Enumerating Generalized Dyck paths and The Sum of Areas with alphabets consi\ sting of integers from, -2, to , 2 By Shalosh B. Ekhad ------------------------------------------------------------- Theorem Number, 1, Let a(n) be the number of generalized Dyck paths of lengt\ h n in the set of steps, {[1, -2], [1, 1]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 3 3 X(t) t - X(t) + 1 = 0 and in Maple notation X(t)^3*t^3-X(t)+1 = 0 Y(t),satisfies the algebraic equation 3 3 3 3 2 3 3 2 3 27 t + (81 t - 9) Y(t) + 9 t (27 t - 4) Y(t) + t (27 t - 4) Y(t) = 0 and in Maple notation 27*t^3+(81*t^3-9)*Y(t)+9*t^3*(27*t^3-4)*Y(t)^2+t^3*(27*t^3-4)^2*Y(t)^3 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 0, 0, 1, 0, 0, 3, 0, 0, 12, 0, 0, 55, 0, 0, 273, 0, 0, 1428, 0, 0, 7752, 0, 0, 43263, 0, 0, 246675, 0, 0, 1430715, 0, 0, 8414640, 0, 0, 50067108, 0, 0, 300830572, 0, 0, 1822766520, 0, 0, 11124755664, 0, 0, 68328754959, 0, 0, 422030545335, 0, 0, 2619631042665, 0, 0, 16332922290300, 0, 0, 102240109897695, 0, 0, 642312451217745, 0, 0, 4048514844039120, 0, 0, 25594403741131680, 0, 0, 162250238001816900, 0, 0, 1031147983159782228, 0, 0, 6568517413771094628, 0, 0, 41932353590942745504, 0, 0, 268225186597703313816, 0, 0, 1718929965542850284040 , 0, 0, 11034966795189838872624, 0, 0, 70956023048640039202464, 0, 0, 456949965738717944767791, 0, 0, 2946924270225408943665279, 0] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 0, 3, 0, 0, 27, 0, 0, 207, 0, 0, 1506, 0, 0, 10692, 0, 0, 74880, 0, 0, 519975, 0, 0, 3590244, 0, 0, 24689547, 0, 0, 169281531, 0, 0, 1158033348, 0, 0, 7907918760, 0, 0, 53924696616, 0, 0, 367292687868, 0, 0, 2499326053911, 0, 0, 16993660693320, 0, 0, 115466864661513, 0, 0, 784109258291889, 0, 0, 5322049255794807, 0, 0, 36107084876982426, 0, 0, 244872769404876048, 0, 0, 1660131806569367904, 0, 0, 11251620871615990692, 0, 0, 76238091836460476496, 0, 0, 516446464947149990052, 0, 0, 3497721807419757013860, 0, 0, 23684314204917226608984, 0, 0, 160346594031622733401200, 0, 0, 1085397606047500614335520, 0, 0, 7346055003689604337793892, 0, 0, 49711972468040184132387063, 0, 0, 336367501598412386611292304, 0, 0, 2275712929901724641832322629, 0] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[192, .8099075721], [195, .8104394563], [198, .8109600284]] ----------------------------- theorem took, 0.785, seconds. ------------------------------------------------------------- Theorem Number, 2, Let a(n) be the number of generalized Dyck paths of lengt\ h n in the set of steps, {[1, -2], [1, 0], [1, 1]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 3 3 1 + (t - 1) X(t) + t X(t) = 0 and in Maple notation 1+(t-1)*X(t)+t^3*X(t)^3 = 0 Y(t),satisfies the algebraic equation 3 3 2 2 27 t + 9 (10 t - 3 t + 3 t - 1) (t - 1) Y(t) 3 3 2 2 - 9 t (t - 1) (31 t - 12 t + 12 t - 4) Y(t) 3 3 2 2 3 + t (31 t - 12 t + 12 t - 4) Y(t) = 0 and in Maple notation 27*t^3+9*(10*t^3-3*t^2+3*t-1)*(t-1)^2*Y(t)-9*t^3*(t-1)*(31*t^3-12*t^2+12*t-4)*Y (t)^2+t^3*(31*t^3-12*t^2+12*t-4)^2*Y(t)^3 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 1, 1, 2, 5, 11, 24, 57, 141, 349, 871, 2212, 5688, 14730, 38403, 100829, 266333, 706997, 1885165, 5047522, 13565203, 36578497, 98934826, 268342933, 729709432, 1989021256, 5433518806, 14873285506, 40790118487, 112064912455, 308390452661, 849969894794, 2346045295997, 6484283432301, 17945109524709, 49723012463106, 137932680852865, 383044179221839, 1064824607532304, 2963004005175517, 8252593204567339, 23005584662234347, 64186354752753085, 179226387970438192, 500835756327776674, 1400580893005660192, 3919465678954889659, 10975858456707879871, 30756084802816699320, 86236913825615976816, 241943701792769918376, 679179186256779959696, 1907628893071342851566, 5360853811656376251062, 15072876802739852001490, 42400645182982320574756, 119331569207334356044159, 335998735787619247732607, 946482395973996299353151, 2667310572579352076400194, 7519944620007080482910241, 21209472582029240721637023, 59843048195329901888172600, 168912137653826279300316793, 476942771415085102551711421, 1347178441550764664809291069, 3806561179579080141345878861, 10759322884306745033005418378, 30421266899468329317175467449, 86041080315779977200559673991, 243425886492438126827823794976, 688900213233826371571568732661, 1950163348229841201208137925801, 5522139523451696072992158445657, 15640917124143150485668188920635, 44313124895564689904726770150964, 125578373214597873138954153759824, 355964550370093377776847708899834, 1009265169810347724862616098686783, 2862251445060031437128664810262425, 8119170553320153561591268337974299, 23036391247957343310806624981906131, 65375306032694223478312123381698043, 185569767671606537631911082511082014, 526856919071338268841215952432590561, 1496125341105351740614574570920434419, 4249433371671825501440141161479907802, 12072016523684286282054950494070982315, 34301446573270766280831230625929339922, 97482576176393694624963634003404352674, 277090540463230421962063352386438868356 , 787761574367667101066809997501401939834, 2239982292072948576310107974878037714287, 6370440130231552439342221519810152713811, 18120399456911834806125588657015946692219, 51551111681996960242346807653660429038444, 146682652064371074776745400765438710116024, 417434634595576859828087098505731563081224, 1188135068897595221475069938481165108172928, 3382278786026287436662562779341272897140360, 9629814566051925778092788747982027061817608] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 0, 3, 15, 45, 132, 426, 1350, 4077, 12177, 36531, 109185, 323958, 957591 , 2826048, 8324016, 24465618, 71790570, 210397527, 615921975, 1801194246, 5262671331, 15364418094, 44825571414, 130696745409, 380856286989, 1109276194086 , 3229394070960, 9397730107404, 27337662034950, 79496925781128, 231101218741320 , 671626844292264, 1951360824780696, 5668127881701183, 16460465494165659, 47791659063738627, 138731808664456794, 402642634801799826, 1168391879539793766, 3389902267281607737, 9833771218113376137, 28522745250426110241, 82718859223955598615, 239862928338293872833, 695456915463705878514, 2016177258955035063168, 5844417573539648877000, 16939877719793374430598, 49095093847691194263558, 142274421571718436972276, 412266878512476914260176, 1194523725191258637400365, 3460808318136067569492027, 10026014816199199381665264, 29043480746981766710962116, 84127873323740299439644614, 243670747018761036123466938, 705732873406206069821309703, 2043863677782016986072949143, 5918875663871788624428113325, 17139701153819989523835723528, 49630080069039911375065265424, 143702829281043630674147866608, 416068783972635987983136171384, 1204606583622418817759729623608, 3487437101254945627352545258839, 10095999352356200096983308687171, 29226358893911003903706885239361, 84602507244770725328669501209260, 244892506491711613701703179697986, 708846320256846007207294534616622, 2051698647213300058515213239661921, 5938277342154174694048676913246717, 17186732225279911541603872133977719, 49740775827849240812276174586522285, 143952320327536239421266289418210136, 416593119578051373779433672428756361, 1205572250900788702928972930099020944, 3488691117164014595062131898434202224, 10095324966442443103713068164176913434, 29212381213603230977568539022037430466, 84528436571792153020644134896355397849, 244584126505053029696923120663004015793 , 707690838703282961170973483501515355424, 2047618568988968623301538191048430158991, 5924408803406772061873427587500039877938, 17140825462000170628307517567290631664194, 49591754278118025488594249715318343777587, 143475746802988694307699455849546349651171, 415086920583441474205600726855553814742884, 1200857225116987847532212243307411130621602, 3474047106986718690148109458170412270940721, 10050143266857461844149726040515089095308895, 29073762406527931126108357182436352644383336, 84105202409022743270564878410052069323243192, 243297316932268637966253636286650977819072808, 703792775424183278982253748401867672052090664, 2035848692944625247644063513383287006712322040, 5888973160951528658428189209290142464603795448] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[190, .6382442071], [191, .6384345427], [192, .6386234636], [193, .6388109873] , [194, .6389971315], [195, .6391819124], [196, .6393653477], [197, .6395474535 ], [198, .6397282454], [199, .6399077400], [200, .6400859524]] ----------------------------- theorem took, 16.820, seconds. Theorem Number, 3, Let a(n) be the number of generalized Dyck paths of lengt\ h n in the set of steps, {[1, -2], [1, 1], [1, 2]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 2 2 2 3 4 4 4 5 6 6 1 - X(t) - t X(t) + t (t + 2) X(t) - t X(t) - t X(t) + t X(t) = 0 and in Maple notation 1-X(t)-t^2*X(t)^2+t^2*(t+2)*X(t)^3-t^4*X(t)^4-t^4*X(t)^5+t^6*X(t)^6 = 0 Y(t),satisfies the algebraic equation 2 8 7 6 5 4 3 t (3638311 t - 1374000 t - 3908416 t + 1269302 t + 1548324 t - 297216 t 2 12 11 - 263680 t + 24576 t + 16384) + (47898156 t - 15856512 t 10 9 8 7 6 - 38179467 t + 6304472 t + 4652272 t + 2287509 t + 3788108 t 5 4 3 2 2 - 995968 t - 1411602 t + 102400 t + 180992 t - 8192) Y(t) + t ( 12 11 10 9 8 163583532 t - 12249441 t - 31967464 t - 112624808 t - 111763326 t 7 6 5 4 3 + 90899222 t + 78656392 t - 23875016 t - 21291031 t + 2493440 t 2 2 2 10 9 + 2653824 t - 65536 t - 126976) Y(t) - t %1 (546439 t - 2462169 t 8 7 6 5 4 3 - 3562264 t + 2680802 t + 2859691 t - 826976 t - 843144 t + 89104 t 2 3 4 + 102592 t - 1024 t - 4096) Y(t) - t 6 5 4 3 2 2 (19725 t - 32178 t - 43080 t + 11785 t + 16794 t - 1024 t - 1920) %1 4 4 4 3 2 3 5 6 4 6 Y(t) + t (27 t + 144 t + 128 t - 8 t - 32) %1 Y(t) + t %1 Y(t) = 0 4 3 2 %1 := 229 t - 144 t - 128 t + 4 t + 16 and in Maple notation t^2*(3638311*t^8-1374000*t^7-3908416*t^6+1269302*t^5+1548324*t^4-297216*t^3-\ 263680*t^2+24576*t+16384)+(47898156*t^12-15856512*t^11-38179467*t^10+6304472*t^ 9+4652272*t^8+2287509*t^7+3788108*t^6-995968*t^5-1411602*t^4+102400*t^3+180992* t^2-8192)*Y(t)+t^2*(163583532*t^12-12249441*t^11-31967464*t^10-112624808*t^9-\ 111763326*t^8+90899222*t^7+78656392*t^6-23875016*t^5-21291031*t^4+2493440*t^3+ 2653824*t^2-65536*t-126976)*Y(t)^2-t^2*(229*t^4-144*t^3-128*t^2+4*t+16)*(546439 *t^10-2462169*t^9-3562264*t^8+2680802*t^7+2859691*t^6-826976*t^5-843144*t^4+ 89104*t^3+102592*t^2-1024*t-4096)*Y(t)^3-t^4*(19725*t^6-32178*t^5-43080*t^4+ 11785*t^3+16794*t^2-1024*t-1920)*(229*t^4-144*t^3-128*t^2+4*t+16)^2*Y(t)^4+t^4* (27*t^4+144*t^3+128*t^2-8*t-32)*(229*t^4-144*t^3-128*t^2+4*t+16)^3*Y(t)^5+t^6*( 229*t^4-144*t^3-128*t^2+4*t+16)^4*Y(t)^6 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 0, 1, 1, 2, 7, 8, 38, 58, 199, 452, 1149, 3277, 7650, 22696, 55726, 157502, 416967, 1128026, 3122336, 8365304, 23402737, 63505268, 176860650, 487957967, 1353427722, 3774616133, 10483218667, 29371164344, 81965145468, 230030965231, 645265199252, 1813615497166, 5107394107927, 14386545035342, 40621735594210, 114720169872202, 324560293765296, 918870098708832, 2604241833793991, 7388579097551618, 20977399823555561, 59621343802550548, 169564084505396074, 482675806553513346, 1374913683174205990, 3919330024586140872, 11180276246544027984, 31912767043682119091, 91151661450077696842, 260504150271472067487, 744947302289612320419, 2131447285649899773482, 6101803306922256585093, 17477041985742079940042, 50083202507295988173638, 143591214804033688415252, 411872960087198439707020, 1181938891881928826882954, 3393237756134243454881923, 9745761652953982997207239, 28002293032810782454429664, 80489913874085151962741069, 231448747966301291146659973, 665775455613130578087964718, 1915825804633865640875510199, 5514859995133063738383715198, 15880307009862092388593834938, 45743033253307915020912523866, 131803971836443495985646934346, 379896890743687513713066562632, 1095299272899961345869825705421, 3158829552927699443639623924172, 9112602113045313890792063470400, 26295302279309798839696626761954, 75897994086190859090327704904386, 219126918794497514604321507948622, 632807420043055365265973336120311, 1827910587314348163667779166891084, 5281328653801629835484380957405308, 15262781202002197316280707499188650, 44118820306298133633541281225090697, 127559089273900630439769019874439578, 368887462905234581317115152477238442, 1067011498596883768561430099061880030, 3086987315483995752005197996854513506, 8932829336110216328563459544355900850, 25854113279862452261954232008644645747, 74843608876778219689608691577566089668, 216701723365174180184832757989599065150, 627553494485772634933093593539396830390, 1817682869249608576889157702251547022020, 5265780349483642346958962385307264328388, 