This is the story of simple random walks that start and end at the origin, in the, 1, -dimensional (cubic) lattice The exponential Generating Function is J[0] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520, 601080390, 2333606220, 9075135300, 35345263800, 137846528820, 538257874440, 2104098963720, 8233430727600, 32247603683100, 126410606437752, 495918532948104, 1946939425648112, 7648690600760440, 30067266499541040, 118264581564861424, 465428353255261088, 1832624140942590534, 7219428434016265740, 28453041475240576740, 112186277816662845432, 442512540276836779204, 1746130564335626209832, 6892620648693261354600, 27217014869199032015600, 107507208733336176461620, 424784580848791721628840, 1678910486211891090247320, 6637553085023755473070800, 26248505381684851188961800, 103827421287553411369671120, 410795449442059149332177040, 1625701140345170250548615520, 6435067013866298908421603100, 25477612258980856902730428600, 100891344545564193334812497256] The sequence is annihilated by the recurrence operator 2 (1 + 2 n) - ----------- + N 1 + n The asymptotic behaviour is n / 1 1 5 \ 0.5641895835 4 |1 - --- + ------ + -------| | 8 n 2 3| \ 128 n 1024 n / -------------------------------------------- 1/2 n This is the story of simple random walks that start and end at the origin, in the, 2, -dimensional (cubic) lattice The exponential Generating Function is 2 J[0] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [4, 36, 400, 4900, 63504, 853776, 11778624, 165636900, 2363904400, 34134779536, 497634306624, 7312459672336, 108172480360000, 1609341595560000, 24061445010950400, 361297635242552100, 5445717990022688400, 82358080713306090000, 1249287673091590440000, 19001665507723090592400, 289721539396666805313600, 4427232449127577876238400, 67789381546187865401760000, 1039907943302284685225610000, 15979641419960227387050813504, 245935191321399712625557194816, 3790573127143000234651249164544, 58502467906161100560306268993600, 904040514754422904734530644281600, 13986511252711760583915116323307776, 216623552013904104610814351046943744, 3358511241965567934376258434786405156, 52120146913882551047712366894297747600, 809575569191760455547338460167829027600, 12585760930357458053423276437090723266624, 195817348302259092738601038640044246873616, 3048971947707052462246909963222193693468224, 47508219406792714958833430867381826941160000, 740765898390201201817024093656033798643360000, 11557799929633114251350118421268267343333024400, 180441940126883670679614817280050213062719745600, 2818740420712248542657293596921237908818767182400, 44057110956508373652134010249336817503981812640000, 688984034772338595398765131357913991090161859240000, 10780133411223099287217024405898344145037376962054400, 168752901282303374664355726096869920349819505903161600, 2642904197719586939741082254435179612685538988784870400, 41410087472950125227714005809509541674456578773929610000, 649108726418971642261659443001541487071989920339697960000, 10179063404211745705290438721372972983668117134799007529536] The sequence is annihilated by the recurrence operator 2 4 (1 + 2 n) - ------------ + N 2 (1 + n) The asymptotic behaviour is n / 1 1 1 \ 0.3183098861 16 |1 - --- + ----- + ------| | 4 n 2 3| \ 32 n 128 n / ------------------------------------------- n This is the story of simple random walks that start and end at the origin, in the, 3, -dimensional (cubic) lattice The exponential Generating Function is 3 J[0] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [6, 90, 1860, 44730, 1172556, 32496156, 936369720, 27770358330, 842090474940, 25989269017140, 813689707488840, 25780447171287900, 825043888527957000, 26630804377937061000, 865978374333905289360, 28342398385058078078010, 932905175625150142902300, 30862498453931119524941700, 1025612461904076314913090600, 34221233837555288931672244980, 1146036523273637310151384256280, 38507502835917076808710010791800, 1297804711481467883526891692041200, 43861095233349763172644603553579100, 1486134894511368488511298785872590056, 50473042959116456206695526548113874936, 1717935171579831227898110571243045115440, 58591130504414522992818338034043897226760, 2002044498542496713444559505063252933261680, 68529473603957996416207510223654247514473456, 2349596714984894432382110228494495236586834656, 80681997360632682235248226291172020032499469370, 2774511796435675001461553546051852814078630633180, 95540165693636170851827063699670877946754920344740, 3294131649118176174850910406317256575226136984268520, 113715685963242228605749065449983286191670924631338820, 3930028637507047671218500394850794924034502049231057400, 135969247090776809839238946061942634518417076025869103000, 4709025755667844340496023085216048877206005830713755934000, 163246703984118797490344652201478353553692393611900875694580, 5664479341216703343854078510029587685311148601499280095932280, 196724532893587305415995590092646261253823748570296428982631400, 6837889531138318706183422168291377920453361981414540691676595600, 237866882568170847968627386837886964145706744904218561602815415800, 8280921245250150903200571527496047270462322900806713570601108119440, 288497166450702182798146228435753944706083873007896790911993667389840, 10057940417526126548843054212711360489062570965029683773091356830586400, 350887917932042036737170173391404927952251239940098702788704651485355100, 12249197735167377456923957901791718691599917703562743723032512167189541800, 427873607843692547458937479389986373816485475359163324762895999681076159640 ] The sequence is annihilated by the recurrence operator 2 36 (2 n + 3) (1 + 2 n) (1 + n) 2 (2 n + 3) (10 n + 30 n + 23) N 2 ------------------------------ - --------------------------------- + N 3 3 (n + 2) (n + 2) The asymptotic behaviour is n / 3 13 27 \ 0.2332905158 36 |1 - --- + ------ + -------| | 8 n 2 3| \ 128 n 1024 n / --------------------------------------------- 3/2 n The expected number of visits to the origin is 1.516386059 The Polya constant, i.e. the probablity of returning to the origin is estimated to be 0.3405373295 This is the story of simple random walks that start and end at the origin, in the, 4, -dimensional (cubic) lattice The exponential Generating Function is 4 J[0] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [8, 168, 5120, 190120, 7939008, 357713664, 16993726464, 839358285480, 42714450658880, 2225741588095168, 118227198981126144, 6380762273973278464, 349019710593278412800, 19310744204362333900800, 1079054103459778710405120, 60818479243449308702049960, 3454039994708417142139594560, 197490435307460298840440788800, 11360014502638218575756079820800, 656992880831611307070049327485120, 38182526849044031197596083174760960, 2228925191153772054359769866962575360, 130642221205751821180454091455388057600, 7685652105392951743099919578297662969600, 453687526340726999466185734291092880939008, 26865624278707814098320171698639130024536064, 1595507883441056800836236333149791985348050944, 95010559630095112452701218196957817465284546560, 5671973497969435568980604951629770694726804930560, 