179, 380652001553129860482798995, 883494697499474927314001201, 2051075452619209350256289939, 4762759874731327435883409954, 11061975317641962863310400262, 25698117521092741572233415672, 59712096957248664358561521889, 138775742402184313315715377316, 322591509897645444074929806223, 750029645171417394902589537216, 1744167142214391071760524674061, 4056767007670417319587124446474, 9437403083691333062035763415892, 21958549988526521939432355828298, 51101285390525603579799232888122, 118942032240129949401856916912786, 276893431455538839876673911173260, 644706688082075915267897498545600, 1501351643055827208280357431912040, 3496809282282403489853994898965247] ------------------------------------------------------------ Theorem Number, 224 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1} No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H H + HH H H HH + H HH HH H + HH H H HH 4+ H H H H + H HH HH H + HH H H HH + H HH H H 3+ H H H H + HH H HH HH + H HH H H + HH HHH HH 2+ H H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 7 6 3 2 2 4 3 2 1 + (x + x - 2 x + x - x + 1) P(x) + (x + 2 x - x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 5, 9, 15, 26, 48, 89, 165, 310, 591, 1138, 2205, 4297, 8427, 16622, 32938, 65530, 130863, 262249, 527197, 1062810, 2148146, 4352276, 8837592, 17982158, 36658592, 74865246, 153145667, 313763676, 643771865, 1322679887, 2721049245, 5604608198, 11557189002, 23857809325, 49300998950, 101977645247, 211133587197, 437515330510, 907391552370, 1883409945578, 3912262282749, 8132599519471, 16917470461058, 35215450619627, 73351807432625, 152882167526053, 318831083771137, 665291972000431, 1389000105612281, 2901494411312661, 6064042668767917, 12679900670970242, 26526198371525921, 55517844071636598, 116247016822421760, 243509692791621693, 510304958692509935, 1069834525683015957, 2243733025635694956, 4707477231914068428, 9880131493978309950, 20743874289698006871, 43567745659235343423, 91534327932366425904, 192372244159797445202, 404423446349987694296, 850475692006620639552, 1789021153134275019838, 3764380495324636198963, 7923054498549552785817, 16680521298024087725246, 35127021615881690891674, 73992001969543494910782, 155896761247337062814246, 328545486973552202438004, 692559921201329410518511, 1460226000605629209724777, 3079507053993080839704110, 6495885392920695093764064, 13705321751718305123493788, 28922220684172584949140422, 61046872393071002819789051, 128879108693661861817539403, 272136603848726764672198432, 574744475893032262155788238, 1214071321413282376714153974, 2565035305811338986244912025, 5420264847804173772189630486, 11455762461878381622441607611, 24215988133284034517908107932, 51198061405255014156095915233, 108262097724071227150484889909, 228965199188889890733294634337] ------------------------------------------------------------ Theorem Number, 225 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} To make it crystal clear here is such a path of semi-length 8 3+ H H H + H H H H H H + HH HH HH HH HH HH 2.5+ H H H H H H + H H H H H H + H H H H H H + H H H H H H 2+ H H H H H H + H H H H H H + H H H H H H 1.5+ HH HH HH HH HH HH + H H H H H H + H H H H H H 1+ H H H H H + H H H H + H H H H + H H H H 0.5+ H H H H + H H H H +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 5 4 3 2 2 4 3 2 3 (x + 2 x + x - x - x + 1) P(x) + (-2 x - 2 x + x + 2 x - 2) P(x) + x - x + 1 = 0 The sequence a(n) satisfies the linear recurrence (5 n - 16) a(n - 1) 3 (n - 4) a(n - 2) (5 n - 16) a(n - 3) a(n) = ------------------- - ------------------ - ------------------- n - 2 n - 2 n - 2 2 (n - 5) a(n - 4) (7 n - 26) a(n - 5) 2 (2 n - 7) a(n - 6) + ------------------ + ------------------- + -------------------- n - 2 n - 2 n - 2 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 3, a(6) = 8] Just for fun, a(1000), equals 4513822755460126660074078189370341261660336817857156196805239383541327668765\ 963201036446242767162378729335963151166140131240054571282001515611884138\ 347707059545419894042376425879663792620809205238834611560388846431423689\ 855628020649544677963687811040497090703256841273530210380101786194509537\ 067113662456127527366723479270300436904011211442668362293097746750575688\ 835502617788079689870672998477081184971149582163166925644575726955020920\ 939105238005483400514361671230605385968066563418828204155098826852134792\ 563845065129442942977648582580202025960544399383623710123012989628552834\ 1200209777585695 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 3, 8, 22, 66, 205, 659, 2176, 7338, 25175, 87600, 308438, 1096911, 3934501, 14217444, 51708228, 189133701, 695297272, 2567616546, 9520266637, 35428877301, 132284362787, 495422056737, 1860573539727, 7005279613890, 26437786566157, 99992915080063, 378956804646499, 1438883017044266, 5472952555956802, 20851091190855195, 79561319082442868, 304018865290849340, 1163288347203209907, 4456856253381250449, 17095950391798323577, 65652723572933964935, 252395084938660470309, 971300228995806201252, 3741520875601046917843, 14425943213758424529359, 55670150878817854299637, 215013056879130310699040, 831103042156988533520420, 3214970394931431167196982, 12445622846771068646936049, 48212567692099479691344384, 186893899244915375271335471, 724952045437841073502437994, 2813788818658344054245134457, 10927761877134616109677428319, 42463765229945814810328526968, 165098811754083923594038455251, 642242003452094064560306356416, 2499622354012123468062896169576, 9733367181217980860848488052798, 37919038999814748506597893341152, 147791661297484637880167152868371, 576280841532182050255873902828269, 2248038071709523897126456069119212, 8773083765348229482115199391560344, 34251076376976414130430307905046705, 133771619285025250764962521215343053, 522656424787518865047777886401309671, 2042801335244081387693443139103231469, 7987092488787586679227198394936689433, 31239168106253639749571060911359828370, 122223298789897367786429938536876432439, 478352617822303714773122930343750423467, 1872741870613410458553776145230519961801, 7333975685590627498781040010814803323006, 28729569562405085730168260855971101193792, 112575364208025175814271086554680276160124, 441244000036893830551313923697840917711081, 1729944755871098067008061615237652096860620, 6784229207252869834114373099052920193725507, 26612192391918503208423771635714944912611046, 104416657090550920598303336040227180137229247, 409793567938702363462329143452206827009074695, 1608659207414294215048604777048043026416995202, 6316317554259676264124909965595673046769547791, 24806322702779024606041295673390995577386633038, 97444418389436645715482691273662631906569013090, 382864762594583821292482001573078227978014313678, 1504615266887581631631661630508586933910344007466, 5914186721689435816247149267666532324208997808507, 23251556901166208107618467179144210914118393238017, 91431214998152372374430145608028518679235984659612, 359600648185260194105199099035757592259894049787112, 1414581918372334687037284067835342112805144391535027, 5565646284619425342154422510400465742007388352401786, 21901871390400918289358355684583192246293979459428060, 86203179780619751935353229364298421064266928243498983, 339343985919548320483293171927251428756964164005897691, 1336073217799584176610547786973269980815347949928534065, 5261287839279045868801487137035902797982956535798369167, 20721638196337016739785554784325774077379467436674334615] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 226 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH H + HH H HH H + H HH H H + HH HHH HH 4+ H H H HH + H H HH H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 2 2 (x + 3 x + 3 x - 2 x - x + 1) P(x) 5 4 3 2 4 3 2 + (-2 x - 4 x - x + 3 x + 2 x - 2) P(x) + x + x - x - x + 1 = 0 The sequence a(n) satisfies the linear recurrence 3 (n - 3) a(n - 1) 3 (n - 4) a(n - 2) (7 n - 29) a(n - 3) a(n) = ------------------ + ------------------ - ------------------- n - 2 n - 2 n - 2 9 (n - 4) a(n - 4) 3 (n - 5) a(n - 5) 2 (7 n - 32) a(n - 6) - ------------------ + ------------------ + --------------------- n - 2 n - 2 n - 2 (11 n - 52) a(n - 7) 3 (n - 5) a(n - 8) + -------------------- + ------------------ n - 2 n - 2 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 2, a(6) = 5, a(7) = 10, a(8) = 23] Just for fun, a(1000), equals 1535875550442896470095395825402152730540485059103186263221372801210323501244\ 111578111330181394715884418663118516141187615047640646477044894473357803\ 280274668210582403538317400444139846335917880226813081544286284810308729\ 072647413532602648950872610016959230151065821816599284663356610746800243\ 319291447553307283277134817529956699076645731800804659833853197106280275\ 021893508153099310089662316073536841861747025996168499724538716524639550\ 578422169119005948677208007890030693 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 2, 5, 10, 23, 54, 129, 317, 794, 2022, 5226, 13676, 36177, 96599, 260049, 705091, 1923871, 5278828, 14556831, 40321752, 112140498, 313018373, 876631744, 2462514328, 6936562032, 19589194824, 55451046515, 157306752376, 447158262791, 1273474767476, 3633119478220, 10381968098066, 29712930309197, 85160125438235, 244407751581020, 702340334430984, 2020705977007378, 5820402230049975, 16783053380443034, 48443202503099686, 139963962492011145, 404762620644449291, 1171564282548823780, 3393872490117955949, 9839475432357248051, 28548274571200159244, 82890502143463936992, 240842477782233628640, 700248347213646076273, 2037275920765430630412, 5930825566325687271777, 17275771005098358805489, 50350846660497998342893, 146829661190705224368913, 428400183540167551362836, 1250564138668682100196765, 3652370722363131893469317, 10672073370795706491503871, 31197566333949098811240969, 91239701690971305877812665, 266951165505227719446991440, 781373227089388454918837340, 2288011399507360177012440001, 6702321896661965308412015488, 19640591793504081872390683473, 57575940058609138924239645255, 168841774108237705791077565384, 495298043850689353798034091939, 1453438944634086434071361342474, 4266446216280524734268914482487, 12527691996807021824410833490622, 36796570730605031661551108337769, 108111389444692259051924439861599, 317731200083074237162198393694548, 934048053501838715982921569892333, 2746605497001170880027951059897851, 8078634368898013673128034690902148, 23767923398552104647316379507412729, 69944452421307104365515113219950301, 205883384600549446793428001836825007, 606167512898831453987377643603758496, 1785109209162200614319479019390848605, 5258177423204439347660160184559334513, 15491790657701878298307393633348809066, 45652193765306071921753371732920059627, 134559115778273182391365815932012805150, 396692424348014937680829010683476694375, 1169720069134376658629373194376390769530, 3449810510114653631841050285956378673623, 10176346501147241736346948992922602590389, 30024101604799559003910287303700174337197, 88598802405588036607244395624654392086364, 261495152070740315272955295758480227482047, 771926076709973082034432567707122551614768, 2279095045901399420133068593235002964846064, 6730111230265473608105224556506581870856631, 19877126509423817034725118069338167548089462] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 227 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H H H + HH H HH H H HH + H HH H H HH H + HH HHH HHH HH 4+ H H H H H + HHH H H + HH H HH HH + H HH H H 3+ H H H + HH HH + H H + HH HH 2+ H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 5 4 3 2 3 (x + x - x - 4 x - 3 x - 3 x + 2 x - x + x - 1) P(x) 8 7 6 5 4 3 2 2 + (-x + x + 7 x + 5 x + 9 x - 5 x + 