15257485740413687597954596861706947065140, 44215771824385306917170261803795365432526, 128157462508714435731132033352842766515845, 371519434327191324719847785949657971272963, 1077180946975716408102603738960550511247786, 3123660784019082236748624127997292996164579, 9059532190599579345068831623354242648370891, 26279255354283451199405178126260914151192874] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 2, 3, 12, 55, 85, 542, 961, 4322, 11026, 34164, 109871, 293816, 987817, 2684484, 8522974, 24638253, 73622709, 221470508, 645182086, 1953599266, 5702670578, 17088987572, 50388845772, 149381314424, 443275492193, 1307893669263 , 3883792129140, 11458936131912, 33955194499352, 100323976024168, 296647340270956, 877183751319631, 2590971833391558, 7660557860546306, 22622122958748207, 66847040564920806, 197412123912173791, 583020785413567540, 1721710552355770384, 5082998791465246421, 15007904386258090962, 44299587533956838088, 130766767635141586866, 385943416745673293877, 1139008591260123525406, 3361242252930426098496, 9918118601874219047904, 29264650849168687035498, 86340707429165767525834, 254724520556263336533125, 751440233380578018388628, 2216643595428557980608619, 6538446188777141928251330, 19285419732647294228900526, 56880708592025701547502405, 167756149127390916131950803, 494736916761043304096827826, 1458984798545256153088626542, 4302391691497156488159438558, 12686806610611138130027096392, 37409178007657121471191214398, 110303403175249548504346884496, 325225334524179475721357318420, 958883416303446449464873406767, 2827048945069117930795396757028, 8334656203551611141591712181546, 24571368693331309939775173718609, 72436707987808749463884054503778, 213538575178539422075054407016488, 629480881687223732014357454517108, 1855572122247614918852242089371602, 5469687730912310069757587138499206, 16122672532804973859962000479690395, 47522762920421304595936409725021280, 140073764698530477479085053126339938, 412859926423414013921191443677872155, 1216857577874469694475892526888188387, 3586478001154390703180397157484685096, 10570322923174811070900274638720531728, 31153028623828517650258841877927701097, 91813063731973267799200751911666778456, 270583376558339938776124716723585989790, 797426027181299393978132870849729543559, 2350025535056343885089767317589524348846, 6925447678053932190934178755960823671820, 20408750759329759694950047625189273713916, 60142085870274244349493901404218368336216, 177228774133123351546755966393005367419006, 522256487338711682533410379242296982088918, 1538960559473816905322645072222818606622594, 4534875190281451448586351435069661282554003, 13362802540730724696509882277262725879914547, 39375336653770824646311195879701721026368556, 116023423019273778414976565521095563028903568, 341870696236398452760416685512733736073558488, 1007332912756072360818273631319182255741106186, 2968104923522366307761012377658469976837136766, 8745420175709977494239641905684133802790117969, 25767805838916270184148568631421473593510802578] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[190, 1.011221371], [191, 1.011440090], [192, 1.011657183], [193, 1.011872671] , [194, 1.012086573], [195, 1.012298909], [196, 1.012509698], [197, 1.012718960 ], [198, 1.012926713], [199, 1.013132974], [200, 1.013337762]] ----------------------------- theorem took, 134.147, seconds. ------------------------------------------------------------- Theorem Number, 4, Let a(n) be the number of generalized Dyck paths of lengt\ h n in the set of steps, {[1, -2], [1, 0], [1, 1], [1, 2]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 2 2 2 3 4 4 4 5 1 + (t - 1) X(t) - t X(t) - (t - 2) t X(t) - t X(t) + t (t - 1) X(t) 6 6 + t X(t) = 0 and in Maple notation 1+(t-1)*X(t)-t^2*X(t)^2-(t-2)*t^2*X(t)^3-t^4*X(t)^4+t^4*(t-1)*X(t)^5+t^6*X(t)^6 = 0 Y(t),satisfies the algebraic equation 2 8 7 6 5 4 3 t (1408261 t + 4194402 t + 533238 t - 2679898 t - 634076 t + 883456 t 2 12 11 + 23040 t - 106496 t + 16384) + (25078366 t + 53046671 t 10 9 8 7 6 - 9924681 t - 32316054 t + 7133967 t + 11473725 t - 6927660 t 5 4 3 2 - 1247728 t + 1756398 t + 94720 t - 359680 t + 98304 t - 8192) Y(t) + 2 12 11 10 9 t (135167090 t + 143330797 t - 163710564 t - 64923588 t 8 7 6 5 4 + 160743194 t + 25580422 t - 89561476 t - 3303696 t + 23650409 t 3 2 2 2 10 + 285440 t - 5005696 t + 1458176 t - 126976) Y(t) + t %1 (380519 t 9 8 7 6 5 - 2024721 t - 2030012 t + 2385850 t + 2320925 t - 1261088 t 4 3 2 3 4 - 631560 t + 276976 t + 72512 t - 39936 t + 4096) Y(t) - t 6 5 4 3 2 2 (12936 t + 28561 t + 3769 t - 27231 t - 6886 t + 10496 t - 1920) %1 4 4 4 3 2 3 5 6 4 6 Y(t) - t (13 t + 8 t + 40 t - 120 t + 32) %1 Y(t) + t %1 Y(t) = 0 4 3 2 %1 := 257 t + 60 t - 44 t - 60 t + 16 and in Maple notation t^2*(1408261*t^8+4194402*t^7+533238*t^6-2679898*t^5-634076*t^4+883456*t^3+23040 *t^2-106496*t+16384)+(25078366*t^12+53046671*t^11-9924681*t^10-32316054*t^9+ 7133967*t^8+11473725*t^7-6927660*t^6-1247728*t^5+1756398*t^4+94720*t^3-359680*t ^2+98304*t-8192)*Y(t)+t^2*(135167090*t^12+143330797*t^11-163710564*t^10-\ 64923588*t^9+160743194*t^8+25580422*t^7-89561476*t^6-3303696*t^5+23650409*t^4+ 285440*t^3-5005696*t^2+1458176*t-126976)*Y(t)^2+t^2*(257*t^4+60*t^3-44*t^2-60*t +16)*(380519*t^10-2024721*t^9-2030012*t^8+2385850*t^7+2320925*t^6-1261088*t^5-\ 631560*t^4+276976*t^3+72512*t^2-39936*t+4096)*Y(t)^3-t^4*(12936*t^6+28561*t^5+ 3769*t^4-27231*t^3-6886*t^2+10496*t-1920)*(257*t^4+60*t^3-44*t^2-60*t+16)^2*Y(t )^4-t^4*(13*t^4+8*t^3+40*t^2-120*t+32)*(257*t^4+60*t^3-44*t^2-60*t+16)^3*Y(t)^5 +t^6*(257*t^4+60*t^3-44*t^2-60*t+16)^4*Y(t)^6 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 1, 2, 5, 13, 38, 116, 368, 1203, 4016, 13642, 46987, 163696, 575816, 2042175, 7294299, 26215927, 94736708, 344015468, 1254647606, 4593682529, 16878510120, 62215957762, 230007985382, 852612196852, 3168359595108, 11800740083576, 44045606325107, 164721039237571, 617148978777583, 2316181581852586, 8706610827498169, 32777540164574119, 123570545078714196, 466475769211533776, 1763136562025613086, 6671991538484571985, 25276100101802859113, 95857089488037503932, 363893671750915858188, 1382735524079117907097, 5258936627524587180296, 20018532472425959709964, 76264882378973750840787, 290776169937515020724622, 1109480587607860328484762, 4236362634599404630254407, 16186979345718023365399126, 61890689908589710188135032, 236788162441057929231090936, 906481251356979376901731572, 3472254168146806439516245852, 13307862200032560528339214444, 51031695790051796244350317755, 195793353373380255061846999352, 751577763515517225449376772964, 2886424260077596338775026932843, 11090453778770790774173089864959, 42631870995974572588259478288830, 163948999685097694281438178407935, 630762921042175124692646870431377, 2427730981044889019800344445037262, 9347732775807755338674379074959572, 36006251774350782839270800703499168, 138742739010253616123822439515958199, 534808627193735170542454600445940932, 2062232491201192827859566546524042872, 7954693048041660294835862000468400054, 30693859825267119442390723675995383969, 118472553290591737110976528087567054533 , 457423229437397341821536612554845001896, 1766644373303497623047263321932390509544, 6825066523538328700903858792328803930489, 26374729289237785639198390721750686036801, 101950456081509435024773999290220794573346, 394191432505587596337242600824038189545744, 1524540499668836620984637670143395646458582, 5897685079375586554211081157332022244875394, 22820869261210325481575538452221488477105381, 88325895684111709972681445199866122830618324, 341937359643471699200377001716059202913031353, 1324052374928654965126768676959967828690662928, 5128158567229393232347808392626679070957720776, 19866119674906259979800256689121320863227842446, 76976419248163567373758873950630055550472610171, 298327461498692526645599606566656958462338358552, 1156425131458233261187280856439700804282193733516, 4483617274224472720624125740286944093765936723826, 17386985933651319524826809633478921821365767791942, 67437727496560803838160280218269486596060393948890, 261614956357265559031747000545153956700527314987588, 1015082755876765614140388021110375872421754684342497, 3939289075302099021727606584808922220241160150590957, 15290092371513229503195174041521719534693827929333908, 59357638055792766898916929671081915237981217442859954, 230470740779357553510102374593612167267787690050665823, 895006432695011857826183798735737954889633147922360376, 3476211750410590813969578694095751054492811383435194552, 13503755842540755030570613849994491941207601422397957448, 52465019750296508337975021669923986213588681571995300488, 203868758531261548665240457937830150237861863452263005464] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 2, 11, 47, 212, 897, 3756, 15577, 63966, 261177, 1061007, 4293193, 17316908, 69666454, 279667076, 1120663484, 4483802537, 17916483702, 71511021581 , 285148949519, 1136063512793, 4522827784119, 17994159319384, 71548226794837, 284340721966841, 1129471727555198, 4484642825956688, 17799753572154674, 70623459009487996, 280121349510269022, 1110752877957989544, 4403247795923950460 , 17451088366844184893, 69147087305818074165, 273926879390853155855, 1084954298251214189170, 4296454992623650733653, 17011261879838965529096, 67343415953064641516496, 266557286637315615159495, 1054937769405692498236924, 4174531054270579276412103, 16517203317602801864329499, 65345608665419831521148900, 258493860473269579544157549, 1022446566609021425374361241, 4043803088385977121798531364, 15991920102261158697295834388, 63237476617959082400886643712, 250042415490281994379985145569, 988598301361619854977898749982, 3908361409272407826348226925317, 15450402690209461842390618219152, 61074027924856571871952887829438, 241405046751457004206128106466640, 954136278172063890696070693262764, 3770942713021327098255729480019918, 14902736713846768447496164377112766, 58892464535780124725738190938361239, 232719090320086971675739475059323867, 919567874394985695131534506101745368, 3633422759452851759810765599972853473, 14355861155670309580147641296129162752, 56718470297871751527583731637489307892, 224079660060091938954900746775421603141 , 885245478018440268343639784551664178953, 3497107617642046478127034707324720025491, 13814618333703687043631417698254876633738, 54570003165714997821920513046945998340897, 215553367735904669760368818713387629924879, 851416209850659554676925714015712597070728, 3362914152498975848911863531127938909384713, 13282410071539478692561656092286087599641547, 52459684337798473305618328822369648753301675, 207187034574604253181265061506686647657287499, 818253733264302795025712297342334597925296640, 3231486333660658514616238335572305029459518158, 12761623526866181598785362588930198502900367095, 50396359271439341863841025315778695196436549804, 199013420704020824118299375412448941425796696800, 785879308101814839768996869031478918741569073799, 3103272691063969572032834638332691318358058123586, 12253916461692274648684948244054227062908477184535, 48386149472760063128047300215591784886848176476479, 191055107157594875868934966530336618607617722703261, 754376133140670297039660053381595802838654872565283, 2978579356834782014014965857835444220293287445197504, 11760413591428997770112447877038095674208809456704157, 46433180010489698923916324800556501413928588904741173, 183327193888798862461074616930269639661686854894681165, 723799360531808470889085503818731936081787087592574100, 2857607268229426551751118075334696981132020654797791385, 11281844905127330383165793052932423361907060021884101946, 44540094201218135356956735023117362209722445409699922825, 175839216388495602998917775327037115045856549373681241880, 694183238414928908358942508163577756269199929258504379168, 2740478805414513027286804821164831290561226542418798227256, 10818645079760057206219006024546787089626281255041459295120, 42708424888813465854650789507194799316261648706566815358260, 168596533226238372284268679807498718281057961068345159718988] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[190, .8594411660], [191, .8596702471], [192, .8598975956], [193, .8601232332] , [194, .8603471815], [195, .8605694611], [196, .8607900938], [197, .8610090995 ], [198, .8612264977], [199, .8614423088], [200, .8616565517]] ----------------------------- theorem took, 289.041, seconds. Theorem Number, 5, Let a(n) be the number of generalized Dyck paths of lengt\ h n in the set of steps, {[1, -1], [1, 1]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 2 2 t X(t) - X(t) + 1 = 0 and in Maple notation t^2*X(t)^2-X(t)+1 = 0 Y(t),satisfies the algebraic