339400613020464943321820563716624850094846472945664, 20353590735349545626273596834297878264431310890074112, 1223093236659569612117465703994253638159195729207494312, 73640006522839054333566645432098046184535441338869176640, 4441751428524916244970239409605289784727778636714056523840, 268371006025652006564567240516462529836748525193696970686464, 16241135215851907274937886413534055087730347896316862389602112, 984370374072341772945167553656530923040003787862740301575286272, 59748499018024410788813503432369278391424468228225775011295283200, 3631528359864014922416588015942340473486412187214078122110960435200, 221012053991085327542689805253246252800998397549518639404243893260480, 13467248319850446115015892105093236675336602927723342449351562182499840, 821584920021912626050659443358728038486609904804251606063464223317834240, 50177940954225299428458290264334511378199399542895147597840022431906611200, 306787555922592490226774237248814845156357503942526256898056214495169105\ 9200, 1877614382828600915112914368537605051646881434525373960589352941102\ 49481338880, 115026902092715874439236513744505736470391943200336890711844\ 22462098793259991040, 705340207968385417220333866606620197533159953859548\ 463644536718744249870204272640, 43289945609188781567656949912522563155569\ 822788377075780700738306500941198052038400, 26591855956723631725677288510\ 71593294639595805327593685189183430284212866999349606400, 163481443380087\ 416632279448361484485900921794816941809380834609069208470347747273095168] The sequence is annihilated by the recurrence operator 2 2 2 256 (2 n + 3) (1 + 2 n) (1 + n) 4 (5 n + 15 n + 12) (2 n + 3) N 2 -------------------------------- - --------------------------------- + N 4 4 (n + 2) (n + 2) The asymptotic behaviour is n / 1 1 1 \ 0.2026423783 64 |1 - --- + ---- + ----| | 2 n 2 3| \ 4 n 8 n / ---------------------------------------- 2 n The expected number of visits to the origin is 1.239467122 The Polya constant, i.e. the probablity of returning to the origin is estimated to be 0.1932016733 This is the story of simple random walks that start and end at the origin, in the, 5, -dimensional (cubic) lattice The exponential Generating Function is 5 J[0] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [10, 270, 10900, 551950, 32232060, 2070891900, 142317232200, 10277494548750, 770878551371500, 59577647564312020, 4717432065143561400, 381091087190569291900, 31308955091335405435000, 2609450031306515140215000, 220199552765301571338488400, 18783738092023856268838548750, 1617638727435662498928811015500, 140490485550992755679150264422500, 12293782273971686129015545115175000, 1083090386083088583352536234005855700, 96006393015807453807557942147810961000, 8557500850900079056484005775576915289000, 766641969179324289735621204502872101790000, 69000383228725934955840875855710619182117500, 6236783650454113027121597421048351804454369560, 565947509654444694407365720533908558720532834600, 51543053701625730737772522025647751529592243641200, 4710084717031098380148037287111553911209738281399000, 431767402588253177020881048618131197389442488020142000, 39695550292118585799560005521816095317642367147975772400, 3659501197431807424613492910042731769601491369176252116000, 338232340050852341337427225083135178435667802074952578196750, 31336716524291411406073840063524916810784053946195637447487500, 2909879323507301078825334522432613220861399941292051147332522500, 270783979705970526653292933585915032498783066288518741281968002200, 25249081196548344093794389242315955133773225171873402074539217446500, 2358822056181276281429750263765433894482946143451672269746854373109000, 220763832188084740662622109203712626029670830989836482884509281521925000, 20696807672737074189673909614781218446708316095327630567580529073894950000, 194349803870299870415553979836261549948087232346486508752566485003781809\ 2500, 1827828797386672013401788144950840183174132866392346794587601229880\ 10013125000, 172157300328376432660977067804740321982417560365706390579944\ 30894310922880475000, 162376809248944453745661853235488529376975062418947\ 7249748244146258763394683250000, 1533569124074157456628427511460828920162\ 96577381391949433304535711233292814985625000, 145023216307313669311939373\ 20159352203942089787586036044945658529840825982852588434000, 137310031451\ 191171600204297303127791172094029743346751989734072309476735158207447817\ 0000, 1301591939127362208703389868193785252570541610304844294031750309867\ 49504579368457983180000, 123519067241564762322811965573010750883113944723\ 47282802515695470408178437164935885843237500, 117343694051093633647986770\ 4217224380848202841530693291180825344726690086975550021446747975000, 1115\ 920723477836890389121628827638646381070953551319056523886716800407954273\ 20890225585532624520] The sequence is annihilated by the recurrence operator 1800 (2 n + 5) (2 n + 3) (1 + 2 n) (n + 2) (1 + n) - -------------------------------------------------- 5 (n + 3) 2 4 (2 n + 5) (2 n + 3) (n + 2) (259 n + 1036 n + 1062) N + -------------------------------------------------------- 5 (n + 3) 4 3 2 2 2 (2 n + 5) (35 n + 350 n + 1323 n + 2240 n + 1433) N 3 - --------------------------------------------------------- + N 5 (n + 3) The asymptotic behaviour is n / 5 65 425 \ 0.1997240968 100 |1 - --- + ------ + -------| | 8 n 2 3| \ 128 n 1024 n / ---------------------------------------------- 5/2 n The expected number of visits to the origin is 1.156308125 The Polya constant, i.e. the probablity of returning to the origin is estimated to be 0.1351786099 This is the story of simple random walks that start and end at the origin, in the, 1, -dimensional (cubic) lattice [0 <= x[1]] The exponential Generating Function is J[0] - J[2] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, 18367353072152, 69533550916004, 263747951750360, 1002242216651368, 3814986502092304, 14544636039226909, 55534064877048198, 212336130412243110, 812944042149730764, 3116285494907301262, 11959798385860453492, 45950804324621742364, 176733862787006701400, 680425371729975800390, 2622127042276492108820, 10113918591637898134020, 39044429911904443959240, 150853479205085351660700, 583300119592996693088040, 2257117854077248073253720, 8740328711533173390046320, 33868773757191046886429490, 131327898242169365477991900, 509552245179617138054608572, 1978261657756160653623774456] The sequence is annihilated by the recurrence operator 2 (1 + 2 n) - ----------- + N n + 2 The asymptotic behaviour is n / 9 145 1155 \ 0.5641896856 4 |1 - --- + ------ - -------| | 8 n 2 3| \ 128 n 1024 n / -------------------------------------------- 3/2 n The expected number of visits to the origin is , 2.000000000 The relative Polya constant, i.e. the probablity of returning to the origin, while staying in [0 <= x[1]] is , 0.5000000000 This is the story of simple random walks that start and end at the origin, in the, 2, -dimensional (cubic) lattice [0 <= x[1], 0 <= x[2]] The exponential Generating Function is 2 2 J[0] - 2 