4 x - 3 x + 3) P(x) 6 5 4 3 2 6 4 3 + (-4 x - 2 x - 9 x + 4 x - 5 x + 3 x - 3) P(x) + x + 3 x - x 2 + 2 x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 6, 14, 36, 100, 281, 808, 2376, 7102, 21521, 65977, 204269, 637838, 2006483, 6352939, 20230008, 64747566, 208172635, 672044179, 2177586477, 7079606329, 23087181572, 75500340946, 247539590355, 813525543281, 2679484452965, 8843331598447, 29241758634574, 96863128374642, 321388886244634, 1068009831332200, 3554270777789155, 11844565513426862, 39522723495313727, 132039029649014016, 441628107381703446, 1478711659537539998, 4956307483439340428, 16628701379883372195, 55842240980342943450, 187694836708367476092, 631406044088105686345, 2125764968083430423559, 7162360381293454456773, 24149983991866410771883, 81486106057418712997102, 275133498762670761680043, 929575153784271264743264, 3142645137300572934738228, 10630784711307320311460598, 35981925995780743779266098, 121854935995676315487432933, 412888270901383278260159356, 1399729835151769284627512716, 4747556660977666047318871311, 16110268903078059179975233456, 54693402863692861806450335500, 185763256662397287120332087752, 631205740531034300160130235999, 2145666259473054422842695661522, 7296719769144177880978263241830, 24823434066024649923177219087201, 84481078889226338895457250984728, 287617549111892991191433080749556, 979545977031190286087710638768041, 3337207231300967396291747599816377, 11373287814232507199615763656398425, 38772979471039644790079349642706166, 132223480003206676024193538221205846, 451045542474369625630215511344029730, 1539079006651777916802774319843750086, 5253232187427421698166565832821468430, 17935522875980973218271681507870125495, 61251959716273408965103875141378434940, 209238356930312245475384484908864272219, 714948846334205987654945562000305207112, 2443532387721049937609640692942992894189, 8353489038144535415396607086324356688834, 28564176792661590059807659743181757848277, 97696034568007522752653886185193897025755, 334218968105069701157350715560918775388102, 1143620129565087598530316728659263648308927, 3914053150314931253981909263552704764508237, 13398730458255539620408344515881553000217397, 45876516908031757897880107229218758376606907, 157110425660715403685260106385885946869273486, 538152485724881982189423842080321763789305025, 1843696953338304699449518866697787148562598378, 6317651957655733829937596559084569571650684807, 21652199401380643190619995994632203196884247610, 74220991579486079127225097822493906010642172096, 254465087845601496348953732219726602104477396149, 872578970003628725661672384330693010860692425830, 2992642036428581929148053661298404503777797127645, 10265420874352423601844287619628336842086268682555, 35218367102121514980733611274817512226345172360074] ------------------------------------------------------------ Theorem Number, 228 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 7+ HH + H HH + HH HH 6+ H HH + H HH + HH HH 5+ HH HH HHH + H HH HH H + HH HHH HH 4+ H H H + HH HH 3+ HH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 7 6 5 4 3 3 (x - 2 x - 5 x - 3 x - 2 x + 3 x + x - 1) P(x) 7 6 5 4 3 2 2 + (3 x + 8 x + 5 x + 6 x - 7 x + x - 3 x + 3) P(x) 7 6 5 4 3 2 6 4 3 + (-x - 4 x - 2 x - 6 x + 5 x - 2 x + 3 x - 3) P(x) + x + 2 x - x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 43, 88, 181, 389, 845, 1831, 4021, 8945, 19980, 44876, 101560, 231010, 527584, 1210453, 2788476, 6445177, 14945402, 34762125, 81073108, 189553176, 444229775, 1043335656, 2455327148, 5789089278, 13673287432, 32347923587, 76645650172, 181868152606, 432131947999, 1028094409010, 2448927381166, 5840049815756, 13942114199763, 33318698172213, 79702566442900, 190836157036246, 457334328621662, 1096917010956494, 2633078585793776, 6325404794268179, 15206619342001349, 36583326860847321, 88069812063175602, 212154504630779797, 511384369072275705, 1233394476770162448, 2976499939131468798, 7187032770060977636, 17362981676250949732, 41968309776585627857, 101492313014921186269, 245556679552748148628, 594388103914626682818, 1439399709664903706555, 3487218741318126496572, 8451957378207794626600, 20493205067305036075159, 49708582948760009479704, 120619210443382870051297, 292792615812251132529932, 710980136125717425588637, 1727046021860570120941582, 4196575501209940797812052, 10200621702455464613659962, 24802455440798284389096014, 60324716048583750707955995, 146765765674535755148007651, 357173774922625517418331537, 869473358467539692845081275, 2117150070857988843015817831, 5156588999541444162109803583, 12562783839142431859361805544, 30613913629937142427400197707, 74620580935239731515082935125, 181929247599493983737210197721, 443657637304923176383168028461, 1082162432233326872644453144656, 2640180374641018514672900497678, 6442717456740989437135783799985, 15725218378158726422687662180370, 38389658112166505912696121858576, 93738863949937324569728179979473, 228934372538591942169266458710404, 559224587509088449210338239175827, 1366292159965124659559109224182072, 3338728826968157284819565578655591, 8160132627279040287185529051465709, 19947574345781438581105186436718539, 48770603600956548080699610819303235, 119261355989728246212509483676650821, 291684525929087013741255678799558755] ------------------------------------------------------------ Theorem Number, 229 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H H + HH H H HH + H HH HH H + HH H H HH 4+ H H H H H + HHH H HH HH H + HH H HH H H HH + H HH H HH H H 3+ H H H H + HH HH + H H + HH HH 2+ H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 2 2 (x + 3 x + 3 x - 2 x - x + 1) P(x) 5 4 3 2 4 3 2 + (-2 x - 4 x - x + 3 x + 2 x - 2) P(x) + x + x - x - x + 1 = 0 The sequence a(n) satisfies the linear recurrence 3 (n - 3) a(n - 1) 3 (n - 4) a(n - 2) (7 n - 29) a(n - 3) a(n) = ------------------ + ------------------ - ------------------- n - 2 n - 2 n - 2 9 (n - 4) a(n - 4) 3 (n - 5) a(n - 5) 2 (7 n - 32) a(n - 6) - ------------------ + ------------------ + --------------------- n - 2 n - 2 n - 2 (11 n - 52) a(n - 7) 3 (n - 5) a(n - 8) + -------------------- + ------------------ n - 2 n - 2 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 2, a(6) = 5, a(7) = 10, a(8) = 23] Just for fun, a(1000), equals 1535875550442896470095395825402152730540485059103186263221372801210323501244\ 111578111330181394715884418663118516141187615047640646477044894473357803\ 280274668210582403538317400444139846335917880226813081544286284810308729\ 072647413532602648950872610016959230151065821816599284663356610746800243\ 319291447553307283277134817529956699076645731800804659833853197106280275\ 021893508153099310089662316073536841861747025996168499724538716524639550\ 578422169119005948677208007890030693 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 2, 5, 10, 23, 54, 129, 317, 794, 2022, 5226, 13676, 36177, 96599, 260049, 705091, 1923871, 5278828, 14556831, 40321752, 112140498, 313018373, 876631744, 2462514328, 6936562032, 19589194824, 55451046515, 157306752376, 447158262791, 1273474767476, 3633119478220, 10381968098066, 29712930309197, 85160125438235, 244407751581020, 702340334430984, 2020705977007378, 5820402230049975, 16783053380443034, 48443202503099686, 139963962492011145, 404762620644449291, 1171564282548823780, 3393872490117955949, 9839475432357248051, 28548274571200159244, 82890502143463936992, 240842477782233628640, 700248347213646076273, 2037275920765430630412, 5930825566325687271777, 17275771005098358805489, 50350846660497998342893, 146829661190705224368913, 428400183540167551362836, 1250564138668682100196765, 3652370722363131893469317, 10672073370795706491503871, 31197566333949098811240969, 91239701690971305877812665, 266951165505227719446991440, 781373227089388454918837340, 2288011399507360177012440001, 6702321896661965308412015488, 19640591793504081872390683473, 57575940058609138924239645255, 168841774108237705791077565384, 495298043850689353798034091939, 1453438944634086434071361342474, 4266446216280524734268914482487, 12527691996807021824410833490622, 36796570730605031661551108337769, 108111389444692259051924439861599, 317731200083074237162198393694548, 934048053501838715982921569892333, 2746605497001170880027951059897851, 8078634368898013673128034690902148, 23767923398552104647316379507412729, 69944452421307104365515113219950301, 205883384600549446793428001836825007, 606167512898831453987377643603758496, 1785109209162200614319479019390848605, 5258177423204439347660160184559334513, 15491790657701878298307393633348809066, 45652193765306071921753371732920059627, 134559115778273182391365815932012805150, 396692424348014937680829010683476694375, 1169720069134376658629373194376390769530, 3449810510114653631841050285956378673623, 10176346501147241736346948992922602590389, 30024101604799559003910287303700174337197, 88598802405588036607244395624654392086364, 261495152070740315272955295758480227482047, 771926076709973082034432567707122551614768, 2279095045901399420133068593235002964846064, 6730111230265473608105224556506581870856631, 19877126509423817034725118069338167548089462] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 230 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 3+ H H H + H H H H H H + HH HH HH HH HH HH 2.5+ H H H H H H + H H H H H H + H H H H H H + H H H H H H 2+ H H H H H H + H H H H H H + H H H H H H 1.5+ HH HH HH HH HH HH + H H H H H H + H H H H H H 1+ H H H H H + H H H H + H H H H + H H H H 0.5+ H H H H + H H H H +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 7 6 5 4 3 2 2 (x + 3 x + 4 x + 2 x - x - 2 x - x + 1) P(x) 6 5 4 3 2 5 4 2 + (-2 x - 4 x - 3 x + x + 3 x + 2 x - 2) P(x) + x + x - x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 4, 8, 16, 33, 70, 150, 327, 724, 1623, 3681, 8437, 19519, 45536, 107030, 253262, 602908, 1443070, 3470954, 8385578, 20340507, 49519730, 120960620, 296371546, 728194151, 1793822734, 4429426335, 10961605329, 27182576415, 67535928339, 168093551092, 419071987208, 1046409830685, 2616670011698, 6552275154848, 16428452398084, 41241152141282, 103649065442512, 260779618021894, 656796504105220, 1655823783167292, 4178326335276288, 10553000222691372, 26675752223358469, 67485051975526502, 170857040626370571, 432890244396224794, 1097560693166690293, 2784660137566683943, 7069629246725243597, 17959344777470420687, 45650238004771051545, 116103213416967047771, 295450394971008048336, 752237693339907678388, 1916228633922016792045, 4883748297887608403258, 12452752489393602069929, 31767014040001686325582, 81073622544524038530886, 206999092435241758311202, 528733513631754741849442, 1351073351501234035819687, 3453736074867883038676328, 8832063790026984856111437, 22593994962609018942548434, 57819831170065015160147258, 148016104724210103267209410, 379040037587504184490060968, 970959159026854695362349530, 2488012618157836350994116103, 6377288789306547119410546353, 16351130742851668466225675508, 41935725885251494891116908522, 107582545160831136156990545003, 276068900975039417577961191741, 708611271260935586778378177294, 1819325982174725461733301456785, 4672206168391097224635813991761, 12001615554273519602977258767978, 30836209618100401406574383084291, 79247078893538383198570456482935, 203706110082593778957038744931844, 523746317307284313148511444896921, 1346888803038138272803743432070913, 3464448567504291062470852345232066, 8913042328640841543950070604839795, 22935342432012856569972142351659937, 59029608373580246804602119121026026, 151956072880316631652959014065460778, 391244137913596746646621211123393927, 1007528749678183420473613625882526153, 