equation 2 2 2 2 2 2 t - (2 t + 1) (2 t - 1) (2 t - 1) Y(t) + t (2 t + 1) (2 t - 1) Y(t) = 0 and in Maple notation t^2-(2*t+1)*(2*t-1)*(2*t^2-1)*Y(t)+t^2*(2*t+1)^2*(2*t-1)^2*Y(t)^2 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, 132, 0, 429, 0, 1430, 0, 4862, 0, 16796, 0, 58786, 0, 208012, 0, 742900, 0, 2674440, 0, 9694845, 0, 35357670, 0, 129644790, 0, 477638700, 0, 1767263190, 0, 6564120420, 0, 24466267020, 0, 91482563640, 0, 343059613650, 0, 1289904147324, 0, 4861946401452, 0, 18367353072152, 0, 69533550916004, 0, 263747951750360, 0, 1002242216651368, 0, 3814986502092304, 0, 14544636039226909, 0, 55534064877048198, 0, 212336130412243110, 0, 812944042149730764, 0, 3116285494907301262, 0, 11959798385860453492, 0, 45950804324621742364, 0, 176733862787006701400, 0, 680425371729975800390, 0, 2622127042276492108820, 0, 10113918591637898134020, 0 , 39044429911904443959240, 0, 150853479205085351660700, 0, 583300119592996693088040, 0, 2257117854077248073253720, 0, 8740328711533173390046320, 0, 33868773757191046886429490, 0, 131327898242169365477991900, 0, 509552245179617138054608572, 0, 1978261657756160653623774456] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 1, 0, 6, 0, 29, 0, 130, 0, 562, 0, 2380, 0, 9949, 0, 41226, 0, 169766, 0 , 695860, 0, 2842226, 0, 11576916, 0, 47050564, 0, 190876696, 0, 773201629, 0, 3128164186, 0, 12642301534, 0, 51046844836, 0, 205954642534, 0, 830382690556, 0 , 3345997029244, 0, 13475470680616, 0, 54244942336114, 0, 218269673491780, 0, 877940640368572, 0, 3530129914546440, 0, 14190053209101764, 0, 57023960788157416, 0, 229098085369281032, 0, 920207327979216432, 0, 3695373947956092637, 0, 14837029856701418746, 0, 59560455557217918094, 0, 239054766271021403140, 0, 959335350578992913822, 0, 3849301200701832108780, 0, 15443155607131950177484, 0, 61949356291314807411336, 0, 248477850536989205445734, 0, 996533529190233313891756, 0, 3996248035352571153701044, 0, 16024036571322189058763416, 0, 64246999764493841586714364, 0, 257571299177568363039945496, 0, 1032542314564350700233035704, 0, 4138909586968935974322189136, 0, 16589507121632934944175186034, 0, 66489356384773909142178736036, 0, 266466977784275253706769552716, 0, 1067846172794857175480701985320] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[190, .5611577769], [192, .5614650763], [194, .5617679488], [196, .5620665009] , [198, .5623608361], [200, .5626510537]] ----------------------------- theorem took, 0.224, seconds. ------------------------------------------------------------- Theorem Number, 6, Let a(n) be the number of generalized Dyck paths of lengt\ h n in the set of steps, {[1, -1], [1, 0], [1, 1]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 2 2 1 + t X(t) + (t - 1) X(t) = 0 and in Maple notation 1+t^2*X(t)^2+(t-1)*X(t) = 0 Y(t),satisfies the algebraic equation 2 2 2 2 2 2 t - (3 t - 1) (t + 1) (t + 2 t - 1) Y(t) + t (3 t - 1) (t + 1) Y(t) = 0 and in Maple notation t^2-(3*t-1)*(t+1)*(t^2+2*t-1)*Y(t)+t^2*(3*t-1)^2*(t+1)^2*Y(t)^2 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572 , 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829, 1697385471211, 4859761676391, 13933569346707, 40002464776083, 114988706524270, 330931069469828, 953467954114363, 2750016719520991, 7939655757745265, 22944749046030949, 66368199913921497, 192137918101841817, 556704809728838604, 1614282136160911722, 4684478925507420069, 13603677110519480289, 39532221379621112004, 114956499435014161638, 334496473194459009429, 973899740488107474693, 2837208756709314025578, 8270140811590103129028, 24119587499879368045581, 70380687801729972163737, 205473381836953330090977, 600161698382141668958313, 1753816895177229449263803, 5127391665653918424581931, 14996791899280244858336604, 43881711243248048262611670, 128453535912993825479057919, 376166554620363320971336899, 1101997131244113831001323618, 3229547920421385142120565580, 9468017265749942384739441267, 27766917351255946264000229811, 81459755507915876737297376646, 239056762740830735069669439852, 701774105036927170410592656651, 2060763101398061220299787957807, 6053261625552368838017538638577, 17785981695172350686294020499397, 52274487460035748810950928411209, 153681622703766437645990598724233, 451929928113276686826984901736388, 1329334277731700374912787442584082, 3911184337415864255099077969308357, 11510402374965653734436362305721089, 33882709435158403490429948661355518, 99762777233730236158474945885114348, 293804991106867190838370294149325217, 865461205861621792586606565768282577, 2549948950073051466077548390833960154, 7514646250637159480132421134685515996, 22150145406114764734833589779994282345, 65303054248346999524711654923215773701, 192564948449128362785882746541078077821 , 567944426681696509718034692302003744197, 1675395722976475387857861526496400455935, 4943221572052274428484817274841589781103, 14587540897567180436019575590444202957764, 43055804394719442101962182766220627765254, 127103430617648266466982424978107271745123, 375281510930976756310181851730346874521559, 1108229819877900763405338193186744667723583, 3273209089476438052473101825635320104642103, 9669131152389329200998265687814683780583133, 28567321136213468215221364999058944720713501, 84414794291793480358891042199686850901302514, 249478578991224378680142561460010030467811580, 737415571391164350797051905752637361193303669] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 1, 4, 16, 56, 190, 624, 2014, 6412, 20219, 63284, 196938, 610052, 1882717, 5792528, 17776102, 54433100, 166374109, 507710420, 1547195902, 4709218604, 14318240578, 43493134160, 132003957436, 400337992056, 1213314272395 , 3674980475284, 11124919273160, 33660499478792, 101799399463446, 307742673568448, 929957353197550, 2809211622529820, 8483281546400477, 25610140323335972, 77292662170517910, 233212294076450364, 703492213702678732, 2121626402922994240, 6397152897821915224, 19284954722600733488, 58125935030294209435, 175164311554822692452, 527776624989874521556, 1589965382829229180688, 4789189184897183557759, 14423698796337447707216, 43434505121125890935500, 130779509898557773457912, 393725844176714426273209, 1185223150103982962023604, 3567473279426398467783388, 10736877109466596425755744, 32311266751247611612788892, 97227853147440200841712352, 292543161745418324302096870, 880143457703633940104648924, 2647785800877149179549343131, 7964886894209167982816908100, 23957703015719529281498291842, 72057779732773932006939586964, 216714499089330202368404036173, 651729888166159472068746075200, 1959841742945638108485323134606, 5893172050475260637056145283356, 17719562167944095530041557117365, 53276237494546887829583922148004, 160173878835222158684633438864902, 481535459150446114907907839474972, 1447585053330713013487433746225420, 4351509204182384353546867492736576, 13080262074121771145680693132061224, 39316458896034426040837028412645200, 118171952319843675767897130524486227, 355170685677409053346391148827755748, 1067439078374033297876659979755694452, 3207989431817843545271830690331068880, 9640668405817979733929634906284855545, 28971185174888061936342806880786676560, 87058417781103200029154413828692130624, 261602047301625340469085484571363248288, 786063835690771710699125152453868566025, 2361898510586983550819561631365678432756, 7096624027401759838991199888738271329936, 21322096495406702565230962736716799449496, 64061326967219466010696696231266275192920, 192464322898891607864018432618443874400704, 578220077995736392638266780764880971091906, 1737100950537223407909215815133438754711092, 5218506454387170899740766730877123035874563, 15676784023883164284875123514330338760689892, 47093135037150770093337567463503933272983342, 141464800226264361531143704896751122430812796, 424941953872927957349395268525470442473777642, 1276443287694247536581289415373642827676912192, 3834108359070191836359289800185876346451098340, 11516444771206950421476670945803925436873063624, 34591062092780056364174880969366107819280011281, 103896522283880394891101026295404315128745722500, 312054167838500898846843604843199581046056970932] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[190, .4453084256], [191, .4454671090], [192, .4456246257], [193, .4457809898] , [194, .4459362155], [195, .4460903166], [196, .4462433074], [197, .4463952008 ], [198, .4465460100], [199, .4466957481], [200, .4468444278]] ----------------------------- theorem took, 0.669, seconds. Theorem Number, 7, Let a(n) be the number of generalized Dyck paths of lengt\ h n in the set of steps, {[1, -1], [1, 2]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 3 3 X(t) t - X(t) + 1 = 0 and in Maple notation X(t)^3*t^3-X(t)+1 = 0 Y(t),satisfies the algebraic equation 3 3 3 3 2 3 3 2 3 27 t + (81 t - 9) Y(t) + 9 t (27 t - 4) Y(t) + t (27 t - 4) Y(t) = 0 and in Maple notation 27*t^3+(81*t^3-9)*Y(t)+9*t^3*(27*t^3-4)*Y(t)^2+t^3*(27*t^3-4)^2*Y(t)^3 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 0, 0, 1, 0, 0, 3, 0, 0, 12, 0, 0, 55, 0, 0, 273, 0, 0, 1428, 0, 0, 7752, 0, 0, 43263, 0, 0, 246675, 0, 0, 1430715, 0, 0, 8414640, 0, 0, 50067108, 0, 0, 300830572, 0, 0, 1822766520, 0, 0, 11124755664, 0, 0, 68328754959, 0, 0, 422030545335, 0, 0, 2619631042665, 0, 0, 16332922290300, 0, 0, 102240109897695, 0, 0, 642312451217745, 0, 0, 4048514844039120, 0, 0, 25594403741131680, 0, 0, 162250238001816900, 0, 0, 1031147983159782228, 0, 0, 6568517413771094628, 0, 0, 41932353590942745504, 0, 0, 268225186597703313816, 0, 0, 1718929965542850284040 , 0, 0, 11034966795189838872624, 0, 0, 70956023048640039202464, 0, 0, 456949965738717944767791, 0, 0, 2946924270225408943665279, 0] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 0, 3, 0, 0, 27, 0, 0, 207, 0, 0, 1506, 0, 0, 10692, 0, 0, 74880, 0, 0, 519975, 0, 0, 3590244, 0, 0, 24689547, 0, 0, 169281531, 0, 0, 1158033348, 0, 0, 7907918760, 0, 0, 53924696616, 0, 0, 367292687868, 0, 0, 2499326053911, 0, 0, 16993660693320, 0, 0, 115466864661513, 0, 0, 784109258291889, 0, 0, 5322049255794807, 0, 0, 36107084876982426, 0, 0, 244872769404876048, 0, 0, 1660131806569367904, 0, 0, 11251620871615990692, 0, 0, 76238091836460476496, 0, 0, 516446464947149990052, 0, 0, 3497721807419757013860, 0, 0, 23684314204917226608984, 0, 0, 160346594031622733401200, 0, 0, 1085397606047500614335520, 0, 0, 7346055003689604337793892, 0, 0, 49711972468040184132387063, 0, 0, 336367501598412386611292304, 0, 0, 2275712929901724641832322629, 0] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[192, .8099075721], [195, .8104394563], [198, .8109600284]] ----------------------------- theorem took, 0.872, seconds. ------------------------------------------------------------- Theorem Number, 8, Let a(n) be the number of generalized Dyck paths of lengt\ h n in the set of steps, {[1, -1], [1, 0], [1, 2]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 3 3 t X(t) + (t - 1) X(t) + 1 = 0 and in Maple notation t^3*X(t)^3+(t-1)*X(t)+1 = 0 Y(t),satisfies the algebraic equation 3 3 2 2 27 t + 9 (10 t - 3 t + 3 t - 1) (t - 1) Y(t) 3 3 2 2 - 9 t (t - 1) (31 t - 12 t + 12 t - 4) Y(t) 3 3 2 2 3 + t (31 t - 12 t + 12 t - 4) Y(t) = 0 and in Maple notation 27*t^3+9*(10*t^3-3*t^2+3*t-1)*(t-1)^2*Y(t)-9*t^3*(t-1)*(31*t^3-12*t^2+12*t-4)*Y (t)^2+t^3*(31*t^3-12*t^2+12*t-4)^2*Y(t)^3 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 1, 1, 2, 5, 11, 24, 57, 141, 349, 871, 2212, 5688, 14730, 38403, 100829, 266333, 706997, 1885165, 5047522, 13565203, 36578497, 98934826, 268342933, 729709432, 1989021256, 5433518806, 14873285506, 40790118487, 112064912455, 308390452661, 849969894794, 2346045295997, 6484283432301, 17945109524709, 49723012463106, 137932680852865, 383044179221839, 1064824607532304, 2963004005175517, 8252593204567339, 23005584662234347, 64186354752753085, 179226387970438192, 500835756327776674, 1400580893005660192, 3919465678954889659, 10975858456707879871, 30756084802816699320, 86236913825615976816, 241943701792769918376, 679179186256779959696, 1907628893071342851566, 5360853811656376251062, 15072876802739852001490, 42400645182982320574756, 119331569207334356044159, 335998735787619247732607, 946482395973996299353151, 2667310572579352076400194, 7519944620007080482910241, 21209472582029240721637023, 59843048195329901888172600, 168912137653826279300316793, 476942771415085102551711421, 1347178441550764664809291069, 3806561179579080141345878861, 10759322884306745033005418378, 30421266899468329317175467449, 86041080315779977200559673991, 243425886492438126827823794976, 688900213233826371571568732661, 1950163348229841201208137925801, 5522139523451696072992158445657, 