J[2] J[0] + J[2] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [2, 10, 70, 588, 5544, 56628, 613470, 6952660, 81662152, 987369656, 12228193432, 154532114800, 1986841476000, 25928281261800, 342787130211150, 4583937702039300, 61923368957373000, 844113292629453000, 11600528392993339800, 160599522947154548400, 2238236829690383152800, 31383972938049937686000, 442514018426504121372600, 6271444827299932255514448, 89301086173347753313564704, 1277147280034703583103520608, 18339331632025423373136761440, 264339331799538861033488492480, 3823540528352039525400523871872, 55487590167495928435147118608336, 807722761415480503697993851559982, 11791888442054875802649856763415780, 172617392151761291161479415814036040, 2533365726722515711236569331137424168, 37270146231872686094137997457183906904, 549562355390600660102182761936228135088, 8121063146460293155356142028612278109600, 120254204284123571723542872346758733546000, 1784161767464204114132466567037398478439800, 26519979442516706449090949041747532784056400, 394892185585913216959553600016984856937344800, 5889988095789889525686365006595831217109868000, 87992852461345925338284180250052872425308028000, 1316577114218746859699196923045718630317217508800, 19727951985305514924521361479643432353263912310400, 296024215694398178734440216670607035471050513976800, 4447914873571442022310849173953763874807366141131000, 66918425404017695078521592062014586296081435086566800, 1008027669262403020923988782072982074041207876292236832, 15202516840052440130043686021037326935653148650144911136] The sequence is annihilated by the recurrence operator 4 (2 n + 3) (1 + 2 n) - --------------------- + N (n + 3) (n + 2) The asymptotic behaviour is n / 15 317 2925 \ 1.273249941 16 |1 - --- + ----- - ------| | 4 n 2 3| \ 32 n 128 n / ------------------------------------------ 3 n The expected number of visits to the origin is , 1.209389095 The relative Polya constant, i.e. the probablity of returning to the origin, while staying in [0 <= x[1], 0 <= x[2]] is , 0.1731362519 This is the story of simple random walks that start and end at the origin, in the, 3, -dimensional (cubic) lattice [0 <= x[1], 0 <= x[2], 0 <= x[3]] The exponential Generating Function is 3 2 2 3 J[0] - 3 J[2] J[0] + 3 J[2] J[0] - J[2] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [3, 24, 285, 4242, 73206, 1403028, 29082339, 640672890, 14818136190, 356665411440, 8874875097270, 227135946200940, 5955171596514900, 159439898653636320, 4347741997166750235, 120493374240909299130, 3387806231071627372590, 96488484001399878973200, 2780207705225528461678410, 80955498783009337288066980, 2379962732750977627873552140, 70581288617804540342161273320, 2110037837716149733627208327250, 63547036534358121314424011217372, 1926902417367481014145849339997556, 58798407382339023426223686465775968, 1804754491589895089349716812827111444, 55698407737654454991572691115695830760, 1727755352524250830508238267193429661592, 53851080282183845712275787735154974539856, 1685972354414977941638960116751341503602403, 53006990308609014328322812517730155887149114, 1673152847273710469284488766946036650872834990, 53010278911076667197081401907493669456098497200, 1685453997998204806878312673235366521661796501570, 53767969801804132718220139623358274964389937358260, 1720696803302390110697432604847465383110126712717660, 55231764622870616085297217263372715767715470964120200, 1777920969487455972446747142658248955271361566052064710, 57387167940313896942283573970430663910823283432313406580, 1857120403531795043080356206043421751751509987907839062140, 60247115934410908148985527348590230041724905101107376391520, 1959092440061040389445046167591402909348737345747528103137100, 63848517159443538220776858407583223557117125443519913093811480, 2085362765854045627443620957161213112555230519488957848529070440, 68250794298399971092450875580595037592402383465390962841925321920, 2238164229442219603501090252950333174060554269699747905394098102210, 73535906799954206214059314113206658212665065255727910588704333950300, 2420457249210357186828234409929417455964071050339310646531660302287380, 79809569210955425842789008181697425145628122449149394321244747662364640] The sequence is annihilated by the recurrence operator 2 36 (2 n + 3) (1 + 2 n) (1 + n) 4 (2 n + 3) (5 n + 30 n + 43) N 2 ------------------------------ - -------------------------------- + N (n + 5) (n + 4) (n + 3) (n + 5) (n + 4) (n + 3) The asymptotic behaviour is n / 63 5061 166401 \ 6.299501907 36 |1 - --- + ------ - -------| | 8 n 2 3| \ 128 n 1024 n / -------------------------------------------- 9/2 n The expected number of visits to the origin is , 1.113463308 The relative Polya constant, i.e. the probablity of returning to the origin, while staying in [0 <= x[1], 0 <= x[2], 0 <= x[3]] is , 0.1019012546 This is the story of simple random walks that start and end at the origin, in the, 4, -dimensional (cubic) lattice [0 <= x[1], 0 <= x[2], 0 <= x[3], 0 <= x[4]] The exponential Generating Function is 4 3 2 2 3 4 J[0] - 4 J[2] J[0] + 6 J[2] J[0] - 4 J[2] J[0] + J[2] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [4, 44, 740, 16016, 410928, 11925672, 380037372, 13034844680, 474427581056, 18134245620608, 722228667002752, 29788160119725952, 1266215117909156800, 55255559436669165840, 2467559432846045635020, 112470462274250431106280, 5220721290405673199827680, 246339034102636678220186400, 11796545568507274169626748640, 572532405600441051736431352320, 28128912903037777348604646842880, 1397537025897658132411070347418880, 70151359510301441611301031053635200, 3554860478412552889030245815683044096, 181724479813289423835491750776055236608, 9365545839087639002929658314561477855232, 486334262154850691321432082191141424637952, 25432971651258852363745840087854419784043520, 1338812990021985019175600653392897181145360128, 70912433396777519634377836052527832705046977824, 3777811348175425286910386804785080645515026288876, 202359672384340053106960425350293658070495706055976, 10895241921370027410017061510215555408203338094590240, 589459398860593330829162861818350683749054685199528288, 32037614893101358337538525145822489402057747971328804384, 1748839809841696343204676079081195751966405079254797223296, 95857474847027118751801713074681557828291068533931506912896, 5274702180716753199586155374004287925392370366181068704139200, 291328162351783153920086807736803350464976940453349528777539360, 16147382624486919900832499593878009227323532607544677365640006080, 898017848174335196800214092717652817391303937262918951477661317120, 50103024005685904587505256312788902861495259908137694300877385502720, 2803990425395148788310146872233844408148052352150013165243253818265600, 157385355671930280194433640866835495736710778225780310538118055196815360, 8858754487175537858243972635383650135344825068693301942423695247233607680, 499977375185782115221982766547543748129072893287088529446180180367592645120 , 28291066923216804901568228618726835132287145338096866295208801095534626\ 773120, 16048130152813728403767306792338524237082766978775910051204823649\ 