2595046421486848446715296247659510435, 6685120019414906923717787261359040175, 17224556501071547042357604518932295461, 44387437195386122034755378429092171929] ------------------------------------------------------------ Theorem Number, 231 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 7+ HHH + H HH + HH HH 6+ H H + H H + HH HH 5+ HH HH + H H + HH HH 4+ HH H H + HH HH HH HH 3+ HH HHH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 2 (x + 3 x + 2 x - x - x - x + 1) P(x) 5 4 3 2 4 + (-2 x - 3 x + x + x + 2 x - 2) P(x) + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 4, 7, 13, 31, 68, 151, 353, 830, 1974, 4770, 11640, 28655, 71137, 177846, 447385, 1131721, 2877047, 7346508, 18834532, 48462272, 125107961, 323947194, 841128592, 2189541500, 5712962262, 14938641202, 39141249503, 102747535602, 270188372857, 711657594649, 1877327468019, 4959442969022, 13119361391251, 34749264755755, 92151139526833, 244652427719736, 650228354697653, 1729918669812748, 4606876307510903, 12279701899270589, 32760484182475877, 87473433250584239, 233748476333341011, 625104546863703460, 1672912767097417426, 4480197378774438801, 12006364182059672476, 32196189195480675152, 86390268339099023784, 231943406867773673559, 623083386995175908592, 1674741886324264405491, 4503792002194003780483, 12117949758427063804038, 32620637607244253751313, 87853837000533097933303, 236715620047918300658849, 638093409766878741244939, 1720784029640200529466109, 4642449192227602970962471, 12529705204132005823084739, 33829995525108470689087028, 91374533506774924198871928, 246891162946274814345255866, 667326361919700464180466164, 1804341836783755969767902081, 4880258183200337657627349597, 13204016231004099310703911640, 35735888322210287302434695483, 96746346945051492129944136673, 261994616605299952299716358913, 709699417012216252826462746398, 1922991879447607695698409355563, 5211924795579765882550153279034, 14129715409831946127753654037827, 38316011030586066637914582753059, 103928796252920215410081803358278, 281966477169083994478108975309047, 765177829541777969692893114202601, 2076959557406565291855683538866509, 5638868945770438919899018896841685, 15312704020048106078050590219312734, 41591579374244913493655901742655268, 112992694770324482050201403155677648, 307032691453092694995591759198423486, 834461137862482346770494050695443513, 2268364636536495630406117530328420694, 6167411942613495429055053735583077660, 16771601256606925288684863153683394607, 45616902005536094888373955476881033944, 124095204548386658478210061054534756445, 337645125563796910137825400887087849425, 918841532591357251637815666765040689673, 2500885592781921396477331864964481145773, 6807984856027371010633861697801309493006] ------------------------------------------------------------ Theorem Number, 232 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 3 2 (x + 4 x + 4 x + 2 x - 3 x - x + 1) P(x) 7 6 5 4 3 2 6 4 3 + (-2 x - 4 x - x - x + 5 x - x + 2 x - 2) P(x) + x + x - 2 x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 16, 30, 56, 110, 216, 427, 860, 1746, 3569, 7364, 15307, 32009, 67334, 142416, 302663, 646031, 1384486, 2977909, 6426645, 13911978, 30200915, 65732898, 143413923, 313595163, 687144819, 1508571730, 3317922569, 7309678595, 16129313045, 35643218886, 78875659428, 174774870620, 387750760231, 861260255477, 1915124365594, 4262999134446, 9498739083878, 21184957996924, 47291188848596, 105658705445218, 236257445857091, 528694503932384, 1183990265577090, 2653392279620791, 5950474873926279, 13353228687666239, 29984296182635404, 67369519671951483, 151455416547760267, 340681282599710033, 766734008185798102, 1726495929499880140, 3889579288927147642, 8766943991330924627, 19769471877891215161, 44600139100704220528, 100661893945840143266, 227287461047749510237, 513406564122610083700, 1160158342765246864333, 2622633765230015042003, 5930856706035176271770, 13416885846141972833882, 30362383327524771174841, 68733011599111138672309, 155645297432095952323778, 352568632584310655173635, 798885376847742072720203, 1810734038572540097719042, 4105354542744134743448103, 9310415132809649662747530, 21120611967784825680218897, 47924750352934920938803331, 108774247977720255491636377, 246946153325595537553849633, 560771040004278719756311221, 1273718100463445848541909110, 2893762556614309851409015460, 6575848332156955986692361543, 14946434073696128773556355402, 33979577457896340549152610437, 77266402617848204240564334985, 175733104543382019162751390086, 399764798964070432229145750542, 909581419077380496710545668751, 2069963664743655297207114411687, 4711574821783439857678518992020, 10726298267792692107831401356403, 24423746955362712321076292407699, 55622647415401209237519460346954, 126696994280216636321751775842705, 288638769125621515697430788944262] ------------------------------------------------------------ Theorem Number, 233 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 7+ HH HHH + H HH H HH + HH HHHH HH 6+ H H H + H H + HH HH 5+ HH HH + H H + HH HH 4+ H H + HH HH 3+ HH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 5 4 3 2 3 (x + x - x - 4 x - 3 x - 3 x + 2 x - x + x - 1) P(x) 8 7 6 5 4 3 2 2 + (-x + x + 7 x + 5 x + 9 x - 5 x + 4 x - 3 x + 3) P(x) 6 5 4 3 2 6 4 3 + (-4 x - 2 x - 9 x + 4 x - 5 x + 3 x - 3) P(x) + x + 3 x - x 2 + 2 x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 6, 14, 36, 100, 281, 808, 2376, 7102, 21521, 65977, 204269, 637838, 2006483, 6352939, 20230008, 64747566, 208172635, 672044179, 2177586477, 7079606329, 23087181572, 75500340946, 247539590355, 813525543281, 2679484452965, 8843331598447, 29241758634574, 96863128374642, 321388886244634, 1068009831332200, 3554270777789155, 11844565513426862, 39522723495313727, 132039029649014016, 441628107381703446, 1478711659537539998, 4956307483439340428, 16628701379883372195, 55842240980342943450, 187694836708367476092, 631406044088105686345, 2125764968083430423559, 7162360381293454456773, 24149983991866410771883, 81486106057418712997102, 275133498762670761680043, 929575153784271264743264, 3142645137300572934738228, 10630784711307320311460598, 35981925995780743779266098, 121854935995676315487432933, 412888270901383278260159356, 1399729835151769284627512716, 4747556660977666047318871311, 16110268903078059179975233456, 54693402863692861806450335500, 185763256662397287120332087752, 631205740531034300160130235999, 2145666259473054422842695661522, 7296719769144177880978263241830, 24823434066024649923177219087201, 84481078889226338895457250984728, 287617549111892991191433080749556, 979545977031190286087710638768041, 3337207231300967396291747599816377, 11373287814232507199615763656398425, 38772979471039644790079349642706166, 132223480003206676024193538221205846, 451045542474369625630215511344029730, 1539079006651777916802774319843750086, 5253232187427421698166565832821468430, 17935522875980973218271681507870125495, 61251959716273408965103875141378434940, 209238356930312245475384484908864272219, 714948846334205987654945562000305207112, 2443532387721049937609640692942992894189, 8353489038144535415396607086324356688834, 28564176792661590059807659743181757848277, 97696034568007522752653886185193897025755, 334218968105069701157350715560918775388102, 1143620129565087598530316728659263648308927, 3914053150314931253981909263552704764508237, 13398730458255539620408344515881553000217397, 45876516908031757897880107229218758376606907, 157110425660715403685260106385885946869273486, 538152485724881982189423842080321763789305025, 1843696953338304699449518866697787148562598378, 6317651957655733829937596559084569571650684807, 21652199401380643190619995994632203196884247610, 74220991579486079127225097822493906010642172096, 254465087845601496348953732219726602104477396149, 872578970003628725661672384330693010860692425830, 2992642036428581929148053661298404503777797127645, 10265420874352423601844287619628336842086268682555, 35218367102121514980733611274817512226345172360074] ------------------------------------------------------------ Theorem Number, 234 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 4 3 2 2 (x + 3 x + 2 x - x - x - x + 1) P(x) 5 4 3 2 4 + (-2 x - 3 x + x + x + 2 x - 2) P(x) + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 4, 7, 13, 31, 68, 151, 353, 830, 1974, 4770, 11640, 28655, 71137, 177846, 447385, 1131721, 2877047, 7346508, 18834532, 48462272, 125107961, 323947194, 841128592, 2189541500, 5712962262, 14938641202, 39141249503, 102747535602, 270188372857, 711657594649, 1877327468019, 4959442969022, 13119361391251, 34749264755755, 92151139526833, 244652427719736, 650228354697653, 1729918669812748, 4606876307510903, 12279701899270589, 32760484182475877, 87473433250584239, 233748476333341011, 625104546863703460, 1672912767097417426, 4480197378774438801, 12006364182059672476, 32196189195480675152, 86390268339099023784, 231943406867773673559, 623083386995175908592, 1674741886324264405491, 4503792002194003780483, 12117949758427063804038, 32620637607244253751313, 87853837000533097933303, 236715620047918300658849, 638093409766878741244939, 1720784029640200529466109, 4642449192227602970962471, 12529705204132005823084739, 33829995525108470689087028, 91374533506774924198871928, 246891162946274814345255866, 667326361919700464180466164, 1804341836783755969767902081, 4880258183200337657627349597, 13204016231004099310703911640, 35735888322210287302434695483, 96746346945051492129944136673, 261994616605299952299716358913, 709699417012216252826462746398, 1922991879447607695698409355563, 5211924795579765882550153279034, 14129715409831946127753654037827, 38316011030586066637914582753059, 103928796252920215410081803358278, 281966477169083994478108975309047, 765177829541777969692893114202601, 2076959557406565291855683538866509, 5638868945770438919899018896841685, 15312704020048106078050590219312734, 41591579374244913493655901742655268, 112992694770324482050201403155677648, 307032691453092694995591759198423486, 834461137862482346770494050695443513, 2268364636536495630406117530328420694, 6167411942613495429055053735583077660, 16771601256606925288684863153683394607, 45616902005536094888373955476881033944, 124095204548386658478210061054534756445, 337645125563796910137825400887087849425, 918841532591357251637815666765040689673, 2500885592781921396477331864964481145773, 6807984856027371010633861697801309493006] ------------------------------------------------------------ Theorem Number, 235 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H H + H HH HHH + HH H H HH + H HH HH H 3+ H H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH H H 1+ H H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The first, 100, terms of the sequence are [1, 0, 0, 1, 1, 2, 4, 9, 23, 55, 136, 351, 919, 2435, 6533, 17727, 48561, 134098, 372932, 1043704, 2937368, 8308221, 23605025, 67337718, 192799408, 553864094, 1595974019, 4611727868, 13360411104, 38797904819, 112915209530, 329294039804, 962146287403, 2816234575297, 8256899960779, 24246015313369, 71301337511760, 209967152438616, 619108941036450, 1827735528658399, 5402083298054717, 15983953933696710, 47343200549589868, 140364972558648429, 416549450423831146, 1237261621784509880, 3678116741650980014, 10943129791900047416, 32583211012822466382, 97088491649862761338, 289500533999000885510, 863825050531386825494, 2579199251870966882018, 7705756241151071442212, 23035953061083403683647, 68904493530162844956986, 206219542187450352838322, 617510021695216154318984, 1850042133180212949868186, 5545422019441507270708224, 16630118494472542280112084, 