15640917124143150485668188920635, 44313124895564689904726770150964, 125578373214597873138954153759824, 355964550370093377776847708899834, 1009265169810347724862616098686783, 2862251445060031437128664810262425, 8119170553320153561591268337974299, 23036391247957343310806624981906131, 65375306032694223478312123381698043, 185569767671606537631911082511082014, 526856919071338268841215952432590561, 1496125341105351740614574570920434419, 4249433371671825501440141161479907802, 12072016523684286282054950494070982315, 34301446573270766280831230625929339922, 97482576176393694624963634003404352674, 277090540463230421962063352386438868356 , 787761574367667101066809997501401939834, 2239982292072948576310107974878037714287, 6370440130231552439342221519810152713811, 18120399456911834806125588657015946692219, 51551111681996960242346807653660429038444, 146682652064371074776745400765438710116024, 417434634595576859828087098505731563081224, 1188135068897595221475069938481165108172928, 3382278786026287436662562779341272897140360, 9629814566051925778092788747982027061817608] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 0, 3, 15, 45, 132, 426, 1350, 4077, 12177, 36531, 109185, 323958, 957591 , 2826048, 8324016, 24465618, 71790570, 210397527, 615921975, 1801194246, 5262671331, 15364418094, 44825571414, 130696745409, 380856286989, 1109276194086 , 3229394070960, 9397730107404, 27337662034950, 79496925781128, 231101218741320 , 671626844292264, 1951360824780696, 5668127881701183, 16460465494165659, 47791659063738627, 138731808664456794, 402642634801799826, 1168391879539793766, 3389902267281607737, 9833771218113376137, 28522745250426110241, 82718859223955598615, 239862928338293872833, 695456915463705878514, 2016177258955035063168, 5844417573539648877000, 16939877719793374430598, 49095093847691194263558, 142274421571718436972276, 412266878512476914260176, 1194523725191258637400365, 3460808318136067569492027, 10026014816199199381665264, 29043480746981766710962116, 84127873323740299439644614, 243670747018761036123466938, 705732873406206069821309703, 2043863677782016986072949143, 5918875663871788624428113325, 17139701153819989523835723528, 49630080069039911375065265424, 143702829281043630674147866608, 416068783972635987983136171384, 1204606583622418817759729623608, 3487437101254945627352545258839, 10095999352356200096983308687171, 29226358893911003903706885239361, 84602507244770725328669501209260, 244892506491711613701703179697986, 708846320256846007207294534616622, 2051698647213300058515213239661921, 5938277342154174694048676913246717, 17186732225279911541603872133977719, 49740775827849240812276174586522285, 143952320327536239421266289418210136, 416593119578051373779433672428756361, 1205572250900788702928972930099020944, 3488691117164014595062131898434202224, 10095324966442443103713068164176913434, 29212381213603230977568539022037430466, 84528436571792153020644134896355397849, 244584126505053029696923120663004015793 , 707690838703282961170973483501515355424, 2047618568988968623301538191048430158991, 5924408803406772061873427587500039877938, 17140825462000170628307517567290631664194, 49591754278118025488594249715318343777587, 143475746802988694307699455849546349651171, 415086920583441474205600726855553814742884, 1200857225116987847532212243307411130621602, 3474047106986718690148109458170412270940721, 10050143266857461844149726040515089095308895, 29073762406527931126108357182436352644383336, 84105202409022743270564878410052069323243192, 243297316932268637966253636286650977819072808, 703792775424183278982253748401867672052090664, 2035848692944625247644063513383287006712322040, 5888973160951528658428189209290142464603795448] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[190, .6382442071], [191, .6384345427], [192, .6386234636], [193, .6388109873] , [194, .6389971315], [195, .6391819124], [196, .6393653477], [197, .6395474535 ], [198, .6397282454], [199, .6399077400], [200, .6400859524]] ----------------------------- theorem took, 16.842, seconds. Theorem Number, 9, Let a(n) be the number of generalized Dyck paths of lengt\ h n in the set of steps, {[1, -1], [1, 1], [1, 2]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 3 3 2 2 X(t) t + X(t) t - X(t) + 1 = 0 and in Maple notation X(t)^3*t^3+X(t)^2*t^2-X(t)+1 = 0 Y(t),satisfies the algebraic equation 2 5 4 3 2 t (29 t + 9) + (-177 t - 81 t + 93 t + 60 t - 2 t - 9) Y(t) 2 2 3 2 2 + t (6 t + 9 t - 1) (31 t + 18 t - t - 4) Y(t) 3 3 2 2 3 + t (31 t + 18 t - t - 4) Y(t) = 0 and in Maple notation t^2*(29*t+9)+(-177*t^5-81*t^4+93*t^3+60*t^2-2*t-9)*Y(t)+t^2*(6*t^2+9*t-1)*(31*t ^3+18*t^2-t-4)*Y(t)^2+t^3*(31*t^3+18*t^2-t-4)^2*Y(t)^3 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 0, 1, 1, 2, 5, 8, 21, 42, 96, 222, 495, 1177, 2717, 6435, 15288, 36374, 87516, 210494, 509694, 1237736, 3014882, 7370860, 18059899, 44379535, 109298070 , 269766655, 667224480, 1653266565, 4103910930, 10203669285, 25408828065, 63364046190, 158229645720, 395632288590, 990419552730, 2482238709888, 6227850849066, 15641497455612, 39322596749218, 98948326105928, 249205769591184, 628164080248656, 1584662249422154, 4000676360466418, 10107612564723278, 25554572385021774, 64651815489864521, 163671444507238419, 414604424265392676, 1050880098774222787, 2665143773089147105, 6762806129129456672, 17169718027841809157, 43613497554655216454, 110838909698132895090, 281818578844741072211, 716879073712295375462, 1824375867825371722519, 4644815138787299518777, 11830473640537847379154, 30144604830858706622385, 76839632990107912235304, 195940365215297486951349, 499827853614467340311790, 1275469563832483663081536, 3255879030421800989036558, 8314010171499430149552262 , 21236980969229242865647188, 54263890285949301341731998, 138695110930263615267329272, 354600851324161940155918554, 906866293102350209637341800, 2319894437764592201079564408, 5936239484437674725175980416, 15193915495855840745715272098, 38899149279980887999850398850, 99613848531935167380079303598, 255155996468127489510984698798, 653725782253127830045543532256, 1675277095431200249501664295920, 4294142129674613532229458321864, 11009371295302268461297595113248, 28232066038503066843010405848816, 72412680255896730050229633088728, 185770290718820984034073194631176, 476678438146741236610807650395544, 1223377154558198677967852498911314, 3140356164337561340111627218826862, 8062677029300855639122980505376604, 20704256212742549059718719679988798, 53176319095921978871991798542333124, 136600877990989505157321565767115566, 350964876670499702831494643590416144, 901876737699549090563341810569964806, 2317941898463865472333981541631229527, 5958380756332587988359308358690090363, 15318733662029548169969883108648446568, 39389910052360574776485124229995908219, 101300891995714947619675809273019879503 , 260559188602979967444178119850227047106] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 1, 3, 6, 29, 56, 195, 497, 1331, 3760, 9652, 26593, 70026, 186546, 497375, 1311729, 3485327, 9196563, 24299861, 64137209, 169002069, 445462250, 1172555762, 3085847580, 8116354038, 21336391530, 56072845336, 147293193938, 386802545800, 1015440174105, 2664986640407, 6992429120597, 18342236192911, 48104271625183, 126131857485133, 330661412757849, 866694444127849, 2271310153792653, 5951419145140725, 15592006346881975, 40843695108544189, 106977688061557833, 280162334632201419, 733631202305520295, 1920879212295646835 , 5028971179983459330, 13164896035224130854, 34460145500271737542, 90194533401333027326, 236052520650462605723, 617737759446892331477, 1616472681561394977324, 4229634401067844517379, 11066466015734948920167, 28952633585709558641222, 75742813713705730903128, 198139107468117492983954, 518293154678858829332391, 1355683388763357286553051, 3545843585637896194724954, 9273854572144693977221669, 24253881608876126903605440, 63428333166807074736033687, 165869746366561281345862205, 433744139400074995626612799, 1134183430197179049958786043, 2965629280756902296931934329, 7754162881798332648251120657, 20273935149217167820001356013, 53006217143932891876156597601, 138580370013565217436775001077, 362295770992857890358949112479, 947135090523965253216713099693, 2475985818484816130834863840017, 6472504621056942309374378514011, 16919400984260754186707852485671, 44226888158240989590163524139399, 115605085525179156002568867851861, 302173938675016934026888955528541, 789817803504410696665714476914503, 2064367451810277755059275382499013, 5395572917664105987740673192256839, 14101941331696864771184979319620709, 36856262983330238546966488164530879, 96324113472851914498347199657454773, 251738900296493416592200785835903161, 657896350241062629949287904322605415, 1719319934673251438753955560492992073, 4493122138937598726495722322054170359, 11741734997300443239909808212858710929, 30683794117167751383440065344355288399, 80182350717351828483126887598017169237, 209527795036271946571856474360993828059, 547517270674404532120551857803599390000, 1430696470741944072773329580228680309998, 3738443878609872910981909966170664235554, 9768504239452296634207803163756274587214, 25524619636770130942861831034000051581897, 66693675205482358529627141013777492895635, 174262657580306000582022973334240290310828] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[190, .6931367586], [191, .6933116102], [192, .6934851811], [193, .6936574872] , [194, .6938285439], [195, .6939983660], [196, .6941669696], [197, .6943343683 ], [198, .6945005769], [199, .6946656096], [200, .6948294801]] ----------------------------- theorem took, 35.318, seconds. ------------------------------------------------------------- Theorem Number, 10, Let a(n) be the number of generalized Dyck paths of leng\ th n in the set of steps, {[1, -1], [1, 0], [1, 1], [1, 2]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 3 3 2 2 t X(t) + t X(t) + (t - 1) X(t) + 1 = 0 and in Maple notation t^3*X(t)^3+t^2*X(t)^2+(t-1)*X(t)+1 = 0 Y(t),satisfies the algebraic equation 2 5 4 3 2 t (20 t + 9) + (-56 t - 124 t - 9 t - 22 t + 43 t - 9) Y(t) 2 3 2 2 2 - t (16 t + 8 t + 11 t - 4) (4 t - 11 t + 1) Y(t) 3 3 2 2 3 + t (16 t + 8 t + 11 t - 4) Y(t) = 0 and in Maple notation t^2*(20*t+9)+(-56*t^5-124*t^4-9*t^3-22*t^2+43*t-9)*Y(t)-t^2*(16*t^3+8*t^2+11*t-\ 4)*(4*t^2-11*t+1)*Y(t)^2+t^3*(16*t^3+8*t^2+11*t-4)^2*Y(t)^3 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 1, 2, 5, 13, 36, 104, 309, 939, 2905, 9118, 28964, 92940, 300808, 980864, 3219205, 10626023, 35252867, 117485454, 393133485, 1320357501, 4449298136, 15038769672, 50973266380, 173214422068, 589998043276, 2014026871496, 6889055189032, 23608722350440, 81049178840528, 278700885572096, 959835173086309 , 3310416757032159, 11432971961630999, 39535937094067710, 136883216842976943, 474465914711874487, 1646380234881262372, 5718752217030650552, 19883643328529880013, 69197975679197263363, 241031745639457967929, 840273329685948718350, 2931673947241147625608, 10236321359743370095136, 35767677070491749898656, 125066933201642573915264, 437609180061455927744044, 1532182548688278855041124, 5367888576523903394511380, 18817189685182373426621128, 66001580256641811595420028, 231628921644551647359330556, 813319571290541676713418240, 2857269732606382755797115744, 10042801630734337232467858248, 35315439977545511682606177912, 124243394901787382519248022472, 437294659453763847748981357872, 1539787452066576966020554717200, 5424091409643814200838410291984, 19114679649447398185620349772448, 67386873318956430623722463729664, 237654786135224610628314797699877, 838447288039106501122769941844175, 2959089476994891590841808281408831, 10446936965400892801709685751598366, 36894702048359142875086189720765107, 130340385637063491885089036351824075, 460606372120512531192108674166911212, 1628219618856550506342349081580617048, 5757374004508255119117099831701564319, 20363892114223305428610898571037799585, 72047452637617934055441873516080986363, 254973308001677159620491719556610891034, 902580464190032824288022869282860897860, 3195871150724847715516750596550071597340, 11318837621251182609620276681508533921640, 40097828938715410513335052746158622746240, 142083510221956300013912488902328321006365, 503579121243900217716520543591447833140175, 1785214865926404191694783396658121449422715, 6330086252914983650635565094744780429138910, 22450334334620365350461231551449561426615245, 79639372649503867006233098200741165461631805, 282567713238969571714901086185968977037902120, 1002778155433588631117401489038102741392404488, 3559367377703023042612404221703378366087665552, 12636431862545244173057624914987400793263041104, 