32627173318400, 912500713931423952728465551179997320832633321905723662030\ 02816683258840069743616, 520038675497124376452363196772724744724097868957\ 0420364054421780533023417075514368] The sequence is annihilated by the recurrence operator 2 256 (2 n + 3) (1 + 2 n) (1 + n) 4 (2 n + 7) (2 n + 3) (5 n + 35 n + 54) N ------------------------------- - ------------------------------------------ (n + 5) (n + 4) (n + 6) (n + 5) (n + 4) (n + 3) (n + 6) 2 + N The asymptotic behaviour is n / 27 110 705\ 51.91492399 64 |1 - --- + --- - ---| | 2 n 2 3 | \ n n / ------------------------------------- 6 n The expected number of visits to the origin is , 1.077781773 The relative Polya constant, i.e. the probablity of returning to the origin, while staying in [0 <= x[1], 0 <= x[2], 0 <= x[3], 0 <= x[4]] is , 0.0721683878 This is the story of simple random walks that start and end at the origin, in the, 5, -dimensional (cubic) lattice [0 <= x[1], 0 <= x[2], 0 <= x[3], 0 <= x[4], 0 <= x[5]] The exponential Generating Function is 2 3 3 2 5 5 4 4 10 J[2] J[0] - 10 J[2] J[0] + J[0] - J[2] + 5 J[2] J[0] - 5 J[2] J[0] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [5, 70, 1525, 43470, 1491210, 58637700, 2561441025, 121647761950, 6184574586190, 332733877657220, 18777547178203450, 1103931695886953500, 67238964580113478500, 4224176650971314371800, 272719884456905867840025, 18039467732299038025008750, 1219414932143088238458969750, 84053651397832245761332816500, 5896996143764203895292650001150, 420413683690639210721673406439700, 30415042050183780874120532445025500, 2230156783119921792726097115207847000, 165558446733560494152816404907474035250, 12431506115281718979969017763231288464220, 943378521846131057119581446924779146396060, 72295412541857680840197761526349321747409800, 5591194237627087427493800843427166842687237700, 436117401622377595561287587883656892217936943000, 34290097175987496245868051973706850506222327957800, 2716341086161050769325404250357118791492664069480400, 216697370369375475876917181622721176171855093286633625, 17401934340418475279767182464966479303341553380146796750, 1406216202306904180070519978310199606357553497784785035750, 114305503047607543983162501465111687845142395309712081690900, 9343392634999332637901494307408001581875332018822043830452950, 767780913974094182434161989635066101989190843647264982583334500, 63408558362614692247401033845842235636350793453822557560967875500, 5261719551209937662421364603398684912891300310025289968133241675000, 438607539974421367618073034989601259214030978174187491817841851103750, 36719738391416266576317782982239220385083057340015424878557171379192500, 3086819059103078106749077929540969059753791140533537501629942175434612500, 260513250956217592927820943788517260229735163885879450530379894157774895000 , 22068857506446606884214432481888283126730305172589006106669192085091512\ 517500, 18762492464658708328110031943280005546069915271393857405256154880\ 60787772113000, 160064723233416086333043548706265429036092695846534073619\ 689992178473512006239000, 13700393547579961601706455228882874274824458078\ 773286161568249252003630549711610000, 11763714335094457466786140654970427\ 14669413693209705962590824074280546457963370501250, 101315314405689888420\ 270526495737249510440008477954934280224133850974219022726584013500, 87513\ 152300979691222874076808622455111442117961017636726916506378222014400524\ 11166033740, 758037236201272430523001504987594983067536018019091782824455\ 132128605657252100831285700520] The sequence is annihilated by the recurrence operator 1800 (2 n + 5) (2 n + 3) (1 + 2 n) (n + 2) (1 + n) - -------------------------------------------------- (n + 5) (n + 4) (n + 8) (n + 7) (n + 6) 2 4 (2 n + 5) (2 n + 3) (n + 2) (259 n + 2331 n + 5102) N + -------------------------------------------------------- (n + 5) (n + 4) (n + 8) (n + 7) (n + 6) 4 3 2 2 2 (2 n + 5) (35 n + 700 n + 5131 n + 16310 n + 18936) N 3 - ----------------------------------------------------------- + N (n + 5) (n + 4) (n + 8) (n + 7) (n + 6) The asymptotic behaviour is n / 165 31745 2337975\ 626.4762897 100 |1 - --- + ------ - -------| | 8 n 2 3| \ 128 n 1024 n / --------------------------------------------- 15/2 n The expected number of visits to the origin is , 1.059219713 The relative Polya constant, i.e. the probablity of returning to the origin, while staying in [0 <= x[1], 0 <= x[2], 0 <= x[3], 0 <= x[4], 0 <= x[5]] is , 0.0559088094 This is the story of simple random walks that start and end at the origin, in the, 1, -dimensional (cubic) lattice [] The exponential Generating Function is J[0] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520, 601080390, 2333606220, 9075135300, 35345263800, 137846528820, 538257874440, 2104098963720, 8233430727600, 32247603683100, 126410606437752, 495918532948104, 1946939425648112, 7648690600760440, 30067266499541040, 118264581564861424, 465428353255261088, 1832624140942590534, 7219428434016265740, 28453041475240576740, 112186277816662845432, 442512540276836779204, 1746130564335626209832, 6892620648693261354600, 27217014869199032015600, 107507208733336176461620, 424784580848791721628840, 1678910486211891090247320, 6637553085023755473070800, 26248505381684851188961800, 103827421287553411369671120, 410795449442059149332177040, 1625701140345170250548615520, 6435067013866298908421603100, 25477612258980856902730428600, 100891344545564193334812497256] The sequence is annihilated by the recurrence operator 2 (1 + 2 n) - ----------- + N 1 + n The asymptotic behaviour is n / 1 1 5 \ 0.5641895835 4 |1 - --- + ------ + -------| | 8 n 2 3| \ 128 n 1024 n / -------------------------------------------- 1/2 n This is the story of simple random walks that start and end at the origin, in the, 2, -dimensional (cubic) lattice [x[2] <= x[1]] The exponential Generating Function is 2 2 J[0] - J[1] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [2, 12, 100, 980, 10584, 121968, 1472328, 18404100, 236390440, 3103161776, 41469525552, 562496897872, 7726605740000, 107289439704000, 1503840313184400, 21252802073091300, 302539888334593800, 4334635827016110000, 62464383654579522000, 904841214653480504400, 13169160881666672968800, 192488367353372951140800, 2824557564424494391740000, 41596317732091387409024400, 614601593075393361040415904, 9108710789681470837983599808, 135377611683678579808973184448, 2017326479522796571045043758400, 30134683825147430157817688142720, 451177782345540663997261816880896, 6769486000434503269087948470216992, 101773067938350543465947225296557732, 1532945497467133854344481379244051400, 23130730548336013015638241719080829360, 349604470287707168150646567696964535184, 5292360764925921425367595638920114780368, 80236103887027696374918683242689307722848, 1218159471969043973303421304291841716440000, 18519147459755030045425602341400844966084000, 281897559259344250032929717591908959593488400, 4296236669687706444752733744763100311017136800, 65552102807261594015285897602819486251599236800, 1001297976284281219366682051121291306908677560000, 