49894953556739489519901553, 149765433948215088440830953, 449731945557689009375506208, 1351066994334330516844923880, 4060459169204450923118378205, 12207955195369457143785540222, 36717650654237364825109492754, 110475489831565432818204540778, 332514985393002227167405968308, 1001166036977168707797035883864, 3015408559363704427127538054583, 9085046406035499228092087001589, 27380729481001250082307594801363, 82545969921350766563696142437067, 248929296740887329265025163283241, 750899579382801920443057070302647, 2265739633768561610990034742953135, 6838441747870879453279966383027860, 20645250631376094893079077689161432, 62344191393789311239915528733945290, 188313628446867377208631774903697956, 568950684799716438864526215149868303, 1719380199627969159125069687767061090, 5197219356812180357777472081048895611, 15713376063523897170250246768245816065, 47518736545298304117164948975425071387, 143732462820473103619762179632542441430, 434847735180697813016850422167843224401, 1315860227609050186823209653265515549563, 3982634260727736827746444305843500329172, 12056387377992960748956137115839762240413, 36504644144719185833533354509524161347137, 110550655387825977856224009480523102087185, 334853607868059828976758493817162445459637, 1014442284884227382395276945100462031766183, 3073807761797684327083159446363267774237007, 9315397973978382113015227457030984922865781, 28235785951857130403413679751044067361706410, 85599368408019286286764214453655767792325282, 259544574521140750886617655056977261749359954] ------------------------------------------------------------ Theorem Number, 236 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 19 17 16 15 14 13 12 11 10 9 8 (x - 3 x - 3 x + x + 6 x + 5 x + 5 x - 4 x - 7 x - 12 x - x 7 6 5 4 3 3 16 14 13 + x + 10 x + 2 x + 2 x - 4 x - x + 1) P(x) + (3 x - 6 x - 6 x 12 11 10 9 8 6 5 4 3 2 - 3 x + 7 x + 7 x + 16 x - x - 18 x - 2 x - 6 x + 10 x - x 2 13 11 10 9 8 7 6 5 + 3 x - 3) P(x) + (3 x - 3 x - 3 x - 6 x + 2 x - x + 10 x - x 4 3 2 10 6 5 4 3 2 + 6 x - 8 x + 2 x - 3 x + 3) P(x) + x - 2 x + x - 2 x + 2 x - x + x - 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 5, 8, 13, 29, 58, 109, 224, 469, 962, 2004, 4252, 9040, 19301, 41563, 89980, 195525, 426833, 935706, 2058178, 4541767, 10053231, 22314028, 49652620, 110747551, 247557767, 554494496, 1244340818, 2797369080, 6299090767, 14206237723, 32085826176, 72567625371, 164336472104, 372610625607, 845820085338, 1922098715982, 4372439993387, 9956343355891, 22692518763045, 51766984724848, 118193127925529, 270074678950222, 617606728681446, 1413389735165284, 3236829392842055, 7417758864228774, 17010163891685398, 39031546594762198, 89615630050729955, 205874421628959923, 473218932824758023, 1088313251325592598, 2504201652272691910, 5765011816121712227, 13278193824888813318, 30596973864393382114, 70536151123238024372, 162679316350017450709, 375347930528973340133, 866385011098516035066, 2000587011422283874074, 4621343255190585868302, 10679186807475470377805, 24686666622984511599253, 57086890984494449675458, 132055217721375981546693, 305573531184986377077384, 707314945608209847639832, 1637732567993388063478190, 3793171352332367061541348, 8787952385570629968379636, 20365508673681019718977166, 47208682141769560467266124, 109462253636911869115750581, 253874894309395675440725721, 588959015626156096858744585, 1366650733136206654486516020, 3172009302049365401821933578, 7363990486154164923351386224, 17099812077001249363177056269, 39716082047248874128446242438, 92264807603841023580831955900, 214386848828249772877195858443, 498253569181261894314960926934, 1158219376625202957075370231454, 2692882159291120533526978907126, 6262215947461674813212714459568, 14565353079098544379221773377774, 33883983435461603108197061482155, 78839999338725373583734755178870, 183474523257584763789771107474471, 427051528522691727246479996839525, 994165194829588457736112300985513, 2314776392397718522798745297649492] ------------------------------------------------------------ Theorem Number, 237 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 7 6 5 4 3 3 (x - 2 x - 5 x - 3 x - 2 x + 3 x + x - 1) P(x) 7 6 5 4 3 2 2 + (3 x + 8 x + 5 x + 6 x - 7 x + x - 3 x + 3) P(x) 7 6 5 4 3 2 6 4 3 + (-x - 4 x - 2 x - 6 x + 5 x - 2 x + 3 x - 3) P(x) + x + 2 x - x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 43, 88, 181, 389, 845, 1831, 4021, 8945, 19980, 44876, 101560, 231010, 527584, 1210453, 2788476, 6445177, 14945402, 34762125, 81073108, 189553176, 444229775, 1043335656, 2455327148, 5789089278, 13673287432, 32347923587, 76645650172, 181868152606, 432131947999, 1028094409010, 2448927381166, 5840049815756, 13942114199763, 33318698172213, 79702566442900, 190836157036246, 457334328621662, 1096917010956494, 2633078585793776, 6325404794268179, 15206619342001349, 36583326860847321, 88069812063175602, 212154504630779797, 511384369072275705, 1233394476770162448, 2976499939131468798, 7187032770060977636, 17362981676250949732, 41968309776585627857, 101492313014921186269, 245556679552748148628, 594388103914626682818, 1439399709664903706555, 3487218741318126496572, 8451957378207794626600, 20493205067305036075159, 49708582948760009479704, 120619210443382870051297, 292792615812251132529932, 710980136125717425588637, 1727046021860570120941582, 4196575501209940797812052, 10200621702455464613659962, 24802455440798284389096014, 60324716048583750707955995, 146765765674535755148007651, 357173774922625517418331537, 869473358467539692845081275, 2117150070857988843015817831, 5156588999541444162109803583, 12562783839142431859361805544, 30613913629937142427400197707, 74620580935239731515082935125, 181929247599493983737210197721, 443657637304923176383168028461, 1082162432233326872644453144656, 2640180374641018514672900497678, 6442717456740989437135783799985, 15725218378158726422687662180370, 38389658112166505912696121858576, 93738863949937324569728179979473, 228934372538591942169266458710404, 559224587509088449210338239175827, 1366292159965124659559109224182072, 3338728826968157284819565578655591, 8160132627279040287185529051465709, 19947574345781438581105186436718539, 48770603600956548080699610819303235, 119261355989728246212509483676650821, 291684525929087013741255678799558755] ------------------------------------------------------------ Theorem Number, 238 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ HH HH + HH HH + H H + HH HH 4+ H H + H H + HH HH 3+ H H H + HH HH HH HH + HH H HH HH 2+ HH H HH HH + H HH H H + HH H HH HH + H HH H H 1+ HH H HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 3 2 (x + 4 x + 4 x + 2 x - 3 x - x + 1) P(x) 7 6 5 4 3 2 6 4 3 + (-2 x - 4 x - x - x + 5 x - x + 2 x - 2) P(x) + x + x - 2 x 2 + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 8, 16, 30, 56, 110, 216, 427, 860, 1746, 3569, 7364, 15307, 32009, 67334, 142416, 302663, 646031, 1384486, 2977909, 6426645, 13911978, 30200915, 65732898, 143413923, 313595163, 687144819, 1508571730, 3317922569, 7309678595, 16129313045, 35643218886, 78875659428, 174774870620, 387750760231, 861260255477, 1915124365594, 4262999134446, 9498739083878, 21184957996924, 47291188848596, 105658705445218, 236257445857091, 528694503932384, 1183990265577090, 2653392279620791, 5950474873926279, 13353228687666239, 29984296182635404, 67369519671951483, 151455416547760267, 340681282599710033, 766734008185798102, 1726495929499880140, 3889579288927147642, 8766943991330924627, 19769471877891215161, 44600139100704220528, 100661893945840143266, 227287461047749510237, 513406564122610083700, 1160158342765246864333, 2622633765230015042003, 5930856706035176271770, 13416885846141972833882, 30362383327524771174841, 68733011599111138672309, 155645297432095952323778, 352568632584310655173635, 798885376847742072720203, 1810734038572540097719042, 4105354542744134743448103, 9310415132809649662747530, 21120611967784825680218897, 47924750352934920938803331, 108774247977720255491636377, 246946153325595537553849633, 560771040004278719756311221, 1273718100463445848541909110, 2893762556614309851409015460, 6575848332156955986692361543, 14946434073696128773556355402, 33979577457896340549152610437, 77266402617848204240564334985, 175733104543382019162751390086, 399764798964070432229145750542, 909581419077380496710545668751, 2069963664743655297207114411687, 4711574821783439857678518992020, 10726298267792692107831401356403, 24423746955362712321076292407699, 55622647415401209237519460346954, 126696994280216636321751775842705, 288638769125621515697430788944262] ------------------------------------------------------------ Theorem Number, 239 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H H + HHH HH H + HH H H HH + H HH H H 3+ H H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 19 17 16 15 14 13 12 11 10 9 8 (x - 3 x - 3 x + x + 6 x + 5 x + 5 x - 4 x - 7 x - 12 x - x 7 6 5 4 3 3 16 14 13 + x + 10 x + 2 x + 2 x - 4 x - x + 1) P(x) + (3 x - 6 x - 6 x 12 11 10 9 8 6 5 4 3 2 - 3 x + 7 x + 7 x + 16 x - x - 18 x - 2 x - 6 x + 10 x - x 2 13 11 10 9 8 7 6 5 + 3 x - 3) P(x) + (3 x - 3 x - 3 x - 6 x + 2 x - x + 10 x - x 4 3 2 10 6 5 4 3 2 + 6 x - 8 x + 2 x - 3 x + 3) P(x) + x - 2 x + x - 2 x + 2 x - x + x - 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 5, 8, 13, 29, 58, 109, 224, 469, 962, 2004, 4252, 9040, 19301, 41563, 89980, 195525, 426833, 935706, 2058178, 4541767, 10053231, 22314028, 49652620, 110747551, 247557767, 554494496, 1244340818, 2797369080, 6299090767, 14206237723, 32085826176, 72567625371, 164336472104, 372610625607, 845820085338, 1922098715982, 4372439993387, 9956343355891, 22692518763045, 51766984724848, 118193127925529, 270074678950222, 617606728681446, 1413389735165284, 3236829392842055, 7417758864228774, 17010163891685398, 39031546594762198, 89615630050729955, 205874421628959923, 473218932824758023, 1088313251325592598, 2504201652272691910, 5765011816121712227, 13278193824888813318, 30596973864393382114, 70536151123238024372, 162679316350017450709, 375347930528973340133, 866385011098516035066, 2000587011422283874074, 4621343255190585868302, 10679186807475470377805, 24686666622984511599253, 57086890984494449675458, 132055217721375981546693, 305573531184986377077384, 707314945608209847639832, 1637732567993388063478190, 3793171352332367061541348, 8787952385570629968379636, 20365508673681019718977166, 47208682141769560467266124, 109462253636911869115750581, 253874894309395675440725721, 588959015626156096858744585, 1366650733136206654486516020, 3172009302049365401821933578, 7363990486154164923351386224, 17099812077001249363177056269, 39716082047248874128446242438, 92264807603841023580831955900, 214386848828249772877195858443, 498253569181261894314960926934, 1158219376625202957075370231454, 2692882159291120533526978907126, 6262215947461674813212714459568, 14565353079098544379221773377774, 33883983435461603108197061482155, 78839999338725373583734755178870, 183474523257584763789771107474471, 427051528522691727246479996839525, 994165194829588457736112300985513, 2314776392397718522798745297649492] ------------------------------------------------------------ Theorem Number, 240 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + HH H + H HH + HH H 4+ H H + H HH + HH H + H HH 3+ H H H + HH H H HH + H HH HH H + HH H H HH 2+ H H H H + H HH HH H + HH H H HH + H HH HH H 1+ H H H H + HH H H HH + H H HH H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-*+-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 13 12 11 10 9 8 7 6 3 2 (x + x - 2 x - 4 x - 3 x + 3 x + 4 x + 6 x - 3 x - x + 1) P(x) 10 9 8 7 6 