44870192393526910558787907361191216110440972464, 159357103416988584841731608141561825289533797408, 566060957037209902245335968335323113169330226976, 2011090274389686280206947135788945406369500601888, 7146195385250021764023389722081536192363438195264, 25397535718611094193567282259082245386204781377536, 90277622587904591779496502813677879253819871453100, 320951198861106168227093715701360010611088890906308, 1141213250970546689543318971259682184898762369302052, 4058468018178676824710597407374680995856014449100424, 14435226902500543320960445108655220043682188575839460, 51351123807469954260389256025959647719598872859788356] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 1, 7, 31, 130, 525, 2057, 7922, 30148, 113716, 426088, 1588348, 5896868, 21820637, 80526709, 296507464, 1089703464, 3998345401, 14650418895, 53616424355 , 196015286006, 715947954020, 2612875362380, 9528858556360, 34728135225488, 126493780157256, 460499141821576, 1675644929737336, 6094632601929356, 22158582231969933, 80534117301132653, 292599952225568180, 1062763184069484192, 3859018908000232451, 14008919768000775285, 50842626222471896901, 184482138475052164942, 669253136297765380917, 2427405546628698664081, 8802694607793221425530, 31916506189897247370644, 115703387279201384145192, 419384173979774531321652, 1519907310511991750066928, 5507630865687288315067024, 19955309757699768254168524, 72293802405577668904028844, 261875818358339202212201344, 948514654944502304316241056, 3435179266966046457050583820, 12439808740609043154019651796, 45044213228557121140767161956, 163089965091958498608104550120, 590446024040118693951850995944, 2137467755787822969360070626664, 7737255131575753439202112574064, 28005524383482990339949528475104, 101361101890092497072062093199952, 366835242822728509383155208988880, 1327529291908742479372017801524576, 4803874406017464127963991157019004, 17382599506065668915341306965690733, 62894745611739486494226320239859549, 227557693289276542212541868422629260, 823279119082426909833881051702415696, 2978392648449432608089654980682136535, 10774493959288977717779164151130334433, 38975586287551797227521719151647456089, 140984055563831748162637570487052402494 , 509952348608084187434800267767852572743, 1844472136245589128856167058734914693371, 6671110368989445793350027398533381647078, 24127274486895580168049135766565455551212, 87257562250153034910051401170585251323860, 315560830358021729122570232023334839616064, 1141165977424923571121183950066334936077740, 4126679435898833171239321963508682564607908, 14922421629777673883089768202790146704397173, 53959134310179326100291206185436805267404653, 195109382205096338629434195820768537753501240, 705471134435493659254766590473253020587950408, 2550754380105159491817544657081440736576525849, 9222458252419483959738889320574997145950742447, 33343698475071706363495235085017570839264053683, 120550826728978620359224219664473182750541419302, 435829164805568907944406192641782328776563104488, 1575623197311202173040812334156439306695886537280, 5696114626369181077788769723203983690709676704480, 20591862764345332996003809036603926733044999007040, 74439489002595514372102028021511870683997422981536, 269092891839238132730567111904839940816922432799136, 972730233854582439514969768615715994271005367991776, 3516205029219849776205749125106990941228804416234320, 12710065443565788783800224963180258706003632515858444, 45942373256307268990999244585472371574188862661118988, 166062405004487615088551175057733786271714999779900272, 600235511762273158117648540344557426594821222691658752, 2169525272149767620806883414764160209059641994128284180, 7841526198092835035735562837960597485721902077328952844, 28341936725208468447407907457698911517636851617585816716] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[190, .5772650412], [191, .5774457687], [192, .5776251545], [193, .5778032157] , [194, .5779799685], [195, .5781554284], [196, .5783296124], [197, .5785025354 ], [198, .5786742126], [199, .5788446592], [200, .5790138898]] ----------------------------- theorem took, 37.915, seconds. Theorem Number, 11, Let a(n) be the number of generalized Dyck paths of leng\ th n in the set of steps, {[1, -2], [1, -1], [1, 1]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 3 3 2 2 X(t) t + X(t) t - X(t) + 1 = 0 and in Maple notation X(t)^3*t^3+X(t)^2*t^2-X(t)+1 = 0 Y(t),satisfies the algebraic equation 2 5 4 3 2 t (29 t + 9) + (-177 t - 81 t + 93 t + 60 t - 2 t - 9) Y(t) 2 2 3 2 2 + t (6 t + 9 t - 1) (31 t + 18 t - t - 4) Y(t) 3 3 2 2 3 + t (31 t + 18 t - t - 4) Y(t) = 0 and in Maple notation t^2*(29*t+9)+(-177*t^5-81*t^4+93*t^3+60*t^2-2*t-9)*Y(t)+t^2*(6*t^2+9*t-1)*(31*t ^3+18*t^2-t-4)*Y(t)^2+t^3*(31*t^3+18*t^2-t-4)^2*Y(t)^3 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 0, 1, 1, 2, 5, 8, 21, 42, 96, 222, 495, 1177, 2717, 6435, 15288, 36374, 87516, 210494, 509694, 1237736, 3014882, 7370860, 18059899, 44379535, 109298070 , 269766655, 667224480, 1653266565, 4103910930, 10203669285, 25408828065, 63364046190, 158229645720, 395632288590, 990419552730, 2482238709888, 6227850849066, 15641497455612, 39322596749218, 98948326105928, 249205769591184, 628164080248656, 1584662249422154, 4000676360466418, 10107612564723278, 25554572385021774, 64651815489864521, 163671444507238419, 414604424265392676, 1050880098774222787, 2665143773089147105, 6762806129129456672, 17169718027841809157, 43613497554655216454, 110838909698132895090, 281818578844741072211, 716879073712295375462, 1824375867825371722519, 4644815138787299518777, 11830473640537847379154, 30144604830858706622385, 76839632990107912235304, 195940365215297486951349, 499827853614467340311790, 1275469563832483663081536, 3255879030421800989036558, 8314010171499430149552262 , 21236980969229242865647188, 54263890285949301341731998, 138695110930263615267329272, 354600851324161940155918554, 906866293102350209637341800, 2319894437764592201079564408, 5936239484437674725175980416, 15193915495855840745715272098, 38899149279980887999850398850, 99613848531935167380079303598, 255155996468127489510984698798, 653725782253127830045543532256, 1675277095431200249501664295920, 4294142129674613532229458321864, 11009371295302268461297595113248, 28232066038503066843010405848816, 72412680255896730050229633088728, 185770290718820984034073194631176, 476678438146741236610807650395544, 1223377154558198677967852498911314, 3140356164337561340111627218826862, 8062677029300855639122980505376604, 20704256212742549059718719679988798, 53176319095921978871991798542333124, 136600877990989505157321565767115566, 350964876670499702831494643590416144, 901876737699549090563341810569964806, 2317941898463865472333981541631229527, 5958380756332587988359308358690090363, 15318733662029548169969883108648446568, 39389910052360574776485124229995908219, 101300891995714947619675809273019879503 , 260559188602979967444178119850227047106] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 1, 3, 6, 29, 56, 195, 497, 1331, 3760, 9652, 26593, 70026, 186546, 497375, 1311729, 3485327, 9196563, 24299861, 64137209, 169002069, 445462250, 1172555762, 3085847580, 8116354038, 21336391530, 56072845336, 147293193938, 386802545800, 1015440174105, 2664986640407, 6992429120597, 18342236192911, 48104271625183, 126131857485133, 330661412757849, 866694444127849, 2271310153792653, 5951419145140725, 15592006346881975, 40843695108544189, 106977688061557833, 280162334632201419, 733631202305520295, 1920879212295646835 , 5028971179983459330, 13164896035224130854, 34460145500271737542, 90194533401333027326, 236052520650462605723, 617737759446892331477, 1616472681561394977324, 4229634401067844517379, 11066466015734948920167, 28952633585709558641222, 75742813713705730903128, 198139107468117492983954, 518293154678858829332391, 1355683388763357286553051, 3545843585637896194724954, 9273854572144693977221669, 24253881608876126903605440, 63428333166807074736033687, 165869746366561281345862205, 433744139400074995626612799, 1134183430197179049958786043, 2965629280756902296931934329, 7754162881798332648251120657, 20273935149217167820001356013, 53006217143932891876156597601, 138580370013565217436775001077, 362295770992857890358949112479, 947135090523965253216713099693, 2475985818484816130834863840017, 6472504621056942309374378514011, 16919400984260754186707852485671, 44226888158240989590163524139399, 115605085525179156002568867851861, 302173938675016934026888955528541, 789817803504410696665714476914503, 2064367451810277755059275382499013, 5395572917664105987740673192256839, 14101941331696864771184979319620709, 36856262983330238546966488164530879, 96324113472851914498347199657454773, 251738900296493416592200785835903161, 657896350241062629949287904322605415, 1719319934673251438753955560492992073, 4493122138937598726495722322054170359, 11741734997300443239909808212858710929, 30683794117167751383440065344355288399, 80182350717351828483126887598017169237, 209527795036271946571856474360993828059, 547517270674404532120551857803599390000, 1430696470741944072773329580228680309998, 3738443878609872910981909966170664235554, 9768504239452296634207803163756274587214, 25524619636770130942861831034000051581897, 66693675205482358529627141013777492895635, 174262657580306000582022973334240290310828] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[190, .6931367586], [191, .6933116102], [192, .6934851811], [193, .6936574872] , [194, .6938285439], [195, .6939983660], [196, .6941669696], [197, .6943343683 ], [198, .6945005769], [199, .6946656096], [200, .6948294801]] ----------------------------- theorem took, 35.605, seconds. ------------------------------------------------------------- Theorem Number, 12, Let a(n) be the number of generalized Dyck paths of leng\ th n in the set of steps, {[1, -2], [1, -1], [1, 0], [1, 1]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 3 3 2 2 t X(t) + t X(t) + (t - 1) X(t) + 1 = 0 and in Maple notation t^3*X(t)^3+t^2*X(t)^2+(t-1)*X(t)+1 = 0 Y(t),satisfies the algebraic equation 2 5 4 3 2 t (20 t + 9) + (-56 t - 124 t - 9 t - 22 t + 43 t - 9) Y(t) 2 3 2 2 2 - t (16 t + 8 t + 11 t - 4) (4 t - 11 t + 1) Y(t) 3 3 2 2 3 + t (16 t + 8 t + 11 t - 4) Y(t) = 0 and in Maple notation t^2*(20*t+9)+(-56*t^5-124*t^4-9*t^3-22*t^2+43*t-9)*Y(t)-t^2*(16*t^3+8*t^2+11*t-\ 4)*(4*t^2-11*t+1)*Y(t)^2+t^3*(16*t^3+8*t^2+11*t-4)^2*Y(t)^3 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 1, 2, 5, 13, 36, 104, 309, 939, 2905, 9118, 28964, 92940, 300808, 980864, 3219205, 10626023, 35252867, 117485454, 393133485, 1320357501, 4449298136, 15038769672, 50973266380, 173214422068, 589998043276, 2014026871496, 6889055189032, 23608722350440, 81049178840528, 278700885572096, 959835173086309 , 3310416757032159, 11432971961630999, 39535937094067710, 136883216842976943, 474465914711874487, 1646380234881262372, 5718752217030650552, 19883643328529880013, 69197975679197263363, 241031745639457967929, 840273329685948718350, 2931673947241147625608, 10236321359743370095136, 35767677070491749898656, 125066933201642573915264, 437609180061455927744044, 1532182548688278855041124, 5367888576523903394511380, 18817189685182373426621128, 66001580256641811595420028, 231628921644551647359330556, 813319571290541676713418240, 2857269732606382755797115744, 10042801630734337232467858248, 35315439977545511682606177912, 124243394901787382519248022472, 437294659453763847748981357872, 1539787452066576966020554717200, 5424091409643814200838410291984, 19114679649447398185620349772448, 67386873318956430623722463729664, 237654786135224610628314797699877, 838447288039106501122769941844175, 2959089476994891590841808281408831, 10446936965400892801709685751598366, 36894702048359142875086189720765107, 130340385637063491885089036351824075, 460606372120512531192108674166911212, 1628219618856550506342349081580617048, 5757374004508255119117099831701564319, 20363892114223305428610898571037799585, 72047452637617934055441873516080986363, 254973308001677159620491719556610891034, 902580464190032824288022869282860897860, 3195871150724847715516750596550071597340, 11318837621251182609620276681508533921640, 40097828938715410513335052746158622746240, 142083510221956300013912488902328321006365, 503579121243900217716520543591447833140175, 1785214865926404191694783396658121449422715, 6330086252914983650635565094744780429138910, 22450334334620365350461231551449561426615245, 79639372649503867006233098200741165461631805, 282567713238969571714901086185968977037902120, 1002778155433588631117401489038102741392404488, 3559367377703023042612404221703378366087665552, 