15310756328274191008861447363509199802003596872000, 234350726330936941026457052302137916196464716566400, 3590487261325603716262887789295104688294032040492800, 55060504119158061244605880300732908597615395599684800, 845103825978573984239061343051215136213399566814890000, 12982174528379432845233188860030829741439798406793959200, 199589478513955798142949778850450450660159159505862892736] The sequence is annihilated by the recurrence operator 2 4 (1 + 2 n) - --------------- + N (n + 2) (1 + n) The asymptotic behaviour is n / 5 41 163 \ 0.3183099512 16 |1 - --- + ----- - ------| | 4 n 2 3| \ 32 n 128 n / ------------------------------------------- 2 n The expected number of visits to the origin is , 1.273239545 The relative Polya constant, i.e. the probablity of returning to the origin, while staying in [x[2] <= x[1]] is equal to , 0.2146018368 This is the story of simple random walks that start and end at the origin, in the, 3, -dimensional (cubic) lattice [x[2] <= x[1], x[3] <= x[2]] The exponential Generating Function is 3 2 2 2 J[0] - 2 J[1] J[0] + 2 J[1] J[2] - J[2] J[0] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [2, 12, 120, 1610, 25956, 474012, 9475752, 202921290, 4587734580, 108376022040, 2654745191280, 67043341981980, 1737717447946200, 46062204663294000, 1245096242017227360, 34239776369652506970, 956050033694583839220, 27060818735432435045400, 775371248506151537209200, 22463243609030102723824980, 657334251145826075591157960, 19411847317987883103962106360, 578067928392949069637824750800, 17347134567675898556456565080700, 524268086976534438376290725056056, 15948694214806165700004818546972112, 488131860407769880364135596438743072, 15024683093154386883233087147674208520, 464904865703381156218295016426242231760, 14456532417147567888553220093623311983856, 451616986174086716248448032084844177265312, 14169669810897334339421247101883188977742202, 446396415875857062556819810795847288507578740, 14117280077686645893079356783005037752609314680, 448081665680484250801065355743863458911980287920, 14270961367318919178084002097749281666248721245540, 455995065173214588556937053238256702894209615652840, 14615207519167480537319258485999453009179236404080600, 469807938092583945293139237169822249265121157638474000, 15144111442301532158321145156019966315467604709031698580, 489460196891036065556927987946389991865406878951922765160, 15859417424010635026934734236489017277980883133895948810160, 515113786217367038338532980416084692254293730380862650463200, 16769434085718780873864800106830585845774080552313258284180600, 547128327149358353237116936427988763125137936089464873105979440, 17888529214834496430804860642428656830393312780463817995205622880, 586052283938040930255539383581083350098506996454932822944989094080, 19237046822725424521579335580030648853234539036246828808186104134300, 632625861215509105259635391734316260079245997887339492102066387740600, 20841557523230235554334396805681614078028131606078081451206665855354640] The sequence is annihilated by the recurrence operator 2 36 (2 n + 3) (1 + 2 n) (1 + n) 2 (2 n + 3) (10 n + 42 n + 41) N 2 ------------------------------ - --------------------------------- + N 2 2 (n + 2) (n + 4) (n + 2) (n + 4) The asymptotic behaviour is n / 45 2649 64359 \ 1.574756973 36 |1 - --- + ------ - -------| | 8 n 2 3| \ 128 n 1024 n / -------------------------------------------- 9/2 n The expected number of visits to the origin is , 1.069341121 The relative Polya constant, i.e. the probablity of returning to the origin, while staying in [x[2] <= x[1], x[3] <= x[2]] is equal to , 0.0648447157 This is the story of simple random walks that start and end at the origin, in the, 4, -dimensional (cubic) lattice [x[2] <= x[1], x[3] <= x[2], x[4] <= x[3]] The exponential Generating Function is 4 2 2 2 4 2 2 J[0] - 3 J[1] J[0] + 4 J[2] J[1] J[0] + J[1] - 2 J[2] J[1] 2 2 3 4 - 2 J[0] J[2] - 2 J[1] J[3] + J[2] + 4 J[0] J[2] J[1] J[3] 2 2 2 2 2 - 2 J[2] J[1] J[3] - J[3] J[0] + J[3] J[1] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [2, 12, 120, 1680, 29988, 641256, 15725424, 428879880, 12729104960, 404742752128, 13628020713216, 481633720573824, 17743276774729800, 677634409667955600, 26709469982646282720, 1082553840134699000040, 44980021321726303779120, 1910979819009785098528800, 82833515924368075250260800, 3656385260445154913341585920, 164091652801145420140284992640, 7476423569680954869000200643840, 345409719705490205231553561177600, 16163407039582642778472265732166400, 765363405259876948488973106167569408, 36640713820089349999716163489683081216, 1772092379492100518848510048612057419776, 86523490761855580866263923082607318338560, 4262218898056034141222903598214128215018640, 211712384006581067900497979895368443930204704, 10598467087879564994662647554654914848375334848, 534470351324911584208738745401686522794323741992, 27139579364513009444256101494008435193163091117840, 1387114674954868679157485613593140819110893029085280, 71333930587349991264505193639282273669676193158135744, 3689875405737934158223175659124464077789558844240528768, 191923158737798928745644193781757687578187797655879961952, 10035097384474766662378861807303453621275311391348726091200, 527329110247180214600930820980592548486566353844474285046400, 27842191691051700305697840075550949353752307326503932176914880, 1476690783039536120268702164102082252354043037123685947212892160, 78659256674464433470778171582284235926450026195331546196400967680, 4207266244022710341915719202097599260055320836852027938861909811200, 225923472109085837044963556190990423651809202801632053972138223923200, 12177560890486020873753851040753118934907555536721171299223943431502080, 658761051921133222849985655569840199244960752373372094827650952992207360, 35760162749675951299834972729297449388527278984089214133894930475718999040, 194766503967922395168303639824284295688310529422393301402128011464921082\ 2400, 1064180053874849349398147679136549537866790947563856183031903964683\ 92737548800, 583241218637273301309086297769456122501193301032195388200221\ 8406771329735891968] The sequence is annihilated by the recurrence operator 3 2 256 (2 n + 3) (1 + 2 n) (1 + n) 4 (2 n + 3) (10 n + 91 n + 255 n + 214) N ------------------------------- - ------------------------------------------- 2 2 (n + 6) (n + 5) (n + 6) (n + 2) (n + 5) 2 + N The asymptotic behaviour is n / 17 339 5175\ 155.8898734 64 |1 - ---- + ---- - ----| | n 2 3| \ 2 n 4 n / ---------------------------------------- 8 n The expected number of visits to the origin is , 1.034781448 The relative Polya constant, i.e. the probablity of returning to the origin, while staying in [x[2] <= x[1], x[3] <= x[2], x[4] <= x[3]] is equal to , 0.0336123614 This is the story of simple random walks that start and end at the origin, in the, 5, -dimensional (cubic) lattice [x[2] <= x[1], x[3] <= x[2], x[4] <= x[3], x[5] <= x[4]] The exponential Generating Function is 2 4 3 4 -2 J[3] J[0] J[2] J[4] + 3 J[1] J[0] - 4 J[1] J[3] J[0] + 2 J[0] J[2] 2 3 2 2 4 2 - J[4] J[0] + 