5 3 2 7 + (2 x + 2 x - 2 x - 4 x - 6 x + x + 5 x - x + 2 x - 2) P(x) + x 6 3 2 + x - 2 x + x - x + 1 = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 6, 10, 19, 33, 58, 108, 200, 369, 693, 1317, 2514, 4829, 9344, 18189, 35592, 69989, 138246, 274200, 545940, 1090783, 2186343, 4395255, 8860180, 17906170, 36272959, 73639706, 149804622, 305325432, 623406188, 1274971376, 2611597375, 5357319501, 11004919758, 22635485192, 46614898206, 96108678206, 198370269901, 409866012138, 847687251598, 1754836188328, 3636009242250, 7540193770668, 15649198235420, 32504074898770, 67562356231747, 140533256797216, 292514736909839, 609254134880606, 1269754726390174, 2647896576486016, 5524994718069158, 11534607248785519, 24093773197898278, 50353464184534819, 105285448864725132, 220249118224532352, 460956131234061995, 965156511973390642, 2021725152698343735, 4236688329900954289, 8881883438230807832, 18627397182916201898, 39080715060247020314, 82022115622422152747, 172207762570687826247, 361678877010846007041, 759867396916920746124, 1596954324667818296589, 3357245924609224890483, 7060019872578142781287, 14851047729721395149780, 31248783492263029171750, 65770414878714109406951, 138466981593815657698333, 291592875976612309859386, 614213871210552443656780, 1294110799316931990023822, 2727279498663073507672105, 5748989295111391492048961, 12121446207030768286650281, 25563244831836776909695275, 53922960715096892884159306, 113769377614144303450467979, 240087037262751143517465894, 506759120361514589789712132, 1069847544922340934122333579, 2259059523710516236809422262, 4771083098316998636523021295, 10078315424868890475447526836, 21293093745583655561384647024, 44995356141694588627315053710, 95098362701380027278219259613, 201026434433379944826868248653] ------------------------------------------------------------ Theorem Number, 241 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H H H H + HHH HHH HHH HH H + HH H HH H HH H H HH + H HH H HH H HH H H 3+ H H H H H + HH HH + H H + HH HH 2+ H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 5 2 2 2 1 + (x - x + 1) P(x) + (x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence 2 (2 n - 7) a(n - 1) 2 (2 n - 7) a(n - 3) a(n) = -------------------- + a(n - 2) - -------------------- - a(n - 5) n - 2 n - 2 2 (2 n - 7) a(n - 6) + -------------------- n - 2 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 2, a(6) = 6] Just for fun, a(1000), equals 3422663465320247263787592109141124468865532061464070704066684132785485227647\ 939805874779430835062333855171641584259581077590686505479346665061038797\ 265714766247988442470318375758813360907912625076779274375374065376278251\ 487484647537672713429602395700511635352456217160831480712452269906007165\ 884380185895725231777751122805958097246555695055652754187249302737427930\ 343929352123156195613442524923025553453213101779081413927976856585717838\ 635930093751427490146968712654482862978806983139361363120347999397404083\ 664648332196994851667052633892263023687661049326883911980789348279461698\ 5587252996313030 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 2, 6, 16, 47, 147, 474, 1571, 5320, 18320, 63959, 225858, 805288, 2894978, 10481813, 38188689, 139900745, 515022101, 1904268957, 7068660708, 26332347288, 98411323603, 368876938837, 1386411201970, 5223754679581, 19727431926834, 74658894271982, 283106506738357, 1075514699721380, 4092869254151080, 15600423307021455, 59552275236927296, 227653447212526208, 871420802686936680, 3339846072865676390, 12815618765240368717, 49231098122250491458, 189321828105034543909, 728785049049539355168, 2808109024308660976339, 10829888021922197120471, 41803307838090854444121, 161494045398902514237262, 624374642382038008176993, 2415803790451841926514643, 9353873465893289201102842, 36242774239804797958500012, 140520277662663752164857480, 545170644777039898004931591, 2116366131628372563862117293, 8220602441825462994034644905, 31949290303161193716046034793, 124237953619945233492423633305, 483363837686337625265650612626, 1881536204097135731921041391984, 7327600163089385659821274432033, 28550539770568826442450216269855, 111291822498591744459827169610808, 434012171060459474547828009024341, 1693261250890614651414678907257720, 6608812529482225739375087581808644, 25804448454165672650683687237261177, 100793371265755613782850645358577961, 393849606158921872655144901941783786, 1539518683165784668003925860695185591, 6019922378189752381909685520209120242, 23547439256668136458711794787514874364, 92138053005884282494312475538551650781, 360639092160725190142840443340631715278, 1412020642743000768316250726868086009790, 5530189094428640997179870951815488311386, 21665363593195732494786537500335212161126, 84901540406500134555893796108626205545105, 332801602812138328442179988575803714473088, 1304885174194135945616338267389720250326335, 5117680344081013598773386326018852687186222, 20076362537410824788332913008976930231415249, 78778022415206971923907644384557555720976717, 309193642109863585914174579049891893781334391, 1213832715188349107909924183121835292138264322, 4766366011754103054778642490230006224593802909, 18720341368267715117856731788028513998303083353, 73542029122119090790742786880514095811858032292, 288968598765552969560585020498147188525193029402, 1135681899329204577655869520740868719242823172490, 4464282306310323601020502206178824640729350540813, 17552276392626363659607157749715726348612110484077, 69024000862951118880726396905444492007599027769521, 271487466097861607214000013508059226246634289481275, 1068022919763976696232469777921245487650817294547831, 4202335417375555969030833084065229911379936618964262, 16537828465863239739701189583949872993316136311899354, 65094187951853020601612670267989434311342066498632741, 256259686517842383301402966193918536773350329510647679, 1008999963009517251757986070945007785660922007438988960, 3973500555506421212896242674489099966571477090270377717, 15650391490673229449035162728018090674380560822026844706] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 242 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ H HH HH + HH H HH HH + H HH H H + HH H HH HH 4+ H HH H H + H HH H H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 3 2 2 3 2 1 + (x + x - x - x + 1) P(x) + (x + x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 7) a(n - 1) (4 n - 17) a(n - 2) (n - 5) a(n - 3) a(n) = ------------------ + ------------------- - ---------------- n - 2 n - 2 n - 2 (5 n - 22) a(n - 4) (4 n - 17) a(n - 5) (n - 5) a(n - 6) - ------------------- - ------------------- + ---------------- n - 2 n - 2 n - 2 (5 n - 22) a(n - 7) 3 (n - 5) a(n - 8) + ------------------- + ------------------ n - 2 n - 2 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 3, a(7) = 6, a(8) = 12] Just for fun, a(1000), equals 8860862023292075968276277514714248593198929839769380856771491476834645411034\ 971894866292077857079552961537030062908551251261647144382496853687782821\ 054184668235733438566888858551955851111944135135787839723906153442696991\ 263785015057838776665886398587178566742546567522735087401082955419046964\ 387465387179652258465254660160763488653122285490983717702179160837248539\ 693902000749455565246146162113746561726289619446359962920995865101594027\ 98072823532463236599122791287290686 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 3, 6, 12, 28, 67, 163, 409, 1047, 2720, 7159, 19048, 51142, 138385, 376995, 1033115, 2845952, 7876302, 21888824, 61058893, 170902575, 479831873, 1350999629, 3813697119, 10791286286, 30602553756, 86962021759, 247586286749, 706142663596, 2017328796314, 5772096786694, 16539476231102, 47457342050583, 136346550591721, 392204416291603, 1129482421173659, 3256256113386519, 9397345776928842, 27146683687948823, 78493250837353534, 227160260729254220, 657959005719580783, 1907282045837204115, 5533054162491377407, 16063278227550962830, 46666904076383085772, 135667712558607666872, 394661414446836273133, 1148794020914769645815, 3345941551324715625346, 9750866064647774999374, 28431988455483862564098, 82947166290697018848452, 242112780921723361735501, 707046117556083065099702, 2065780034773677339223036, 6038367709611593213853662, 17658238896863824381246792, 51660799040420898435104415, 151200983620991436647323170, 442712766405318283383365368, 1296754766161140895530674505, 3799765063841219444556153664, 11138165760379116840837129933, 32660575398596894269004630395, 95803772582109903302659971705, 281116036367240287000597160307, 825141933187631607642170430582, 2422744979523191074317651806222, 7115720853948264721550464469138, 20905404259902466570682178134101, 61436033484557854416515787554557, 180596489848062297043580473736583, 531023478945026185132223045188242, 1561828335230849070577015587599862, 4594783181095102005982648845629943, 13520912751397068669158402972820535, 39797288428151734415529516833294183, 117166853197429318351440194184640001, 345030340472240118667348975322651831, 1016268735970876894277016612352986956, 2994030447810228732421196508522801508, 8822627125900727689992098802555040065, 26003480448270289308764833189852136625, 76657514628388286510106161064786323442, 226029756946856262771695312945809615724, 666595122574573979910207473435766415800, 1966266336978008980717249705195918553528, 5801020343999533654167963129230758635937, 17117741362043104820828161774821249467025, 50520403804121740187565594126636709015553, 149129567444169474699296647095777703816561, 440286794637589018357948150453173148034809, 1300112504445214435657315221574732403366232, 3839706689852153907055350498280218948390511, 11341892306305967810005135303941131373501596] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 243 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ H HH HH + HH H HH HH + H HH H H + HH HHH HH 4+ HH H H H + H HH H H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 6 5 3 2 3 6 5 3 2 2 1 + (x - 2 x - x + x + x - 1) P(x) + (2 x + x - 2 x - 2 x + 3) P(x) 3 2 + (x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 5, 11, 28, 76, 211, 603, 1765, 5264, 15937, 48839, 151202, 472225, 1485993, 4706844, 14994745, 48013004, 154436696, 498783089, 1616850496, 5258677510, 17155495909, 56122636054, 184069825882, 605132075407, 1993718666131, 6581959257798, 21770219454604, 72132609144399, 239393313735202, 795717620574009, 2648693476182953, 8828624078450717, 29465173088811459, 98457437552634323, 329369272090148464, 1103027614595877015, 3697721671613957100, 12408068651321649605, 41674994654944877763, 140097075013073845341, 471353613687260606177, 1587128931879860902479, 5348224467495188034491, 18035349882379680910115, 60861612575446205158642, 205519739753068972678883, 694453035266610905992136, 2348011630987864114638090, 7943557937841427302485261, 26889170203032263300230997, 91070528550580309538274714, 308608282745862412163668173, 1046304728325015826334785857, 3549128824657861954759780787, 12044556132054484371730517178, 40893867252310281572689219400, 138904770482112069033827176260, 472021002594407564000257953774, 1604665713022310408351788221724, 5457341266348644083436776116764, 18567157426246872236210550366105, 63193474271403094358924753152457, 215157522081650694011677394062788, 732813623056840252859387665367371, 2496770705583635702188829698635748, 8509568535300028664477282084567852, 29011888168716459915175295573805305, 98941849113584962386331971475792390, 337532653082144472910275106796152219, 1151806792788709943749157992804603915, 3931588267430959334859999826417811290, 13423865794966053710448317354049404630, 45846386205274160315105968060094314811, 