12636431862545244173057624914987400793263041104, 44870192393526910558787907361191216110440972464, 159357103416988584841731608141561825289533797408, 566060957037209902245335968335323113169330226976, 2011090274389686280206947135788945406369500601888, 7146195385250021764023389722081536192363438195264, 25397535718611094193567282259082245386204781377536, 90277622587904591779496502813677879253819871453100, 320951198861106168227093715701360010611088890906308, 1141213250970546689543318971259682184898762369302052, 4058468018178676824710597407374680995856014449100424, 14435226902500543320960445108655220043682188575839460, 51351123807469954260389256025959647719598872859788356] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 1, 7, 31, 130, 525, 2057, 7922, 30148, 113716, 426088, 1588348, 5896868, 21820637, 80526709, 296507464, 1089703464, 3998345401, 14650418895, 53616424355 , 196015286006, 715947954020, 2612875362380, 9528858556360, 34728135225488, 126493780157256, 460499141821576, 1675644929737336, 6094632601929356, 22158582231969933, 80534117301132653, 292599952225568180, 1062763184069484192, 3859018908000232451, 14008919768000775285, 50842626222471896901, 184482138475052164942, 669253136297765380917, 2427405546628698664081, 8802694607793221425530, 31916506189897247370644, 115703387279201384145192, 419384173979774531321652, 1519907310511991750066928, 5507630865687288315067024, 19955309757699768254168524, 72293802405577668904028844, 261875818358339202212201344, 948514654944502304316241056, 3435179266966046457050583820, 12439808740609043154019651796, 45044213228557121140767161956, 163089965091958498608104550120, 590446024040118693951850995944, 2137467755787822969360070626664, 7737255131575753439202112574064, 28005524383482990339949528475104, 101361101890092497072062093199952, 366835242822728509383155208988880, 1327529291908742479372017801524576, 4803874406017464127963991157019004, 17382599506065668915341306965690733, 62894745611739486494226320239859549, 227557693289276542212541868422629260, 823279119082426909833881051702415696, 2978392648449432608089654980682136535, 10774493959288977717779164151130334433, 38975586287551797227521719151647456089, 140984055563831748162637570487052402494 , 509952348608084187434800267767852572743, 1844472136245589128856167058734914693371, 6671110368989445793350027398533381647078, 24127274486895580168049135766565455551212, 87257562250153034910051401170585251323860, 315560830358021729122570232023334839616064, 1141165977424923571121183950066334936077740, 4126679435898833171239321963508682564607908, 14922421629777673883089768202790146704397173, 53959134310179326100291206185436805267404653, 195109382205096338629434195820768537753501240, 705471134435493659254766590473253020587950408, 2550754380105159491817544657081440736576525849, 9222458252419483959738889320574997145950742447, 33343698475071706363495235085017570839264053683, 120550826728978620359224219664473182750541419302, 435829164805568907944406192641782328776563104488, 1575623197311202173040812334156439306695886537280, 5696114626369181077788769723203983690709676704480, 20591862764345332996003809036603926733044999007040, 74439489002595514372102028021511870683997422981536, 269092891839238132730567111904839940816922432799136, 972730233854582439514969768615715994271005367991776, 3516205029219849776205749125106990941228804416234320, 12710065443565788783800224963180258706003632515858444, 45942373256307268990999244585472371574188862661118988, 166062405004487615088551175057733786271714999779900272, 600235511762273158117648540344557426594821222691658752, 2169525272149767620806883414764160209059641994128284180, 7841526198092835035735562837960597485721902077328952844, 28341936725208468447407907457698911517636851617585816716] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[190, .5772650412], [191, .5774457687], [192, .5776251545], [193, .5778032157] , [194, .5779799685], [195, .5781554284], [196, .5783296124], [197, .5785025354 ], [198, .5786742126], [199, .5788446592], [200, .5790138898]] ----------------------------- theorem took, 38.018, seconds. Theorem Number, 13, Let a(n) be the number of generalized Dyck paths of leng\ th n in the set of steps, {[1, -2], [1, -1], [1, 2]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 2 2 2 3 4 4 4 5 6 6 1 - X(t) - t X(t) + t (t + 2) X(t) - t X(t) - t X(t) + t X(t) = 0 and in Maple notation 1-X(t)-t^2*X(t)^2+t^2*(t+2)*X(t)^3-t^4*X(t)^4-t^4*X(t)^5+t^6*X(t)^6 = 0 Y(t),satisfies the algebraic equation 2 8 7 6 5 4 3 t (3638311 t - 1374000 t - 3908416 t + 1269302 t + 1548324 t - 297216 t 2 12 11 - 263680 t + 24576 t + 16384) + (47898156 t - 15856512 t 10 9 8 7 6 - 38179467 t + 6304472 t + 4652272 t + 2287509 t + 3788108 t 5 4 3 2 2 - 995968 t - 1411602 t + 102400 t + 180992 t - 8192) Y(t) + t ( 12 11 10 9 8 163583532 t - 12249441 t - 31967464 t - 112624808 t - 111763326 t 7 6 5 4 3 + 90899222 t + 78656392 t - 23875016 t - 21291031 t + 2493440 t 2 2 2 10 9 + 2653824 t - 65536 t - 126976) Y(t) - t %1 (546439 t - 2462169 t 8 7 6 5 4 3 - 3562264 t + 2680802 t + 2859691 t - 826976 t - 843144 t + 89104 t 2 3 4 + 102592 t - 1024 t - 4096) Y(t) - t 6 5 4 3 2 2 (19725 t - 32178 t - 43080 t + 11785 t + 16794 t - 1024 t - 1920) %1 4 4 4 3 2 3 5 6 4 6 Y(t) + t (27 t + 144 t + 128 t - 8 t - 32) %1 Y(t) + t %1 Y(t) = 0 4 3 2 %1 := 229 t - 144 t - 128 t + 4 t + 16 and in Maple notation t^2*(3638311*t^8-1374000*t^7-3908416*t^6+1269302*t^5+1548324*t^4-297216*t^3-\ 263680*t^2+24576*t+16384)+(47898156*t^12-15856512*t^11-38179467*t^10+6304472*t^ 9+4652272*t^8+2287509*t^7+3788108*t^6-995968*t^5-1411602*t^4+102400*t^3+180992* t^2-8192)*Y(t)+t^2*(163583532*t^12-12249441*t^11-31967464*t^10-112624808*t^9-\ 111763326*t^8+90899222*t^7+78656392*t^6-23875016*t^5-21291031*t^4+2493440*t^3+ 2653824*t^2-65536*t-126976)*Y(t)^2-t^2*(229*t^4-144*t^3-128*t^2+4*t+16)*(546439 *t^10-2462169*t^9-3562264*t^8+2680802*t^7+2859691*t^6-826976*t^5-843144*t^4+ 89104*t^3+102592*t^2-1024*t-4096)*Y(t)^3-t^4*(19725*t^6-32178*t^5-43080*t^4+ 11785*t^3+16794*t^2-1024*t-1920)*(229*t^4-144*t^3-128*t^2+4*t+16)^2*Y(t)^4+t^4* (27*t^4+144*t^3+128*t^2-8*t-32)*(229*t^4-144*t^3-128*t^2+4*t+16)^3*Y(t)^5+t^6*( 229*t^4-144*t^3-128*t^2+4*t+16)^4*Y(t)^6 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 0, 1, 1, 2, 7, 8, 38, 58, 199, 452, 1149, 3277, 7650, 22696, 55726, 157502, 416967, 1128026, 3122336, 8365304, 23402737, 63505268, 176860650, 487957967, 1353427722, 3774616133, 10483218667, 29371164344, 81965145468, 230030965231, 645265199252, 1813615497166, 5107394107927, 14386545035342, 40621735594210, 114720169872202, 324560293765296, 918870098708832, 2604241833793991, 7388579097551618, 20977399823555561, 59621343802550548, 169564084505396074, 482675806553513346, 1374913683174205990, 3919330024586140872, 11180276246544027984, 31912767043682119091, 91151661450077696842, 260504150271472067487, 744947302289612320419, 2131447285649899773482, 6101803306922256585093, 17477041985742079940042, 50083202507295988173638, 143591214804033688415252, 411872960087198439707020, 1181938891881928826882954, 3393237756134243454881923, 9745761652953982997207239, 28002293032810782454429664, 80489913874085151962741069, 231448747966301291146659973, 665775455613130578087964718, 1915825804633865640875510199, 5514859995133063738383715198, 15880307009862092388593834938, 45743033253307915020912523866, 131803971836443495985646934346, 379896890743687513713066562632, 1095299272899961345869825705421, 3158829552927699443639623924172, 9112602113045313890792063470400, 26295302279309798839696626761954, 75897994086190859090327704904386, 219126918794497514604321507948622, 632807420043055365265973336120311, 1827910587314348163667779166891084, 5281328653801629835484380957405308, 15262781202002197316280707499188650, 44118820306298133633541281225090697, 127559089273900630439769019874439578, 368887462905234581317115152477238442, 1067011498596883768561430099061880030, 3086987315483995752005197996854513506, 8932829336110216328563459544355900850, 25854113279862452261954232008644645747, 74843608876778219689608691577566089668, 216701723365174180184832757989599065150, 627553494485772634933093593539396830390, 1817682869249608576889157702251547022020, 5265780349483642346958962385307264328388, 15257485740413687597954596861706947065140, 44215771824385306917170261803795365432526, 128157462508714435731132033352842766515845, 371519434327191324719847785949657971272963, 1077180946975716408102603738960550511247786, 3123660784019082236748624127997292996164579, 9059532190599579345068831623354242648370891, 26279255354283451199405178126260914151192874] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 2, 3, 12, 55, 85, 542, 961, 4322, 11026, 34164, 109871, 293816, 987817, 2684484, 8522974, 24638253, 73622709, 221470508, 645182086, 1953599266, 5702670578, 17088987572, 50388845772, 149381314424, 443275492193, 1307893669263 , 3883792129140, 11458936131912, 33955194499352, 100323976024168, 296647340270956, 877183751319631, 2590971833391558, 7660557860546306, 22622122958748207, 66847040564920806, 197412123912173791, 583020785413567540, 1721710552355770384, 5082998791465246421, 15007904386258090962, 44299587533956838088, 130766767635141586866, 385943416745673293877, 1139008591260123525406, 3361242252930426098496, 9918118601874219047904, 29264650849168687035498, 86340707429165767525834, 254724520556263336533125, 751440233380578018388628, 2216643595428557980608619, 6538446188777141928251330, 19285419732647294228900526, 56880708592025701547502405, 167756149127390916131950803, 494736916761043304096827826, 1458984798545256153088626542, 4302391691497156488159438558, 12686806610611138130027096392, 37409178007657121471191214398, 110303403175249548504346884496, 325225334524179475721357318420, 958883416303446449464873406767, 2827048945069117930795396757028, 8334656203551611141591712181546, 24571368693331309939775173718609, 72436707987808749463884054503778, 213538575178539422075054407016488, 629480881687223732014357454517108, 1855572122247614918852242089371602, 5469687730912310069757587138499206, 16122672532804973859962000479690395, 47522762920421304595936409725021280, 140073764698530477479085053126339938, 412859926423414013921191443677872155, 1216857577874469694475892526888188387, 3586478001154390703180397157484685096, 10570322923174811070900274638720531728, 31153028623828517650258841877927701097, 91813063731973267799200751911666778456, 270583376558339938776124716723585989790, 797426027181299393978132870849729543559, 2350025535056343885089767317589524348846, 6925447678053932190934178755960823671820, 20408750759329759694950047625189273713916, 60142085870274244349493901404218368336216, 177228774133123351546755966393005367419006, 522256487338711682533410379242296982088918, 1538960559473816905322645072222818606622594, 4534875190281451448586351435069661282554003, 13362802540730724696509882277262725879914547, 39375336653770824646311195879701721026368556, 116023423019273778414976565521095563028903568, 341870696236398452760416685512733736073558488, 1007332912756072360818273631319182255741106186, 2968104923522366307761012377658469976837136766, 8745420175709977494239641905684133802790117969, 25767805838916270184148568631421473593510802578] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[190, 1.011221371], [191, 1.011440090], [192, 1.011657183], [193, 1.011872671] , [194, 1.012086573], [195, 1.012298909], [196, 1.012509698], [197, 1.012718960 ], [198, 1.012926713], [199, 1.013132974], [200, 1.013337762]] ----------------------------- theorem took, 124.681, seconds. ------------------------------------------------------------- Theorem Number, 14, Let a(n) be the number of generalized Dyck paths of leng\ th n in the set of steps, {[1, -2], [1, -1], [1, 0], [1, 2]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 2 2 2 3 4 4 4 5 1 + (t - 1) X(t) - t X(t) - t (t - 2) X(t) - t X(t) + t (t - 1) X(t) 6 6 + t X(t) = 0 and in Maple notation 1+(t-1)*X(t)-t^2*X(t)^2-t^2*(t-2)*X(t)^3-t^4*X(t)^4+t^4*(t-1)*X(t)^5+t^6*X(t)^6 = 0 Y(t),satisfies the algebraic equation 2 8 7 6 5 4 3 t (1408261 t + 4194402 t + 533238 t - 2679898 t - 634076 t + 883456 t 2 12 11 + 23040 t - 106496 t + 16384) + (25078366 t + 53046671 t 10 9 8 7 6 - 9924681 t - 32316054 t + 7133967 t + 11473725 t - 6927660 t 5 4 3 2 - 1247728 t + 1756398 t + 94720 t - 359680 t + 98304 t - 8192) Y(t) + 2 12 11 10 9 t (135167090 t + 143330797 t - 163710564 t - 64923588 t 8 7 6 5 4 + 160743194 t + 25580422 t - 89561476 t - 3303696 t + 23650409 t 