6 J[0] J[2] J[1] - 4 J[1] J[2] + 4 J[4] J[2] J[1] J[3] 2 2 2 2 2 + 8 J[0] J[3] J[2] J[1] - 2 J[1] J[2] J[0] - 2 J[4] J[1] J[2] 2 3 2 2 4 2 3 + 2 J[1] J[2] + J[4] J[2] J[0] + J[3] J[0] - 4 J[1] J[0] 4 2 2 - 2 J[4] J[2] + 2 J[4] J[1] J[3] - 4 J[4] J[1] J[0] J[2] J[3] 2 3 2 3 - 6 J[1] J[0] J[2] J[4] + 2 J[2] J[3] - 4 J[4] J[1] J[3] 2 2 2 2 2 - 8 J[1] J[3] J[0] J[2] + 2 J[3] J[0] J[2] + 2 J[4] J[2] J[0] 3 2 2 2 2 2 - 2 J[0] J[3] + 4 J[4] J[1] J[2] + 4 J[1] J[3] J[2] 2 5 3 2 3 + 4 J[4] J[0] J[3] J[1] + J[0] - 3 J[0] J[2] - 4 J[2] J[1] J[3] 3 4 3 + 4 J[1] J[3] J[2] + 2 J[1] J[4] - 4 J[3] J[1] J[2] 2 2 + 2 J[4] J[1] J[0] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [2, 12, 120, 1680, 30240, 664356, 17170296, 507451230, 16765974940, 608141002040, 23868372139248, 1001975473774548, 44573971432523400, 2085675636255328800, 102024369293541120000, 5191184452316394948030, 273595496197869862475580, 14882994221440971694206600, 833098779545619828829064400, 47862668814325241116481841600, 2815860020527049409221218035840, 169309993809447511846603707959640, 10386251181340366713942180991035600, 649044387787210075409342360788511100, 41260847361958268226934643439898491960, 2665153852115032139617722587303508220080, 174724429761864550272180371679376797040480, 11614652749872514172945079430069422429642800, 782159337124441253524503679407869392834533600, 53317741959819357877874646512201255417523951600, 3676363608900622959992531418763830934865190531360, 256239401184776872423537826875330553270172666102510, 18042193157822109466307983012437473480793314177022300, 1282643219471863657397844098326382447556426569995770600, 92017888465021518644339687314350011888394337559508560400, 6658597285744435027498513515620344976028863063806002515200, 485788567501068631886248825249227725865956051406852360147200, 35718235886533113282273499228787038566182196968015879527755000, 2645743739678431720726732127702905130617330191816449611438266000, 197364518203376515074962114834183408048753626056445029248956672500, 14822206552493607426337279775662720645257504738471376612635610242600, 1120334278431492151421361454863849774520130427775364352394223735643600, 85202152920888176720362252664025117963195461576116647724899585760932000, 6517887252533493372390873385218791243717919792427313374434210236989563000, 501428335555448009547931532690356119091621529730360485853611896691659902000 , 38784281428141149633419343569960570160069605363422351608790072249751720\ 516160, 30154539545613204034045305705561824172151995663729536167908747767\ 58374157231360, 235619155716816017958472807698098881658385245097466345043\ 084793444090647796927900, 18498882876362485030395485316650698969170346001\ 583542512133857173232797142367350200, 14590772910961709070429675897033860\ 44026134393643267773888031034460588162257750191120] The sequence is annihilated by the recurrence operator 1800 (2 n + 5) (2 n + 3) (1 + 2 n) (n + 2) (1 + n) - -------------------------------------------------- 2 2 (n + 3) (n + 9) (n + 7) 2 4 (2 n + 5) (2 n + 3) (n + 2) (259 n + 2176 n + 4242) N + -------------------------------------------------------- 2 2 (n + 3) (n + 9) (n + 7) 4 3 2 2 2 (2 n + 5) (35 n + 742 n + 5631 n + 17932 n + 19941) N 3 - ----------------------------------------------------------- + N 2 2 (n + 3) (n + 9) (n + 7) The asymptotic behaviour is n / 325 117225 15498975\ 582068.2631 100 |1 - --- + ------ - --------| | 8 n 2 3 | \ 128 n 1024 n / ---------------------------------------------- 25/2 n The expected number of visits to the origin is , 1.021340738 The relative Polya constant, i.e. the probablity of returning to the origin, while staying in [x[2] <= x[1], x[3] <= x[2], x[4] <= x[3], x[5] <= x[4]] is equal to , 0.0208948270 This is the story of simple random walks that start and end at the origin, in the, 1, -dimensional (cubic) lattice [0 <= x[1]] The exponential Generating Function is J[0] - J[2] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, 18367353072152, 69533550916004, 263747951750360, 1002242216651368, 3814986502092304, 14544636039226909, 55534064877048198, 212336130412243110, 812944042149730764, 3116285494907301262, 11959798385860453492, 45950804324621742364, 176733862787006701400, 680425371729975800390, 2622127042276492108820, 10113918591637898134020, 39044429911904443959240, 150853479205085351660700, 583300119592996693088040, 2257117854077248073253720, 8740328711533173390046320, 33868773757191046886429490, 131327898242169365477991900, 509552245179617138054608572, 1978261657756160653623774456] The sequence is annihilated by the recurrence operator 2 (1 + 2 n) - ----------- + N n + 2 The asymptotic behaviour is n / 9 145 1155 \ 0.5641896856 4 |1 - --- + ------ - -------| | 8 n 2 3| \ 128 n 1024 n / -------------------------------------------- 3/2 n The expected number of visits to the origin is , 2.000000000 The relative Polya constant, i.e. the probablity of returning to the origin, while staying in [0 <= x[1]] is equal to, 0.5000000000 This is the story of simple random walks that start and end at the origin, in the, 2, -dimensional (cubic) lattice [x[2] <= x[1], 0 <= x[2]] The exponential Generating Function is 2 2 2 J[0] - J[1] - J[0] J[2] + 2 J[3] J[1] - J[4] J[0] - J[3] + J[4] J[2] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [1, 3, 14, 84, 594, 4719, 40898, 379236, 3711916, 37975756, 403127256, 4415203280, 49671036900, 571947380775, 6721316278650, 80419959684900, 977737404590100, 12058761323277900, 150656212896017400, 1904342169333848400, 24328661192286773400, 313839729380499376860, 4084744785475422658824, 53602092541025062012944, 708738779153553597726704, 9437048867251504308646704, 126478149186382230159563872, 1705415043867992651828958016, 23126253195677658419761233096, 315270398678954138836063173911, 4319373055697756704267346799786, 59454899707839709929327008891172, 821987581675053767435616265781124, 11411557327578899600164726716835244, 159047565712116725864031283601069304, 2224948807249395385029080007838980304, 31234858255616512135985161648508761960, 439954405917525262403205630536922195900, 6216591524265519561437165738806266475400, 88106243995072114448807139673579843136400, 1252300799955327326510212262210311385636400, 17848448775120877350564742444229791566999600, 255051746264770798081983131159573543261762400, 3653775525121406641163358713355370851944174400, 52467957407727433309897237977775086045914660400, 755163815546934129424592389465834274160843147900, 10892852751603531483210242874988809489324161978280, 157455118597688694302403746028269614814309259027216, 2280605586566522671773730276183217362084180715593296, 33096916197501683882533423848412177653816724202057136] The sequence is annihilated by the recurrence operator 4 (2 n + 3) (1 + 2 n) - --------------------- + N (n + 4) (n + 3) The asymptotic behaviour is n / 35 1525 26705 \ 7.640529279 16 |1 - --- + ----- - ------| | 4 n 2 3| \ 32 n 128 n / ------------------------------------------ 5 n The expected number of visits to the origin is , 1.080180413 