156620078987872854793233932094448257435, 535182127030444009092860857528364660887, 1829214861034953381096946003245244760703, 6253656635830852640527004901838412684886, 21384882714487230010667560715941555639300, 73144332311382758073193270954887001912118, 250237841171995850710686988931368133141850, 856291055935496524320477080966931768867563, 2930782940632295758122099872732927105983101, 10033154805261385069977353492775158958494793, 34354279676122797433331027757543454548418875, 117655326756676047434274855936575922349640998, 403021081540695451691200672776973956592503536, 1380789361838233623157088485298286083558419854, 4731607803480864352462590028891014127013768721, 16216976478813348552294470551221507198158908641, 55591595066663998204221299846137505201070752220, 190600836132803636720816502467271281512353886715, 653604834893454714182973008142627835759411284024, 2241707379109917718335468588579714695453663057368, 7689786265286437722107208751585796001746302957504, 26382735148028890878132386512999786745614107129861] ------------------------------------------------------------ Theorem Number, 244 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 6 5 3 2 3 1 + (x - 3 x - x + 2 x + x - 1) P(x) 6 5 3 2 2 3 2 + (3 x + x - 4 x - 2 x + 3) P(x) + (2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 24, 50, 100, 207, 447, 965, 2097, 4641, 10355, 23208, 52434, 119238, 272302, 624715, 1439586, 3329035, 7723353, 17974332, 41946862, 98136543, 230136785, 540859834, 1273645217, 3004841530, 7101517437, 16810654036, 39854525348, 94621732620, 224950428542, 535465514846, 1276128784737, 3044720948720, 7272175007841, 17386856915967, 41609801199087, 99670684111928, 238955727084254, 573360774386420, 1376837089630860, 3308764126191171, 7957256958283794, 19149694328506228, 46115679249370442, 111125023198040762, 267941826636175204, 646433889341639098, 1560460708781195478, 3768924211315976522, 9107707171651752850, 22020112055371883983, 53264882178765929245, 128903877018767159586, 312095509354848105514, 755960176581781422429, 1831866563301055461943, 4440851470520127715854, 10769869026085811825291, 26128886252691178913799, 63415069809483920442764, 153964356091782976724423, 373937747991975936017966, 908500784048788194813896, 2207973525844179375897501, 5367855788092394722780239, 13053960160887471570162186, 31755167086049496587497855, 77270472173378202155837143, 188077261542308295134223872, 457909065550704597273291014, 1115165194681535923031110438, 2716521249676194665104160483, 6619085378281378272215833397, 16132103128989793305097961362, 39326870592610533254429937069, 95893803896808211685222661778, 233879375049458391590317811366, 570546544663074707446614357597, 1392148793303975476592593878598, 3397608858830076013984401770568, 8293771449881197142382854212644, 20249746445904137513663328249672, 49450867782510794412659596235458, 120785019511666197733598404565551, 295076860334680412091808688807665, 721005064284596647207054443970788, 1762060340462307761691050403140135, 4307058934161988158319017861316542, 10529718506027909698423650877921477, 25747021917792169515323378704085982, 62966561145792275331841952840077732, 154015405894044114004769528686025159] ------------------------------------------------------------ Theorem Number, 245 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H H + HH H H HH + H HH HH H + HH H H HH 4+ H H H H H + HHH H HH HH H + HH H HH H H HH + H HH H HH H H 3+ H H H H + HH HH + H H + HH HH 2+ H H + H H + HH HH + H H 1+ H H + HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 6 5 3 2 2 3 2 1 + (x + x - x - x + 1) P(x) + (x + x - 2) P(x) = 0 The sequence a(n) satisfies the linear recurrence (2 n - 7) a(n - 1) (4 n - 17) a(n - 2) (n - 5) a(n - 3) a(n) = ------------------ + ------------------- - ---------------- n - 2 n - 2 n - 2 (5 n - 22) a(n - 4) (4 n - 17) a(n - 5) (n - 5) a(n - 6) - ------------------- - ------------------- + ---------------- n - 2 n - 2 n - 2 (5 n - 22) a(n - 7) 3 (n - 5) a(n - 8) + ------------------- + ------------------ n - 2 n - 2 subject to the initial conditions [a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 3, a(7) = 6, a(8) = 12] Just for fun, a(1000), equals 8860862023292075968276277514714248593198929839769380856771491476834645411034\ 971894866292077857079552961537030062908551251261647144382496853687782821\ 054184668235733438566888858551955851111944135135787839723906153442696991\ 263785015057838776665886398587178566742546567522735087401082955419046964\ 387465387179652258465254660160763488653122285490983717702179160837248539\ 693902000749455565246146162113746561726289619446359962920995865101594027\ 98072823532463236599122791287290686 For the sake of the OEIS here are the first, 100, terms. [1, 0, 0, 1, 1, 1, 3, 6, 12, 28, 67, 163, 409, 1047, 2720, 7159, 19048, 51142, 138385, 376995, 1033115, 2845952, 7876302, 21888824, 61058893, 170902575, 479831873, 1350999629, 3813697119, 10791286286, 30602553756, 86962021759, 247586286749, 706142663596, 2017328796314, 5772096786694, 16539476231102, 47457342050583, 136346550591721, 392204416291603, 1129482421173659, 3256256113386519, 9397345776928842, 27146683687948823, 78493250837353534, 227160260729254220, 657959005719580783, 1907282045837204115, 5533054162491377407, 16063278227550962830, 46666904076383085772, 135667712558607666872, 394661414446836273133, 1148794020914769645815, 3345941551324715625346, 9750866064647774999374, 28431988455483862564098, 82947166290697018848452, 242112780921723361735501, 707046117556083065099702, 2065780034773677339223036, 6038367709611593213853662, 17658238896863824381246792, 51660799040420898435104415, 151200983620991436647323170, 442712766405318283383365368, 1296754766161140895530674505, 3799765063841219444556153664, 11138165760379116840837129933, 32660575398596894269004630395, 95803772582109903302659971705, 281116036367240287000597160307, 825141933187631607642170430582, 2422744979523191074317651806222, 7115720853948264721550464469138, 20905404259902466570682178134101, 61436033484557854416515787554557, 180596489848062297043580473736583, 531023478945026185132223045188242, 1561828335230849070577015587599862, 4594783181095102005982648845629943, 13520912751397068669158402972820535, 39797288428151734415529516833294183, 117166853197429318351440194184640001, 345030340472240118667348975322651831, 1016268735970876894277016612352986956, 2994030447810228732421196508522801508, 8822627125900727689992098802555040065, 26003480448270289308764833189852136625, 76657514628388286510106161064786323442, 226029756946856262771695312945809615724, 666595122574573979910207473435766415800, 1966266336978008980717249705195918553528, 5801020343999533654167963129230758635937, 17117741362043104820828161774821249467025, 50520403804121740187565594126636709015553, 149129567444169474699296647095777703816561, 440286794637589018357948150453173148034809, 1300112504445214435657315221574732403366232, 3839706689852153907055350498280218948390511, 11341892306305967810005135303941131373501596] ------------------------------------------------------------ ------------------------------------------------------------ Theorem Number, 246 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 7 6 5 4 3 2 2 4 3 2 1 + (x + x + x - x - x - x + 1) P(x) + (x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 13, 27, 57, 122, 269, 605, 1379, 3182, 7420, 17453, 41357, 98634, 236566, 570211, 1380506, 3355528, 8185280, 20031389, 49166375, 121002963, 298537692, 738233460, 1829385845, 4542215715, 11298528028, 28152340251, 70258317672, 175601508371, 439507839770, 1101481096871, 2763928226505, 6943617188432, 17463308610473, 43966681055061, 110803591314883, 279508718029179, 705713141395523, 1783338434049627, 4510189067496665, 11415502088207975, 28914756126315351, 73291605054009739, 185902808224083404, 471848246294267687, 1198372494643665002, 3045394236766909042, 7743675807786471786, 19701218942356810935, 50150147311769770266, 127724997572390171097, 325458808262111252105, 829707261507611876254, 2116192588906284015186, 5399828958003920513740, 13784555460579860793317, 35203612522652740364321, 89940949867922712140206, 229878222115467330486132, 587763931062371858642318, 1503376118181573715492116, 3846689074287382715336194, 9845926103531341883449308, 25209930548939165325681225, 64569593220925809983353741, 165432828463614467696091458, 423983228232161138542127417, 1086939209804757271772782924, 2787326502503975028357540702, 7149785357374383356755014728, 18344988147682731284390067244, 47082332510245985594850527490, 120868063607826687877377441737, 310366824810174497951080379711, 797161550726939334098105323361, 2047962538520677057918137011882, 5262591774393682820710492185268, 13526232782451771125033589219572, 34773716938692040062180157785472, 89417006035041206300195848318547, 229975535400889609582190376440923, 591607252505314635177965446303011, 1522206540138937137410957478162416, 3917417870346613394306707106257430, 10083481058857910286087790951922170, 25959926405844391913481266197640325, 66846240398637627465766894999925009, 172158833514831085495533759750852249, 443464362514862318990873457216395456, 1142519379979763280935871522685254468, 2944030220631010667545133900797281562, 7587404552052090177381410902098091913, 19557575074829519830141191357108835960] ------------------------------------------------------------ Theorem Number, 247 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 4 3 2 1 + (x + x + x + x - x - x - x + 1) P(x) + (x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 9, 20, 43, 94, 216, 505, 1198, 2888, 7046, 17359, 43137, 107986, 272046, 689198, 1754652, 4486913, 11519069, 29677863, 76709558, 198858270, 516900481, 1346928260, 3517812058, 9206982999, 24144187305, 63430675750, 166926209959, 439987494362, 1161457799001, 3070255463861, 8126765806794, 21537753472145, 57146877005927, 151798160546375, 403642671170599, 1074387128983240, 2862437766174048, 7633132414976498, 20372410447459630, 54417377540703801, 145469599771072044, 389162759164454635, 1041839048797994230, 2791046978146480666, 7482013275489230337, 20069799089461878199, 53867882389413538721, 144666793314662499979, 388731932228847260560, 1045116905170070363120, 2811280699994517254169, 7565891179843013798830, 20371584272556101353559, 54877087351154823046446, 147894461371360500600404, 398750098993789333077497, 1075552191594669068647971, 2902271444012572374471826, 7834562869658274177153008, 21157109326212823185371492, 57155440767212455161258473, 154459142393974795917404387, 417560856723240437282759617, 1129201890148987098172382917, 3054673524396563025177593345, 8265998275897394297578700464, 22374802833318826872122652538, 60583279272141959914982521230, 164086330649285086653717627671, 444543898955742904386559063447, 1204692560613252897219505954158, 3265532089955788106865668972679, 8854107516472847109120249657859, 24012966115006271427327717275224, 65140965498791000199417379317401, 176753178774204821202519803270294, 479713926425134540506383824683065, 1302257781851035311627912843109992, 3535971195076524443101782142887928, 9603184357286987364921458068688114, 26086412423311750697378497706485168, 70876753808269982532179552022657879, 192611209804903641890607349672943792, 523534801206376171916739816450584025, 1423291500980846384191200772841684210, 3870121247506880066064768564504438295, 10525333776394036585049608115707480862, 28630307203934076060493508135753907692, 77892072069382575356484169897529778150, 211951244648000788109732632639813482287, 576836257741440102366311224036503009129, 