3 2 2 2 10 + 285440 t - 5005696 t + 1458176 t - 126976) Y(t) + t %1 (380519 t 9 8 7 6 5 - 2024721 t - 2030012 t + 2385850 t + 2320925 t - 1261088 t 4 3 2 3 4 - 631560 t + 276976 t + 72512 t - 39936 t + 4096) Y(t) - t 6 5 4 3 2 2 (12936 t + 28561 t + 3769 t - 27231 t - 6886 t + 10496 t - 1920) %1 4 4 4 3 2 3 5 6 4 6 Y(t) - t (13 t + 8 t + 40 t - 120 t + 32) %1 Y(t) + t %1 Y(t) = 0 4 3 2 %1 := 257 t + 60 t - 44 t - 60 t + 16 and in Maple notation t^2*(1408261*t^8+4194402*t^7+533238*t^6-2679898*t^5-634076*t^4+883456*t^3+23040 *t^2-106496*t+16384)+(25078366*t^12+53046671*t^11-9924681*t^10-32316054*t^9+ 7133967*t^8+11473725*t^7-6927660*t^6-1247728*t^5+1756398*t^4+94720*t^3-359680*t ^2+98304*t-8192)*Y(t)+t^2*(135167090*t^12+143330797*t^11-163710564*t^10-\ 64923588*t^9+160743194*t^8+25580422*t^7-89561476*t^6-3303696*t^5+23650409*t^4+ 285440*t^3-5005696*t^2+1458176*t-126976)*Y(t)^2+t^2*(257*t^4+60*t^3-44*t^2-60*t +16)*(380519*t^10-2024721*t^9-2030012*t^8+2385850*t^7+2320925*t^6-1261088*t^5-\ 631560*t^4+276976*t^3+72512*t^2-39936*t+4096)*Y(t)^3-t^4*(12936*t^6+28561*t^5+ 3769*t^4-27231*t^3-6886*t^2+10496*t-1920)*(257*t^4+60*t^3-44*t^2-60*t+16)^2*Y(t )^4-t^4*(13*t^4+8*t^3+40*t^2-120*t+32)*(257*t^4+60*t^3-44*t^2-60*t+16)^3*Y(t)^5 +t^6*(257*t^4+60*t^3-44*t^2-60*t+16)^4*Y(t)^6 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 1, 2, 5, 13, 38, 116, 368, 1203, 4016, 13642, 46987, 163696, 575816, 2042175, 7294299, 26215927, 94736708, 344015468, 1254647606, 4593682529, 16878510120, 62215957762, 230007985382, 852612196852, 3168359595108, 11800740083576, 44045606325107, 164721039237571, 617148978777583, 2316181581852586, 8706610827498169, 32777540164574119, 123570545078714196, 466475769211533776, 1763136562025613086, 6671991538484571985, 25276100101802859113, 95857089488037503932, 363893671750915858188, 1382735524079117907097, 5258936627524587180296, 20018532472425959709964, 76264882378973750840787, 290776169937515020724622, 1109480587607860328484762, 4236362634599404630254407, 16186979345718023365399126, 61890689908589710188135032, 236788162441057929231090936, 906481251356979376901731572, 3472254168146806439516245852, 13307862200032560528339214444, 51031695790051796244350317755, 195793353373380255061846999352, 751577763515517225449376772964, 2886424260077596338775026932843, 11090453778770790774173089864959, 42631870995974572588259478288830, 163948999685097694281438178407935, 630762921042175124692646870431377, 2427730981044889019800344445037262, 9347732775807755338674379074959572, 36006251774350782839270800703499168, 138742739010253616123822439515958199, 534808627193735170542454600445940932, 2062232491201192827859566546524042872, 7954693048041660294835862000468400054, 30693859825267119442390723675995383969, 118472553290591737110976528087567054533 , 457423229437397341821536612554845001896, 1766644373303497623047263321932390509544, 6825066523538328700903858792328803930489, 26374729289237785639198390721750686036801, 101950456081509435024773999290220794573346, 394191432505587596337242600824038189545744, 1524540499668836620984637670143395646458582, 5897685079375586554211081157332022244875394, 22820869261210325481575538452221488477105381, 88325895684111709972681445199866122830618324, 341937359643471699200377001716059202913031353, 1324052374928654965126768676959967828690662928, 5128158567229393232347808392626679070957720776, 19866119674906259979800256689121320863227842446, 76976419248163567373758873950630055550472610171, 298327461498692526645599606566656958462338358552, 1156425131458233261187280856439700804282193733516, 4483617274224472720624125740286944093765936723826, 17386985933651319524826809633478921821365767791942, 67437727496560803838160280218269486596060393948890, 261614956357265559031747000545153956700527314987588, 1015082755876765614140388021110375872421754684342497, 3939289075302099021727606584808922220241160150590957, 15290092371513229503195174041521719534693827929333908, 59357638055792766898916929671081915237981217442859954, 230470740779357553510102374593612167267787690050665823, 895006432695011857826183798735737954889633147922360376, 3476211750410590813969578694095751054492811383435194552, 13503755842540755030570613849994491941207601422397957448, 52465019750296508337975021669923986213588681571995300488, 203868758531261548665240457937830150237861863452263005464] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 2, 11, 47, 212, 897, 3756, 15577, 63966, 261177, 1061007, 4293193, 17316908, 69666454, 279667076, 1120663484, 4483802537, 17916483702, 71511021581 , 285148949519, 1136063512793, 4522827784119, 17994159319384, 71548226794837, 284340721966841, 1129471727555198, 4484642825956688, 17799753572154674, 70623459009487996, 280121349510269022, 1110752877957989544, 4403247795923950460 , 17451088366844184893, 69147087305818074165, 273926879390853155855, 1084954298251214189170, 4296454992623650733653, 17011261879838965529096, 67343415953064641516496, 266557286637315615159495, 1054937769405692498236924, 4174531054270579276412103, 16517203317602801864329499, 65345608665419831521148900, 258493860473269579544157549, 1022446566609021425374361241, 4043803088385977121798531364, 15991920102261158697295834388, 63237476617959082400886643712, 250042415490281994379985145569, 988598301361619854977898749982, 3908361409272407826348226925317, 15450402690209461842390618219152, 61074027924856571871952887829438, 241405046751457004206128106466640, 954136278172063890696070693262764, 3770942713021327098255729480019918, 14902736713846768447496164377112766, 58892464535780124725738190938361239, 232719090320086971675739475059323867, 919567874394985695131534506101745368, 3633422759452851759810765599972853473, 14355861155670309580147641296129162752, 56718470297871751527583731637489307892, 224079660060091938954900746775421603141 , 885245478018440268343639784551664178953, 3497107617642046478127034707324720025491, 13814618333703687043631417698254876633738, 54570003165714997821920513046945998340897, 215553367735904669760368818713387629924879, 851416209850659554676925714015712597070728, 3362914152498975848911863531127938909384713, 13282410071539478692561656092286087599641547, 52459684337798473305618328822369648753301675, 207187034574604253181265061506686647657287499, 818253733264302795025712297342334597925296640, 3231486333660658514616238335572305029459518158, 12761623526866181598785362588930198502900367095, 50396359271439341863841025315778695196436549804, 199013420704020824118299375412448941425796696800, 785879308101814839768996869031478918741569073799, 3103272691063969572032834638332691318358058123586, 12253916461692274648684948244054227062908477184535, 48386149472760063128047300215591784886848176476479, 191055107157594875868934966530336618607617722703261, 754376133140670297039660053381595802838654872565283, 2978579356834782014014965857835444220293287445197504, 11760413591428997770112447877038095674208809456704157, 46433180010489698923916324800556501413928588904741173, 183327193888798862461074616930269639661686854894681165, 723799360531808470889085503818731936081787087592574100, 2857607268229426551751118075334696981132020654797791385, 11281844905127330383165793052932423361907060021884101946, 44540094201218135356956735023117362209722445409699922825, 175839216388495602998917775327037115045856549373681241880, 694183238414928908358942508163577756269199929258504379168, 2740478805414513027286804821164831290561226542418798227256, 10818645079760057206219006024546787089626281255041459295120, 42708424888813465854650789507194799316261648706566815358260, 168596533226238372284268679807498718281057961068345159718988] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[190, .8594411660], [191, .8596702471], [192, .8598975956], [193, .8601232332] , [194, .8603471815], [195, .8605694611], [196, .8607900938], [197, .8610090995 ], [198, .8612264977], [199, .8614423088], [200, .8616565517]] ----------------------------- theorem took, 245.889, seconds. Theorem Number, 15, Let a(n) be the number of generalized Dyck paths of leng\ th n in the set of steps, {[1, -2], [1, -1], [1, 1], [1, 2]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 2 2 3 4 4 1 + (-2 t - 1) X(t) + t (3 t + 2) X(t) - t (2 t + 1) X(t) + t X(t) = 0 and in Maple notation 1+(-2*t-1)*X(t)+t*(3*t+2)*X(t)^2-t^2*(2*t+1)*X(t)^3+t^4*X(t)^4 = 0 Y(t),satisfies the algebraic equation 2 4 3 2 t (81 t - 144 t + 358 t - 168 t + 24) 6 5 4 3 2 - (4 t - 1) (162 t + 594 t + 391 t - 476 t + 38 t + 40 t - 8) Y(t) 5 4 3 2 2 2 + t (243 t + 1206 t + 561 t - 352 t - 68 t + 32) (4 t - 1) Y(t) 2 3 2 3 3 - 2 t (9 t + 4) (9 t + 23 t + 5 t - 4) (4 t - 1) Y(t) 4 2 4 4 + t (9 t + 4) (4 t - 1) Y(t) = 0 and in Maple notation t^2*(81*t^4-144*t^3+358*t^2-168*t+24)-(4*t-1)*(162*t^6+594*t^5+391*t^4-476*t^3+ 38*t^2+40*t-8)*Y(t)+t*(243*t^5+1206*t^4+561*t^3-352*t^2-68*t+32)*(4*t-1)^2*Y(t) ^2-2*t^2*(9*t+4)*(9*t^3+23*t^2+5*t-4)*(4*t-1)^3*Y(t)^3+t^4*(9*t+4)^2*(4*t-1)^4* Y(t)^4 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 0, 2, 2, 11, 24, 93, 272, 971, 3194, 11293, 39148, 139687, 497756, 1798002, 6517194, 23807731, 87336870, 322082967, 1192381270, 4431889344, 16527495396, 61831374003, 231973133544, 872598922407, 3290312724374, 12434632908623, 47089829065940, 178672856753641, 679155439400068, 2585880086336653, 9861191391746256, 37660870323158835, 144029959800495438, 551546279543420059, 2114684919809270434, 8117356580480783638, 31193334574672753772, 119994768635233629431, 462054434301743595662, 1780873197452044558004, 6870085192588231176696, 26525413368458894971320, 102498149078904263186092, 396376387809028169260770, 1533990030181715109390144, 5940819914799609711208621, 23023231953092569848963708, 89283253961720586067595703, 346454385040363614359308410, 1345190037713910576952235253, 5226038144996982795627301998, 20314365865347330357708328558, 79007200442487739347572093244, 307435653473446231048643096847, 1196899964358115792546873957924, 4661974359967246584457211409153, 18166993097773828593546359274018, 70825635249096059428913247959357, 276239400690328919028087352434180, 1077859789424649612050263190848671, 4207406847476763120078646546475340, 16429967829511848960435267122989362, 64183518431911827259692722310744426, 250824422659309684041099915681954387, 980552244573444785667033316872784254, 3834613660197593378144406044210346051, 15000923724468763422493060423296021450, 58702371619748128538961120228244523322, 229789658007412342606995920067121896060, 899784691777811683828487590911757184679, 3524327660995500923314394240876627529666, 13808291144605449156112867267105174260978, 54116056174220175567728544043777003031508, 212144411220338720127210592679905848015798, 831865553887187487019327030449843431699124, 3262779604741124610385860524847406304799636, 12800663089560825245598001156707695698208952, 50232468096169454775367333419723565209570159, 197170158538966858096222843007903930749039022, 774104996763583376262064205135293538225433260, 3039891383145269395161868470307942665881954240, 11940248276612158030892902676253554427215119072, 46909782093278416533525922170391659114339304192, 184334232810121482305598457302715870545851071192, 724500903559396556083836246305640635198157727056, 2848132117700716828107062827017680854530538549704, 11198699102069668154302936865474952658336995867420, 44041217250750367888832696898555022966322046396730, 173234130336223732600055340099650311580809549131220, 681535043731071292988601178704899499814852312169530, 2681771340108510395970569860401129350232076752294320, 10554372428038158055387757107356315861285006611370410, 41544979561504552826597843172141328058908696421305880, 163560545687403946504274192577922763424783943494504435, 644037076656191537576599711875544763776519494139546920, 2536378172533074139312752782660485383627956991656120055, 9990483526075716162794929791613271384893217244892795170, 39357450970796372868828851360049000832080917735346418977, 155072220357382194252045200359802826534940387996568255010, 611091591810684456067863161251870728926938257559235085144] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 3, 6, 53, 168, 883, 3332, 14855, 59306, 249552, 1011324, 4162533, 16904164, 68916311, 279491766, 1133987621, 4590010542, 18570201382, 75033966062 , 302991796682, 1222486925180, 4929596954168, 19866453776480, 80024291224217, 322199379242118, 1296749667248358, 5217093650868012, 20982599499502855, 84364465645323916, 339110711826931075, 1362745202856167492, 5475050273836411703 , 21992246242879951838, 88321454402666081050, 354637163035072119658, 