The relative Polya constant, i.e. the probablity of returning to the origin, while staying in [x[2] <= x[1], 0 <= x[2]] is equal to, 0.0742287233 This is the story of simple random walks that start and end at the origin, in the, 3, -dimensional (cubic) lattice [x[2] <= x[1], x[3] <= x[2], 0 <= x[3]] The exponential Generating Function is 2 2 2 -2 J[0] J[1] + 2 J[0] J[3] J[1] - J[0] J[4] - 2 J[2] J[1] J[3] + J[6] J[3] 2 2 + J[5] J[2] + J[6] J[4] J[0] + 3 J[2] J[0] J[4] - J[2] J[0] 2 3 2 3 2 + 2 J[5] J[1] J[4] - J[6] J[0] + J[0] - J[0] J[4] + J[4] - J[0] J[2] 2 2 - 2 J[2] J[4] + J[6] J[1] + 2 J[5] J[3] J[2] + J[6] J[0] J[2] 2 2 - 4 J[5] J[1] J[2] - J[5] J[0] - 2 J[4] J[5] J[3] + J[2] J[4] 2 + 2 J[5] J[1] J[0] + 2 J[1] J[3] J[4] - J[0] J[3] - J[6] J[4] J[2] 2 2 - 2 J[6] J[1] J[3] + 3 J[1] J[2] - 2 J[1] J[4] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [1, 3, 15, 104, 909, 9449, 112398, 1489410, 21562086, 336086022, 5577242292, 97671172836, 1792348213025, 34268124834495, 679376016769260, 13911118850603610, 293220749128031010, 6344131671525765150, 140550723100846328310, 3181697782193052729360, 73459767901858232196510, 1727044595730442160807934, 41285980044993776079738060, 1002302425948893564797245004, 24683359974561809758475626836, 616003023279061114538256738036, 15564772611605225152799127493188, 397860703469983248426632641950240, 10280833434499876198908694501019409, 268376721591420990284565592446408999, 7073214783664487802731048185883509164, 188106389344792576966396652549505193338, 5045271517797653091722112838141173176010, 136413567951591903895613325974136963848046, 3716520683841159658794729873561387038109462, 101988203936049930526524846323605300399661232, 2817988503276525719550584299077723573227580666, 78371399321134177234015392365450622823071876450, 2193153030872343221790777811862765817187836757260, 61737163896749523173948703990005759753146159026180, 1747724587563983631224683706486501860180746774127980, 49743692637554708037626849672091908137119627438583980, 1423112109846766077410824981418785600869610851429652200, 40914783577753618651861000970933187540407593632152428680, 1181874868309286672286493878125866910016938721648489923910, 34294812947915297537599452838682494340819652744266353655610, 999474682215283873338304683331948691519889984695175694331880, 29250075974168261304729781512944611785398278878312848349107100, 859456313201228742075421534397894820873388398401875490457880620, 25351052517140054578361171143782468180761403724521702102099356340] The sequence is annihilated by the recurrence operator 2 36 (2 n + 3) (1 + 2 n) (1 + n) 4 (2 n + 3) (5 n + 42 n + 79) N 2 ------------------------------ - -------------------------------- + N (n + 5) (n + 8) (n + 7) (n + 5) (n + 8) (n + 7) The asymptotic behaviour is n / 273 83377 9415035\ 53112.45938 36 |1 - --- + ------ - -------| | 8 n 2 3| \ 128 n 1024 n / -------------------------------------------- 21/2 n The expected number of visits to the origin is , 1.030497741 The relative Polya constant, i.e. the probablity of returning to the origin, while staying in [x[2] <= x[1], x[3] <= x[2], 0 <= x[3]] is equal to, 0.0295951556 This is the story of simple random walks that start and end at the origin, in the, 4, -dimensional (cubic) lattice [x[2] <= x[1], x[3] <= x[2], x[4] <= x[3], 0 <= x[4]] The exponential Generating Function is 2 -J[8] J[6] J[4] J[0] - J[8] J[6] J[3] - 4 J[1] J[7] J[0] J[2] 2 2 2 - 2 J[6] J[1] J[3] J[2] + 3 J[2] J[6] + 2 J[8] J[2] J[4] 2 2 + J[8] J[0] J[4] - 2 J[2] J[7] J[5] - 4 J[6] J[7] J[1] J[2] 2 2 - 2 J[7] J[4] J[2] J[5] - 3 J[8] J[2] J[0] J[4] - J[7] J[0] 2 2 3 2 + 6 J[5] J[1] J[3] + 4 J[6] J[4] J[2] + J[0] J[4] + 2 J[8] J[1] J[4] 2 2 2 + J[6] J[4] + 3 J[0] J[6] J[2] + 2 J[2] J[7] J[6] J[3] 2 2 2 2 + J[7] J[0] J[2] - J[0] J[6] + 2 J[6] J[1] J[5] J[4] + J[5] J[0] J[2] 2 2 3 2 - J[0] J[4] - J[8] J[0] - J[3] J[6] J[4] - 4 J[3] J[0] J[6] J[5] 2 3 2 2 3 3 - J[0] J[2] J[6] - J[8] J[4] + J[3] J[5] + J[6] J[0] - 2 J[2] J[4] 2 2 2 3 - 2 J[1] J[4] J[6] - 4 J[5] J[0] J[3] J[4] - J[5] J[1] - J[0] J[4] 2 2 + 4 J[3] J[0] J[6] - 4 J[3] J[5] J[6] J[2] - J[7] J[4] J[2] 2 2 + 2 J[6] J[1] J[3] - 2 J[2] J[6] J[4] - 4 J[3] J[0] J[6] J[1] 2 2 - 4 J[3] J[1] J[5] - 2 J[5] J[1] J[4] + 4 J[0] J[2] J[5] J[3] 2 2 + 2 J[5] J[7] J[4] - J[8] J[6] J[1] - J[8] J[6] J[0] J[2] 2 2 2 - 2 J[5] J[6] J[1] - J[8] J[2] J[4] + 2 J[0] J[1] J[3] 2 2 2 + 2 J[5] J[6] J[2] + 4 J[3] J[6] J[4] J[5] - 3 J[0] J[1] 2 + 2 J[6] J[3] J[5] - 2 J[8] J[1] J[3] J[4] - 2 J[8] J[1] J[5] J[0] 2 - 2 J[0] J[2] J[3] - 10 J[5] J[1] J[2] J[0] + 2 J[3] J[7] J[0] J[4] 2 4 3 - J[0] J[6] J[4] J[2] + 2 J[5] J[7] J[1] + J[2] + J[0] J[2] 2 2 - 2 J[8] J[2] J[5] J[3] - 2 J[6] J[7] J[5] J[0] - 2 J[5] J[2] 2 + 2 J[6] J[7] J[5] J[2] + 2 J[8] J[4] J[5] J[3] - 2 J[7] J[3] J[1] 2 2 2 4 3 - 3 J[8] J[1] J[2] - 2 J[3] J[0] + J[0] - 3 J[2] J[6] 2 3 3 - 2 J[1] J[7] J[4] J[0] - 2 J[2] J[7] J[3] - J[6] J[2] - 4 J[1] J[3] 2 2 3 2 + J[8] J[5] J[0] + J[0] J[2] J[4] - J[0] J[2] - 2 J[6] J[4] J[2] 2 2 + J[8] J[0] J[3] + 4 J[2] J[7] J[3] J[4] + J[5] J[0] J[6] 2 2 2 3 + 2 J[3] J[6] J[2] + 3 J[0] J[2] J[4] - 2 J[0] J[4] J[2] - 2 J[1] J[7] 4 2 2 + J[5] + J[1] J[4] J[2] - 2 J[2] J[4] J[5] J[3] + J[0] J[6] J[4] + 4 J[5] J[0] J[6] J[1] + 4 J[5] J[7] J[0] J[2] - 2 J[8] J[4] J[5] J[1] 3 2 2 + 2 J[1] J[5] - 4 J[1] J[7] J[3] + J[8] J[2] J[0] 2 2 + 2 J[8] J[1] J[3] J[2] + J[8] J[0] J[4] - 2 J[3] J[1] J[5] 2 2 2 3 + 4 J[1] J[2] J[6] + 2 J[1] J[7] J[0] + J[7] J[4] J[0] - J[0] J[6] 2 3 2 2 2 2 - 2 J[5] J[7] J[3] - 2 J[5] J[1] - 2 J[0] J[2] - 2 J[5] J[0] 2 2 - J[0] J[6] J[1] + 2 J[3] J[5] J[2] + 4 J[0] J[4] J[5] J[1] 2 2 2 + 4 J[3] J[1] + J[0] J[4] J[5] - 2 J[7] J[4] J[6] J[3] 2 - 2 J[1] J[2] J[6] J[5] + 2 J[0] J[2] J[1] J[3] + 2 J[3] J[1] J[4] 4 2 2 + J[1] - 2 J[7] J[4] J[0] J[5] - J[0] J[6] J[2] - J[8] J[5] J[2] 2 2 - 2 J[2] J[7] J[0] J[3] + J[7] J[3] - 2 J[3] J[1] J[2] J[4] 2 2 3 + J[3] J[0] J[4] + 4 J[5] J[1] J[2] - 2 J[3] J[5] 2 + 2 J[8] J[6] J[3] J[1] + 2 J[5] J[1] J[0] + J[8] J[6] J[4] J[2] 2 2 2 + 2 J[8] J[1] J[0] + 4 J[2] J[7] J[1] + J[8] J[0] J[2] 2 2 2 + 6 J[1] J[7] J[3] + J[7] J[1] + 2 J[7] J[4] J[6] J[1] 2 2 2 + 2 J[3] J[7] J[5] + 4 J[8] J[1] J[5] J[2] - 4 J[1] J[2] 2 2 + 6 J[0] J[2] J[1] + 2 J[6] J[7] J[1] J[0] - J[1] J[4] J[0] 2 2 2 2 + 2 J[3] J[0] J[5] - 2 J[1] J[7] J[5] + J[2] J[4] 2 2 2 - 2 J[8] J[0] J[1] J[3] - 2 J[6] J[3] - 2 J[7] J[4] J[3] 2 2 2 - 3 J[6] J[4] J[5] + J[8] J[6] J[0] + 2 J[2] J[4] J[5] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [1, 3, 15, 105, 944, 10340, 133055, 1958060, 32279090, 586453658, 11589971918, 246518371679, 5594169454700, 134456679614850, 3402014360391645, 90146180439817440, 2490533922180210720, 71468389947184389600, 2123114263550335500000, 65104931224768470927840, 2055564904550412149017344, 66672923072742905317067232, 2217186549442555777478959460, 75459913496130915391681205360, 2624212491876184655951319842468, 93116618969306756155330342214948, 3366962631775833546329700765469244, 123914627348490251595203982116380595, 