1570151092397116337614826283811806160336, 4274655842190291606027844557863134575971] ------------------------------------------------------------ Theorem Number, 248 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + H H 5+ HH HH H + HH H H HH + H HH HH H + HH H H HH 4+ H HH HH H + H HH HH H + HH HH H HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 1 + (x + 2 x + x + x - 2 x - x - x + 1) P(x) 4 3 2 + (2 x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 5, 9, 15, 27, 52, 99, 191, 380, 767, 1561, 3212, 6676, 13980, 29462, 62457, 133095, 284904, 612325, 1320835, 2858528, 6204794, 13504896, 29467046, 64443065, 141232079, 310127824, 682241562, 1503390100, 3318125585, 7334286838, 16234066006, 35980324422, 79843282903, 177385141221, 394524626312, 878382600262, 1957594535824, 4366863711975, 9749996633366, 21787566703548, 48726502251068, 109058113252455, 244271064473613, 547511721413636, 1228030388231890, 2756175501803298, 6189779845643445, 13909237626615638, 31273831159967158, 70355639977282178, 158360839466453771, 356631087517226215, 803535502075715446, 1811330859313674816, 4084981786161499396, 9216695405000005915, 20803989056321610682, 46978386063151976380, 106126487292393541621, 239838068767767185118, 542220278033115896654, 1226285592006812766558, 2774347696927967271630, 6278832855959591570828, 14214818277513329553301, 32191694467833655881358, 72926011196323997616532, 165254507682930495490441, 374586998426594973124882, 849331174613651352346800, 1926295937143221477067260, 4370058634352064414375042, 9916689051600915661994716, 22509108319009393216650851, 51104501330215476319608970, 116055707226039146541380078, 263619574451311192836750119, 598949259458766324199929106, 1361134842968359225485601500, 3093916712876970178692847268, 7034125719967269865497972628, 15995707321136515987498854278, 36381989171910295666778485227, 82766964402142036623410315006, 188327268860574074593683978712, 428600872006951760458663393481, 975606498851851793659981066972, 2221141899698881971264987257058, 5057736078329066998909302286110, 11518941426334046103018614433700, 26238795102693120770431186038224, 59778984596015548358887856377059, 136215011845586736829636505040690] ------------------------------------------------------------ Theorem Number, 249 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 8+ HH + HH HH + HH HHH + HH H + HH HH 6+ HH HH + HH HH + HH HH + HH HH + HH HH 4+ HH HH + HH HH + HH HH + HHH HHH + HH HH 2+ HH HH + HH HH + HH HH + HHH HHH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 6 5 3 2 3 6 5 3 2 2 1 + (x - 2 x - x + x + x - 1) P(x) + (2 x + x - 2 x - 2 x + 3) P(x) 3 2 + (x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 5, 11, 28, 76, 211, 603, 1765, 5264, 15937, 48839, 151202, 472225, 1485993, 4706844, 14994745, 48013004, 154436696, 498783089, 1616850496, 5258677510, 17155495909, 56122636054, 184069825882, 605132075407, 1993718666131, 6581959257798, 21770219454604, 72132609144399, 239393313735202, 795717620574009, 2648693476182953, 8828624078450717, 29465173088811459, 98457437552634323, 329369272090148464, 1103027614595877015, 3697721671613957100, 12408068651321649605, 41674994654944877763, 140097075013073845341, 471353613687260606177, 1587128931879860902479, 5348224467495188034491, 18035349882379680910115, 60861612575446205158642, 205519739753068972678883, 694453035266610905992136, 2348011630987864114638090, 7943557937841427302485261, 26889170203032263300230997, 91070528550580309538274714, 308608282745862412163668173, 1046304728325015826334785857, 3549128824657861954759780787, 12044556132054484371730517178, 40893867252310281572689219400, 138904770482112069033827176260, 472021002594407564000257953774, 1604665713022310408351788221724, 5457341266348644083436776116764, 18567157426246872236210550366105, 63193474271403094358924753152457, 215157522081650694011677394062788, 732813623056840252859387665367371, 2496770705583635702188829698635748, 8509568535300028664477282084567852, 29011888168716459915175295573805305, 98941849113584962386331971475792390, 337532653082144472910275106796152219, 1151806792788709943749157992804603915, 3931588267430959334859999826417811290, 13423865794966053710448317354049404630, 45846386205274160315105968060094314811, 156620078987872854793233932094448257435, 535182127030444009092860857528364660887, 1829214861034953381096946003245244760703, 6253656635830852640527004901838412684886, 21384882714487230010667560715941555639300, 73144332311382758073193270954887001912118, 250237841171995850710686988931368133141850, 856291055935496524320477080966931768867563, 2930782940632295758122099872732927105983101, 10033154805261385069977353492775158958494793, 34354279676122797433331027757543454548418875, 117655326756676047434274855936575922349640998, 403021081540695451691200672776973956592503536, 1380789361838233623157088485298286083558419854, 4731607803480864352462590028891014127013768721, 16216976478813348552294470551221507198158908641, 55591595066663998204221299846137505201070752220, 190600836132803636720816502467271281512353886715, 653604834893454714182973008142627835759411284024, 2241707379109917718335468588579714695453663057368, 7689786265286437722107208751585796001746302957504, 26382735148028890878132386512999786745614107129861] ------------------------------------------------------------ Theorem Number, 250 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 7+ HH + H HH + HH HH 6+ H HH + H HH + HH HH 5+ HH HH + H HH + HH HH 4+ H H HH + HH HH HH HH 3+ HH HHH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 4 3 2 1 + (x + x + x + x - x - x - x + 1) P(x) + (x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 3, 5, 9, 20, 43, 94, 216, 505, 1198, 2888, 7046, 17359, 43137, 107986, 272046, 689198, 1754652, 4486913, 11519069, 29677863, 76709558, 198858270, 516900481, 1346928260, 3517812058, 9206982999, 24144187305, 63430675750, 166926209959, 439987494362, 1161457799001, 3070255463861, 8126765806794, 21537753472145, 57146877005927, 151798160546375, 403642671170599, 1074387128983240, 2862437766174048, 7633132414976498, 20372410447459630, 54417377540703801, 145469599771072044, 389162759164454635, 1041839048797994230, 2791046978146480666, 7482013275489230337, 20069799089461878199, 53867882389413538721, 144666793314662499979, 388731932228847260560, 1045116905170070363120, 2811280699994517254169, 7565891179843013798830, 20371584272556101353559, 54877087351154823046446, 147894461371360500600404, 398750098993789333077497, 1075552191594669068647971, 2902271444012572374471826, 7834562869658274177153008, 21157109326212823185371492, 57155440767212455161258473, 154459142393974795917404387, 417560856723240437282759617, 1129201890148987098172382917, 3054673524396563025177593345, 8265998275897394297578700464, 22374802833318826872122652538, 60583279272141959914982521230, 164086330649285086653717627671, 444543898955742904386559063447, 1204692560613252897219505954158, 3265532089955788106865668972679, 8854107516472847109120249657859, 24012966115006271427327717275224, 65140965498791000199417379317401, 176753178774204821202519803270294, 479713926425134540506383824683065, 1302257781851035311627912843109992, 3535971195076524443101782142887928, 9603184357286987364921458068688114, 26086412423311750697378497706485168, 70876753808269982532179552022657879, 192611209804903641890607349672943792, 523534801206376171916739816450584025, 1423291500980846384191200772841684210, 3870121247506880066064768564504438295, 10525333776394036585049608115707480862, 28630307203934076060493508135753907692, 77892072069382575356484169897529778150, 211951244648000788109732632639813482287, 576836257741440102366311224036503009129, 1570151092397116337614826283811806160336, 4274655842190291606027844557863134575971] ------------------------------------------------------------ Theorem Number, 251 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 7+ HHH + H HH + HH HH 6+ H H + H H + HH HH 5+ HH HH + H H + HH HH 4+ HH H H + HH HH HH HH 3+ HH HHH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 16 15 14 13 12 11 10 9 8 7 1 + (x - x + 5 x - 4 x + 9 x - 10 x + 11 x - 15 x + 10 x - 9 x 6 5 4 3 2 4 13 12 11 + 11 x - 3 x + 3 x - 4 x + x - x + 1) P(x) + (x - 2 x + 9 x 10 9 8 7 6 5 4 3 2 - 9 x + 17 x - 16 x + 16 x - 24 x + 7 x - 8 x + 13 x - 3 x + 3 x 3 - 4) P(x) 9 8 7 6 5 4 3 2 2 + (-x + 6 x - 7 x + 13 x - 5 x + 7 x - 14 x + 3 x - 3 x + 6) P(x) 5 4 3 2 + (x - 2 x + 5 x - x + x - 4) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 2, 4, 8, 19, 45, 109, 274, 706, 1853, 4935, 13310, 36294, 99887, 277080, 773890, 2174564, 6142994, 17435967, 49699752, 142207256, 408311561, 1176056410, 3397136562, 9838815780, 28564499121, 83115843915, 242349891944, 708010477632, 2072130084832, 6074676800886, 17836576344636, 52449296429420, 154443347079891, 455370624288193, 1344296224042833, 3973101538881707, 11755524179079394, 34818258116193514, 103229008060692388, 306340787185880424, 909905907985941303, 2704944605526137569, 8047730208573057475, 23962142966274349035, 71400333638264115197, 212903724121787138250, 635274875582057982342, 1896807634533229819937, 5667044361896249801975, 16941461018536018552034, 50675247253133035903840, 151663852459302068188567, 454151178363641840946220, 1360637463611326749125506, 4078495115483496523670002, 12231095766329072899789213, 36697074224144886683988623, 110151723217308796647530048, 330779364211208261048905057, 993725827356903839035757398, 2986550802827264657190871788, 8979306222659152141008830522, 27007218475295484991166837452, 81259853248056922960743254955, 244583100660660153829137673619, 736421486674089753064633215785, 2218051427528819913261839518920, 6682787989701003052006455028430, 20140980458431291139609889355048, 60720662190831894650419213659464, 183114035391954655906522002346728, 552373000705795000525835783987615, 1666730964015327379511768432941102, 5030573883836617758315970625139016, 15187467902598696614512679454444238, 45863373600840634895077218659510962, 138534037449538073999509058682449166, 418556465258795426215264715221405589, 1264899413596602083001491538207902240, 3823487778569295944720069873113936211, 11560129540993990762501709865572681491, 34959286970484081744420684201522549420, 105744303499391353349596937150871992058, 319921705303491560691981225089334416683, 968100804279936279532054055475556884777, 2930120764108356154566020492379108770732, 8870263262264521475883094855050261705051, 26857872251900514412080420502807689627855, 81337144677435763254775297744856216182180, 246369333028750642755806698872490495383761, 746385203567840868272929012734727819752572, 2261602845499645031050459062952094686254883, 6854013617806049203504746064284314278331466, 20775299351072062541291263978452724295707271, 62982773259528325110667531595783021832436437, 190970785698740806851338590811851515473877661] ------------------------------------------------------------ Theorem Number, 252 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 7+ HH + H HH + HH HH 6+ H HH + H HH + HH HH 5+ HH HH + H HH + HH HH 4+ H H HH + HH HH HH HH 3+ HH HHH HH + HH HH + HH HH 2+ HH HH + HH HH + H H 1+ HH HH + HH HH + HH HH -**+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+**- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 13 12 11 10 9 8 7 6 4 3 2 1 + (x + x + 2 x + x + x - 3 x - x - 4 x + x + 2 x + x - 1) 3 9 8 7 6 5 4 3 2 2 P(x) + (x + 3 x + x + 4 x - x - 2 x - 4 x - 2 x + 3) P(x) 5 4 3 2 + (x + x + 2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 6, 10, 20, 38, 