1423735741661734994248, 5714875209461723183892, 22936174250793337112850, 92039928707114368069470, 369297581908481301739202, 1481579770392117341879520, 5943266318450848666253008, 23838550377713547974750796, 95607436983449868438619018, 383409756579495470070846656, 1537434314544091871931394176, 6164445338718734937488412852, 24714817661409460151917221909, 99080590015508752791607623930, 397181668902543971505671568792, 1592065017125816198714231305854, 6381236469954397848136374733208, 25575413257589287932638603372580, 102498030472580492368874344125862, 410756641707957998504265022628948, 1646004359896048697333262642644903, 6595621276808382032381654221643522, 26427723086025668113147928848126742, 105887330560719864283992105167692860, 424237758611523403867956614496647905, 1699638300775897047377951236555239220, 6809046881293645074134562669316006183, 27277180508882640314220245073767137654, 109268917773758929296886902314927416741, 437701920236026396984287806143834888006, 1753256707095909153621630493685860217746, 7022607265886078203293327501885540603442, 28127919339916387910466307808676557573692, 112658431123027669382327741299150737547012, 451208361055444657389055689082758482403022, 1807084192524511726134961176125338201125730, 7237155107066250782270738680700248066966168, 28983180059236917249191546989809483621916108, 116068198884488513433471224358966774219552100, 464804033120846126472110179490659713932688676, 1861299758986506670319979930065154084681340456, 7453373535435751573726541622262868647920790552, 29845572333138757458513367139901393846209801866, 119508200217606690679293871554229079323559627630, 478527132925468644793725153779002033111573607138, 1916049747670191254359802196143294217118518895592, 7671823749390471179060078891553966173550454304388, 30717250264318782070200349938227016709365613520496, 122986710923868404981820707641587655501628961328176, 492409483466335864233315050526595884969927130723888, 1971456635210738943729208211474810718532317949081960, 7892977529967243178756821827761781123816425847273692, 31600032702377988476130169045709201076364735692881178, 126510749050179937067284169887408954777847012069512452, 506478186228070029497542257863248538291009251375649868, 2027625163755028065509953799903293464806647075964297312, 8117240008399677988864139533265977071096490749502672958, 32495487878878957895363090331860469724352036416729877192, 130086389777930681608851041378788614337838282295661490968, 520756793727797039208534756975030036338063797834185378832, 2084646710721651334945607161233719042611179948845227274713, 8344965964098281736826502491925168836459043898373465904242, 33404994241895062187648499208297800474682808914240855389772, 133718992670765909650863972094615889832637336406599824228322, 535266159037113317442359973587785117219151190390550658723722] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[190, .9050469037], [191, .9052547646], [192, .9054610835], [193, .9056658798] , [194, .9058691725], [195, .9060709792], [196, .9062713195], [197, .9064702104 ], [198, .9066676692], [199, .9068637138], [200, .9070583606]] ----------------------------- theorem took, 54.786, seconds. ------------------------------------------------------------- Theorem Number, 16, Let a(n) be the number of generalized Dyck paths of leng\ th n in the set of steps, {[1, -2], [1, -1], [1, 0], [1, 1], [1, 2]} also Let b(n) be the sum of the areas under these generalized Dyck paths of \ length n in the set of steps Let X(t), Y(t) be the ordinary generating functions of the sequences a(n), \ b(n), in other words infinity ----- \ n X(t) = ) a(n) t / ----- n = 0 infinity ----- \ n Y(t) = ) b(n) t / ----- n = 0 X(t),satisfies the algebraic equation 2 2 3 4 4 1 + (-t - 1) X(t) + t (t + 2) X(t) - t (t + 1) X(t) + t X(t) = 0 and in Maple notation 1+(-t-1)*X(t)+t*(t+2)*X(t)^2-t^2*(t+1)*X(t)^3+t^4*X(t)^4 = 0 Y(t),satisfies the algebraic equation 2 4 3 2 t (775 t - 1460 t + 1006 t - 264 t + 24) + (t - 1) (5 t - 1) 6 5 4 3 2 (425 t - 1520 t + 1527 t - 68 t - 282 t + 88 t - 8) Y(t) - t 5 4 3 2 2 2 2 (150 t + 540 t - 889 t - 240 t + 228 t - 32) (t - 1) (5 t - 1) Y(t) 2 3 2 3 3 3 - 2 t (5 t + 4) (5 t - t - 17 t + 4) (t - 1) (5 t - 1) Y(t) 4 2 4 4 4 + t (5 t + 4) (t - 1) (5 t - 1) Y(t) = 0 and in Maple notation t^2*(775*t^4-1460*t^3+1006*t^2-264*t+24)+(t-1)*(5*t-1)*(425*t^6-1520*t^5+1527*t ^4-68*t^3-282*t^2+88*t-8)*Y(t)-t*(150*t^5+540*t^4-889*t^3-240*t^2+228*t-32)*(t-\ 1)^2*(5*t-1)^2*Y(t)^2-2*t^2*(5*t+4)*(5*t^3-t^2-17*t+4)*(t-1)^3*(5*t-1)^3*Y(t)^3 +t^4*(5*t+4)^2*(t-1)^4*(5*t-1)^4*Y(t)^4 = 0 Using these algebraic equations, we can get many more terms. Here are the fi\ rst 101 terms of the enuerating sequence, a(n), starting at n=0, are [1, 1, 3, 9, 32, 120, 473, 1925, 8034, 34188, 147787, 647141, 2864508, 12796238 , 57615322, 261197436, 1191268350, 5462080688, 25162978925, 116414836445, 540648963645, 2519574506595, 11779011525030, 55225888341334, 259612579655392, 1223396051745310, 5778116086462293, 27347124593409513, 129681868681425643, 616072123886855885, 2931681447103047687, 13972949818523099259, 66696500485420585110, 318803423221000803432, 1525852728670173719609, 7312059310463342118721, 35081215570214126170473, 168496226788080483702535, 810142199984165279526260, 3899102778065263063546530, 18783607897859472881263225 , 90569997329067191894683145, 437081298169980061730084955, 2111034934785413366875488795, 10203940232459352070358671815, 49358812944657330842426387235, 238930623633883719698052914430, 1157380389499151041370438447700, 5610031665986465785678823529804, 27209956176926345181672284632318, 132054390039098720357658664965527, 641253999173660901171561096973791, 3115659065615590120086821780931555, 15146185403763094626935616111599177, 73668446150550500763124675518650564, 358489565929896574328232621809841986, 1745342246184829265406121720836504123, 8501321841064199095413175513927793595, 41427351638917971637647467563291734469, 201965025778414615775531934775637840853, 985024903834036403146646401376067228757, 4806117198883383966476023752849646832041, 23459132951299387338605409746398132395783, 114549876489982832134489155806946799054653, 559547848525897172134211882758649658289622, 2734228930964839384539804108547155300035896, 13365428343958127305091127736057893792933945, 65354694030251517093087279720909851994079233, 319677680132890888108671556737159164004051585, 1564175154205329760497269536308362056569554895, 7655826427536241331351735233336327147280979560, 37482508997161134858870416726506157702532740750, 183565692623143213908041164950353238984416341885, 899243327254521500394985907378654230512893266133, 4406385280263713995394799912808252217011931796827, 21597529454568227789073290172856310351091804367971, 105886110569423960459541980572738896743139462951515, 519259319516919564004420658059586364130542486058527, 2547048224785155254512743403701863465103957915967688, 12496686864588185992857819933635943656248425587270652, 61327441734666485828092115260260221528115494770364861, 301033297570046497081222525805244524883574375910686245, 1477990106786705784046732451350049902261276760786827735, 7258109001458181792939964373913601879866230581784071325, 35650708125468185782254786588958197581212392854437300560, 175147271941802088083410424582414833655411642091936558120, 860650921592136238653645814899768270926114854099185048705, 4229968347622487939468524823426617041986588284987964669343, 20793694484582873173229931539619271918031544971737007397953, 102237158441083706530806689431776532717600319595338536116697, 502766837530152221698395426946609159183630645890006765321621, 2472882329953180364947900996964265585354190269326677780225195, 12165152054050346713667462495534515861584744131856674212077188, 59855937720461953779656642516867867001810770153187852217115580, 294558079922278097378610398631889781547665001363693286479230300, 1449796652386643632262108407314973221008909034851905602576823300, 7136975268590578065283643331284519899578669168374831010715496700, 35139111804141564321929388337021665334134331661092716062145790150, 173035592007238799636514926173400324675015168482071702569259553895, 852210362966131248349989409574499000644098753693729562104772997195, 4197817497650525071046495415575336122415123086111821102811635556145] The first 101 terms of the sum of areas sequence, b(n), starting at n=0, ar\ e [0, 0, 3, 18, 113, 636, 3487, 18656, 98429, 514012, 2664690, 13737758, 70522801 , 360806214, 1840913908, 9371761174, 47621259557, 241601881822, 1224111502194, 6195045902854, 31321134873744, 158217553824544, 798622703316154, 4028438371631942, 20308239308212037, 102323623873153810, 515313296262175206, 2594054240062008690, 13053194513626873348, 65659889953142043376, 330173873558299929392, 1659802785989257457816, 8341653442425350379973, 41912071242528366149892, 210535758897546519782006, 1057352845368672621901200, 5309188148971984465502484, 26653639602123776232914638, 133786003742616538077923218, 671422062290481196833070418, 3369115139401799021775723788, 16903458272506945284122947878, 84796618310136726261050582163, 425332159026439916093375165798, 2133181496179112160759035148538, 10697448787865688773808783427678, 53639921716221066347896974043243, 268939162046244880914402407636158, 1348280475484786853040729046469941, 6758788444176998609852853653955094, 33878317826194203635351191191039492, 169801307121974854204298237775179230, 850997190989965683538539951664283768, 4264662959105477325397518312078390672, 21370381193066698778991707822692701736, 107080938948650105409228231951445976940 , 536519744970631343188712275051607042344, 2688029888048708310889295912243996248428, 13466615321821727460616757964268120939001, 67462099980725776121652921628789535120664, 337939763581800608618026814759294594358762, 1692769064453896691899693020762985474260340, 8478831279411613930891086968304073423691783, 42467331101470728452745574807761694332704726, 212694090923589485452518375843263245830279685, 1065217038296219837371380293071354401023983006, 5334622711598059375830428219832762845085733434, 26714860064298972978038230289147126777368651734, 133778497768461997530267543943599311636772962356, 669891610017613949473171747460308626768443113376, 3354348173112544015904310934316046065974737537686, 16795682369062257315424431579607017351192264031216, 84095660932812325735108189266965500266768163166616, 421052672999326398581928519713153703608948004218672, 2108077799610471121813137299948494437336884261892211, 10554183572741119625220019703017656248467621158673092, 52838547786319251385984251342123336537044301003701830, 264524388895930715777155452016640158911907279576525640, 1324248723960009765467286385728271467204501990148615219, 6629225080270547714498782091782551658451519184070046118, 33185294031636448910872920448671992863203109376816136764, 166118732496804807248128601696428238055437473073875507042, 831537609132834589000736405790126062751130798734530899467, 4162323536544599809545567616861886620960122776821549281032, 20834386308089675078977663546267170338827563319229229102937, 104283789917471994058151310137507523928076757097034487524862, 521968602758566884274798079964519739480667598500369656729407, 2612544454598301963127580127748445198148084526502369635700272, 13076001925393958066692544048495175624818564642341620162814390, 65445301721372453225755901711588900680786198682983844637109324, 327547590121027409241012768073953674889796020428034337897386845, 1639317193910258800246806369296439687462236245236953409106113998, 8204354879962082456711373346808404319379245719327441590132261423, 41059999343213393012206154616399855369209843822558842235034425580, 205488104883576826615457427201454048818393967813471699335809908735, 1028366375127290431375942249559880512511603280343372573412739921110, 5146389570835805371628886840139764674764954626447356066458105355285, 25754387662132296331013339361888706459171109254392336026659322439510, 128882438016567107885097700115336502619316353351989645079131319350400, 644956404276831241323019728204795743218699631915505918658669157443530, 3227462695569465530843411790534937107217581032224816015981777984545570] Finally, here are the average areas for n from 190 to 200 divided by n^(3/2) [[190, .7991316761], [191, .7993463220], [192, .7995593560], [193, .7997707982] , [194, .7999806687], [195, .8001889865], [196, .8003957716], [197, .8006010424 ], [198, .8008048174], [199, .8010071150], [200, .8012079528]] ----------------------------- theorem took, 58.171, seconds. -------------------------------- This ends this book that took, 1089.784, seconds to create