71, 141, 287, 583, 1202, 2521, 5325, 11323, 24282, 52406, 113664, 247757, 542480, 1192313, 2629716, 5818748, 12912436, 28729813, 64079490, 143246101, 320883803, 720195496, 1619321024, 3647055270, 8226810087, 18584803194, 42041964361, 95229445722, 215968599608, 490356105984, 1114569328616, 2536017260869, 5775957109341, 13167432219753, 30044425752651, 68611060035416, 156810360349206, 358665774457819, 820965478557937, 1880467407759439, 4310223033012889, 9885862452845120, 22688139458714478, 52100512493277179, 119710934944822697, 275210416296780580, 633032733279400641, 1456831279717572075, 3354332676055975953, 7726969805373657763, 17807843238326490566, 41058742911104314735, 94707794486141835055, 218547278951873016885, 504520534310020995281, 1165146383227358880327, 2691813944679198933254, 6221103655060471930962, 14382781044804077557878, 33263407167250290261739, 76954607918587159600611, 178091158645065231338585, 412273954575974881103440, 954687857089931041054750, 2211389126498755280117222, 5123817785575377937814225, 11875270986534273727976053, 27530333413891900687626772, 63840220572849222278104788, 148077560084497124425798169, 343552566984827422999062338, 797266374037095566105098803, 1850620324705013667862682500, 4296672645787599932254592314, 9978052601349627898050764057, 23176908557720826553542054224, 53846703494421872456717997242, 125127973383987991144787652593, 290830021503801381930721105937, 676100921983193825969329450941, 1572060634527419283928666538120, 3656037156311472113594424613186, 8504202409975272430583173001536, 19785018164064980155520108020417, 46038106439011476416261131849967, 107145737836783124653205894881028, 249406152369906834599933074660180, 580647632830318254723094356140953, 1352040998078175397319979202051475] ------------------------------------------------------------ Theorem Number, 253 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 6+ H + HHHH + HH H 5+ H HH HH + HH H HH HH + H HH H H + HH H HH HH 4+ H HH H H + H HH H H + HH HH HH HH 3+ H H H + HH HH + HH HH 2+ HH HH + H H + HH HH + H H 1+ HH HH + H H +HH HH -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 6 5 3 2 3 1 + (x - 3 x - x + 2 x + x - 1) P(x) 6 5 3 2 2 3 2 + (3 x + x - 4 x - 2 x + 3) P(x) + (2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 24, 50, 100, 207, 447, 965, 2097, 4641, 10355, 23208, 52434, 119238, 272302, 624715, 1439586, 3329035, 7723353, 17974332, 41946862, 98136543, 230136785, 540859834, 1273645217, 3004841530, 7101517437, 16810654036, 39854525348, 94621732620, 224950428542, 535465514846, 1276128784737, 3044720948720, 7272175007841, 17386856915967, 41609801199087, 99670684111928, 238955727084254, 573360774386420, 1376837089630860, 3308764126191171, 7957256958283794, 19149694328506228, 46115679249370442, 111125023198040762, 267941826636175204, 646433889341639098, 1560460708781195478, 3768924211315976522, 9107707171651752850, 22020112055371883983, 53264882178765929245, 128903877018767159586, 312095509354848105514, 755960176581781422429, 1831866563301055461943, 4440851470520127715854, 10769869026085811825291, 26128886252691178913799, 63415069809483920442764, 153964356091782976724423, 373937747991975936017966, 908500784048788194813896, 2207973525844179375897501, 5367855788092394722780239, 13053960160887471570162186, 31755167086049496587497855, 77270472173378202155837143, 188077261542308295134223872, 457909065550704597273291014, 1115165194681535923031110438, 2716521249676194665104160483, 6619085378281378272215833397, 16132103128989793305097961362, 39326870592610533254429937069, 95893803896808211685222661778, 233879375049458391590317811366, 570546544663074707446614357597, 1392148793303975476592593878598, 3397608858830076013984401770568, 8293771449881197142382854212644, 20249746445904137513663328249672, 49450867782510794412659596235458, 120785019511666197733598404565551, 295076860334680412091808688807665, 721005064284596647207054443970788, 1762060340462307761691050403140135, 4307058934161988158319017861316542, 10529718506027909698423650877921477, 25747021917792169515323378704085982, 62966561145792275331841952840077732, 154015405894044114004769528686025159] ------------------------------------------------------------ Theorem Number, 254 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1} To make it crystal clear here is such a path of semi-length 8 5+ H + H HH + HH H + H HH 4+ H H + HH H + H HH + H H 3+ H H H + HH H HH HH + H HH H H + HH H HH HH 2+ H H H H + H HH H H + HH H HH HH + H HH H H 1+ H H H H + HH H HH HH + H HH H H +HH HHH HH -*-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-++-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 8 7 6 5 4 3 2 2 1 + (x + 2 x + x + x - 2 x - x - x + 1) P(x) 4 3 2 + (2 x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 5, 9, 15, 27, 52, 99, 191, 380, 767, 1561, 3212, 6676, 13980, 29462, 62457, 133095, 284904, 612325, 1320835, 2858528, 6204794, 13504896, 29467046, 64443065, 141232079, 310127824, 682241562, 1503390100, 3318125585, 7334286838, 16234066006, 35980324422, 79843282903, 177385141221, 394524626312, 878382600262, 1957594535824, 4366863711975, 9749996633366, 21787566703548, 48726502251068, 109058113252455, 244271064473613, 547511721413636, 1228030388231890, 2756175501803298, 6189779845643445, 13909237626615638, 31273831159967158, 70355639977282178, 158360839466453771, 356631087517226215, 803535502075715446, 1811330859313674816, 4084981786161499396, 9216695405000005915, 20803989056321610682, 46978386063151976380, 106126487292393541621, 239838068767767185118, 542220278033115896654, 1226285592006812766558, 2774347696927967271630, 6278832855959591570828, 14214818277513329553301, 32191694467833655881358, 72926011196323997616532, 165254507682930495490441, 374586998426594973124882, 849331174613651352346800, 1926295937143221477067260, 4370058634352064414375042, 9916689051600915661994716, 22509108319009393216650851, 51104501330215476319608970, 116055707226039146541380078, 263619574451311192836750119, 598949259458766324199929106, 1361134842968359225485601500, 3093916712876970178692847268, 7034125719967269865497972628, 15995707321136515987498854278, 36381989171910295666778485227, 82766964402142036623410315006, 188327268860574074593683978712, 428600872006951760458663393481, 975606498851851793659981066972, 2221141899698881971264987257058, 5057736078329066998909302286110, 11518941426334046103018614433700, 26238795102693120770431186038224, 59778984596015548358887856377059, 136215011845586736829636505040690] ------------------------------------------------------------ Theorem Number, 255 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} No downward-run can belong to, {2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 13 12 11 10 9 8 7 6 4 3 2 1 + (x + x + 2 x + x + x - 3 x - x - 4 x + x + 2 x + x - 1) 3 9 8 7 6 5 4 3 2 2 P(x) + (x + 3 x + x + 4 x - x - 2 x - 4 x - 2 x + 3) P(x) 5 4 3 2 + (x + x + 2 x + x - 3) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 4, 6, 10, 20, 38, 71, 141, 287, 583, 1202, 2521, 5325, 11323, 24282, 52406, 113664, 247757, 542480, 1192313, 2629716, 5818748, 12912436, 28729813, 64079490, 143246101, 320883803, 720195496, 1619321024, 3647055270, 8226810087, 18584803194, 42041964361, 95229445722, 215968599608, 490356105984, 1114569328616, 2536017260869, 5775957109341, 13167432219753, 30044425752651, 68611060035416, 156810360349206, 358665774457819, 820965478557937, 1880467407759439, 4310223033012889, 9885862452845120, 22688139458714478, 52100512493277179, 119710934944822697, 275210416296780580, 633032733279400641, 1456831279717572075, 3354332676055975953, 7726969805373657763, 17807843238326490566, 41058742911104314735, 94707794486141835055, 218547278951873016885, 504520534310020995281, 1165146383227358880327, 2691813944679198933254, 6221103655060471930962, 14382781044804077557878, 33263407167250290261739, 76954607918587159600611, 178091158645065231338585, 412273954575974881103440, 954687857089931041054750, 2211389126498755280117222, 5123817785575377937814225, 11875270986534273727976053, 27530333413891900687626772, 63840220572849222278104788, 148077560084497124425798169, 343552566984827422999062338, 797266374037095566105098803, 1850620324705013667862682500, 4296672645787599932254592314, 9978052601349627898050764057, 23176908557720826553542054224, 53846703494421872456717997242, 125127973383987991144787652593, 290830021503801381930721105937, 676100921983193825969329450941, 1572060634527419283928666538120, 3656037156311472113594424613186, 8504202409975272430583173001536, 19785018164064980155520108020417, 46038106439011476416261131849967, 107145737836783124653205894881028, 249406152369906834599933074660180, 580647632830318254723094356140953, 1352040998078175397319979202051475] ------------------------------------------------------------ Theorem Number, 256 Theorem: Let a(n) be the number of Dyck paths of semi-length n obeying the f\ ollowing restrictions The height of a peak can not belong to , {1, 2} The height of a valley can not belong to , {1, 2} No upward-run can belong to, {1, 2} No downward-run can belong to, {1, 2} To make it crystal clear here is such a path of semi-length 8 4+ H H + HH HH + HH HH HH HH + H H H H + H H H H 3+ H H H H + H H H H + H HH HH H + H H H H + H H H H 2+ H H H H + H H H H + H H H H + HH H HH HH + H H H H 1+ H H H H + H H H H + H H H H + HH HH HH HH +H HH H -*-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-+*-+-+-+-+-++-+-+-+-++-+-+-+-++-+-+-+-*- 2 4 6 8 10 12 14 16 The ordinary generating function of the sequence a(n), let's call it , P(x), satisfies the algebraic equation 9 8 7 6 4 3 2 2 1 + (x + 2 x + 2 x + x - 2 x - x - x + 1) P(x) 5 4 3 2 + (x + 2 x + x + x - 2) P(x) = 0 For the sake of the OEIS, here are the first, 100, terms [1, 0, 0, 1, 1, 1, 2, 3, 4, 7, 12, 19, 32, 57, 102, 184, 338, 631, 1193, 2278, 4386, 8512, 16640, 32727, 64698, 128497, 256288, 513091, 1030652, 2076551, 4195389, 8497633, 17251501, 35097685, 71545640, 146109780, 298889934, 612391857, 1256577012, 2581964630, 5312226040, 10942958613, 22568085609, 46593874760, 96296775298, 199214136675, 412508168198, 854927060291, 1773336983575, 3681307349391, 7647959447370, 15900359496772, 33080520927256, 68869800315583, 143471049030427, 299066191297844, 623774018719246, 1301769563196978, 2718180873152235, 5678728121019830, 11869812600023930, 24822684507644365, 51934768256973512, 108708894175429681, 227647361551092841, 476918760095371805, 999549561565926862, 2095736135278881599, 4395777411189686495, 9223512701958825628, 19360365984653297116, 40652059395072526234, 85388375869205205243, 179414611502079962724, 377099148730245402023, 792844459482558248785, 1667444254566230073271, 3507856998483416898418, 7381696118116215545446, 15537845884622259716287, 32714673959131873020032, 68898287436832289381615, 145139401140260625493636, 305823205702351606859661, 644556519941560063235675, 1358796405023640572966082, 2865154456516923094879448, 6042817833843456908071545, 12747540906403185592012971, 26897164309201694344507690, 56764595715167086292090332, 119822245028973861534159847, 252978817559414921672077464, 534214521684417321803286560, 1128314411252867525432280150, 2383557588085094296518911183, 5036170748773190613622743871, 10642723994868396804864305247, 22494743116136658770993174434, 47553615795218186780095160580, 100544600599431729835469608976] This concludes this exciting paper with its, 256, theorems that took, 14261.195, to generate. --------------------------------------------------