4636782766543219223154598763754079140, 176237558954937059707146854549885138682, 6798007055513783969640068411036650958655, 265896266073431480376415085668311954136120, 10538167674670634167753951594608011144976280, 422902857934753875906446775418645393605696520, 17173634548464870449382225212338201051659084904, 705300250904904893120911452123258614627912549944, 29277832018318390881793272997004149164662879333632, 1227825309148985032248191739449201665024573258844800, 51995062397511056103395041403499997095727431013434400, 2222411540830399093104874289658221989587431596775447680, 95839694172685804878834775836691495567205474913283652160, 4168292635202632907854302179120019339365026615601138038080, 182770763496262654325170987016050030298479272284209948363200, 8076898290094516576662149121925862987313125180034323122999200, 359612197898924346157201642135184719685905650655086277564974720, 16126751468985946047453626875993516840369720367283117160363287744, 728216287460325174115966961988013993131320970702113968342131496032, 33102456939773731231458680351083421873175578948558671506730526722560, 1514391661826531432462938503097493261787467414439956009877066573442560, 69709522750684468832444030610938144831564817376670899450477684762034688] The sequence is annihilated by the recurrence operator 256 (n + 5) (2 n + 3) (1 + 2 n) (1 + n) --------------------------------------- (n + 11) (n + 9) (n + 6) (n + 12) 3 2 4 (2 n + 3) (10 n + 197 n + 1189 n + 2102) N 2 - ---------------------------------------------- + N (n + 11) (n + 9) (n + 6) (n + 12) The asymptotic behaviour is 12 n / 189 19113 1374615\ 0.3694804063 10 64 |1 - --- + ----- - -------| | 2 n 2 3 | \ 4 n 8 n / - ------------------------------------------------- 18 n The expected number of visits to the origin is , 1.016421970 The relative Polya constant, i.e. the probablity of returning to the origin, while staying in [x[2] <= x[1], x[3] <= x[2], x[4] <= x[3], 0 <= x[4]] is equal to, 0.0161566460 This is the story of simple random walks that start and end at the origin, in the, 5, -dimensional (cubic) lattice [x[2] <= x[1], x[3] <= x[2], x[4] <= x[3], x[5] <= x[4], 0 <= x[5]] The exponential Generating Function is 2 2 -4 J[3] J[1] J[2] J[7] J[5] + 2 J[0] J[2] J[5] J[1] J[4] + 2 J[9] J[4] J[2] 2 + 6 J[9] J[1] J[2] J[6] + 8 J[8] J[3] J[7] J[6] J[0] + 2 J[7] J[9] J[0] J[6] J[2] - 4 J[0] J[8] J[4] J[5] J[1] 3 2 2 - 4 J[6] J[7] J[1] J[3] J[5] + 12 J[2] J[1] J[3] - 3 J[8] J[2] J[7] 2 3 + 4 J[10] J[0] J[5] J[6] J[3] + 4 J[0] J[6] J[4] J[2] - 2 J[7] J[1] J[6] 2 2 + 4 J[9] J[2] J[0] J[7] - 2 J[10] J[8] J[2] J[4] 2 - 4 J[8] J[3] J[7] J[1] + 4 J[4] J[8] J[2] J[7] J[1] 2 + 8 J[7] J[1] J[4] J[3] - 8 J[8] J[1] J[4] J[0] J[3] 2 2 3 - 4 J[9] J[2] J[0] J[7] + 4 J[7] J[8] J[4] J[3] + 8 J[1] J[6] J[3] 2 3 3 - 2 J[8] J[6] J[4] J[2] + 2 J[4] J[1] J[5] + 3 J[10] J[2] J[6] 2 2 3 - 4 J[3] J[0] J[7] J[2] + 2 J[10] J[2] J[7] J[3] - 2 J[6] J[7] J[1] 3 2 - 10 J[1] J[4] J[3] - 4 J[10] J[1] J[5] J[2] 2 - 4 J[3] J[9] J[0] J[1] J[5] + 2 J[4] J[6] J[7] J[5] 2 + J[10] J[8] J[6] J[0] J[2] - 2 J[10] J[7] J[5] J[4] 2 2 + 12 J[3] J[1] J[2] J[7] - 6 J[10] J[3] J[5] J[1] 2 2 3 - 6 J[4] J[6] J[1] J[5] - 4 J[1] J[0] - 10 J[4] J[6] J[1] J[3] J[2] + 2 J[3] J[9] J[8] J[0] J[4] + 6 J[6] J[7] J[2] J[4] J[5] 2 2 + 2 J[8] J[9] J[7] J[1] - 2 J[10] J[6] J[1] J[3] 2 2 3 + 8 J[4] J[6] J[7] J[1] - 2 J[7] J[0] + 6 J[1] J[6] J[8] J[5] J[0] 2 2 3 2 - 4 J[7] J[2] J[6] - 2 J[6] J[1] J[3] - 6 J[9] J[2] J[6] J[3] 2 3 2 - 7 J[0] J[8] J[2] J[4] - 2 J[6] J[8] J[2] + J[9] J[6] J[4] J[2] 2 - 4 J[10] J[2] J[7] J[4] J[3] + 4 J[9] J[2] J[4] J[5] 2 2 2 2 + 3 J[7] J[8] J[6] J[2] - 3 J[0] J[2] J[6] - 2 J[7] J[8] J[1] J[4] 2 2 2 - 4 J[1] J[5] J[4] - 4 J[3] J[0] J[4] J[7] + 4 J[4] J[2] J[7] J[1] J[6] 3 2 2 2 + 2 J[7] J[5] J[4] - 4 J[7] J[5] J[0] J[3] + 2 J[6] J[7] J[2] 3 2 + 4 J[0] J[5] J[6] J[4] J[3] - 2 J[3] J[7] J[0] - 2 J[7] J[9] J[3] J[4] 2 2 2 2 2 - 5 J[0] J[6] J[8] J[4] + 4 J[4] J[6] J[0] + J[8] J[5] J[0] 2 2 2 - 6 J[0] J[6] J[4] J[2] + 4 J[9] J[2] J[4] J[3] - 6 J[3] J[0] J[2] J[4] 2 4 2 + 6 J[3] J[5] J[0] J[1] - 5 J[2] J[1] - 6 J[9] J[2] J[5] J[1] 3 + 3 J[0] J[8] J[2] + 12 J[6] J[7] J[1] J[0] J[4] 2 2 3 + 4 J[0] J[8] J[5] J[3] J[4] + J[9] J[5] J[0] - 2 J[7] J[1] J[2] 2 2 - 4 J[9] J[6] J[4] J[5] - 4 J[4] J[2] J[1] J[3] 2 2 - 14 J[9] J[1] J[2] J[3] - 6 J[4] J[6] J[7] J[5] - 4 J[4] J[8] J[2] J[5] J[3] - 4 J[10] J[5] J[6] J[0] J[1] 2 2 2 2 + J[4] J[8] J[0] - 4 J[0] J[1] J[7] J[4] + 2 J[4] J[1] J[0] J[3] 2 - 4 J[9] J[1] J[6] J[4] J[2] - 2 J[10] J[7] J[5] J[3] 2 2 2 2 - J[10] J[8] J[0] J[4] - 2 J[0] J[5] J[1] J[4] + J[7] J[0] J[6] 2 2 + 6 J[8] J[6] J[4] J[2] + J[10] J[0] J[4] J[1] 2 + 2 J[10] J[3] J[7] J[0] J[2] - 12 J[3] J[0] J[1] J[2] 2 + 2 J[8] J[9] J[7] J[4] J[0] + 2 J[9] J[2] J[7] J[3] 2 3 2 + 2 J[10] J[8] J[5] J[3] J[2] - J[9] J[4] + 4 J[3] J[5] J[7] J[4] 2 2 2 2 - 6 J[0] J[5] J[6] J[1] + J[7] J[8] J[2] J[0] - J[3] J[8] J[4] 2 2 + 2 J[9] J[6] J[2] J[5] + 4 J[3] J[5] J[7] J[6] 2 2 2 + 2 J[10] J[7] J[1] J[5] - 8 J[3] J[1] J[2] J[7] - J[0] J[8] J[2] J[1] 2 2 2 2 - 4 J[7] J[0] J[2] J[3] - J[8] J[2] J[0] + 4 J[9] J[6] J[4] J[5] 3 4 2 2 + 2 J[3] J[9] J[6] - 3 J[1] J[6] - 4 J[0] J[2] J[4] 2 2 2 + 2 J[9] J[1] J[6] J[0] J[4] - J[9] J[5] J[2] - 4 J[9] J[1] J[6] J[5] 2 2 - 3 J[9] J[0] J[4] J[2] - 2 J[7] J[5] J[8] J[1] 3 2 - 6 J[0] J[6] J[7] J[3] J[4] - J[10] J[0] J[2] + 4 J[6] J[0] J[3] J[5] 2 3 + 2 J[2] J[3] J[4] J[5] - 2 J[9] J[6] J[1] - 4 J[9] J[1] J[5] J[7] J[2] + 4 J[7] J[9] J[1] J[5] J[0] - 2 J[8] J[9] J[4] J[6] J[3] 2 2 2 2 + 2 J[1] J[6] J[5] J[7] + 10 J[6] J[0] J[3] J[1] - 5 J[4] J[2] J[6] 2 3 + 3 J[3] J[0] J[6] J[2] - J[4] J[6] J[0] - 12 J[9] J[1] J[4] J[5] J[3] 2 2 - 2 J[7] J[5] J[2] J[4] + 3 J[10] J[8] J[2] J[1] - 4 J[10] J[0] J[2] J[5] J[3] - 6 J[9] J[6] J[4] J[7] J[2] 2 2 + 4 J[7] J[4] J[6] + 2 J[10] J[6] J[1] J[3] J[2] 2 2 - 4 J[9] J[1] J[2] J[0] + 2 J[10] J[5] J[7] J[3] 2 2 + 4 J[8] J[2] J[5] J[3] - 6 J[3] J[5] J[0] J[1] 2 2 3 - 10 J[0] J[2] J[1] J[4] - 8 J[9] J[1] J[6] J[3] + 2 J[2] J[1] J[5] 3 2 2 2 3 - J[4] J[7] + 2 J[8] J[1] J[5] J[3] + 2 J[0] J[2] 2 2 + 2 J[10] J[2] J[6] J[4] + 10 J[2] J[1] J[4] J[3] 2 - 4 J[6] J[7] J[8] J[2] J[3] - 6 J[7] J[5] J[6] J[3] 2 2 - 10 J[0] J[2] J[5] J[1] + 8 J[1] J[6] J[5] J[4] 3 2 3 - 4 J[9] J[1] J[2] J[0] J[6] - J[8] J[4] J[0] + 2 J[3] J[6] 2 2 2 + 4 J[4] J[2] J[6] J[3] - 8 J[8] J[1] J[5] J[3] - 7 J[8] J[3] J[0] J[2] 2 2 + J[10] J[8] J[5] J[2] + 4 J[9] J[1] J[5] J[0] 2 + 12 J[9] J[1] J[2] J[0] J[4] + 2 J[3] J[9] J[8] J[5] 2 2 2 2 - 10 J[0] J[1] J[7] J[2] + 6 J[2] J[5] J[6] J[3] + 9 J[1] J[0] J[2] 3 2 2 2 + J[6] J[2] + 3 J[10] J[0] J[1] - 6 J[3] J[0] J[6] J[1] J[4] 2 2 2 2 + 2 J[4] J[2] J[7] J[3] - 2 J[6] J[9] J[0] J[5] + 5 J[3] J[0] J[2] 3 2 - 4 J[7] J[4] J[3] - 2 J[10] J[5] J[6] J[2] 2 - 2 J[10] J[0] J[1] J[3] J[2] + 2 J[0] J[5] J[1] J[4] 2 2 2 - 4 J[8] J[5] J[7] J[3] + 4 J[6] J[7] J[2] J[3] + 2 J[7] J[5] J[1] J[6] 2 + 2 J[10] J[6] J[7] J[3] J[4] - 2 J[9] J[2] J[5] J[3] 2 2 + 2 J[7] J[5] J[0] J[1] - 4 J[7] J[0] J[2] J[5] 2 - 2 J[3] J[5] J[8] J[6] J[4] + 2 J[7] J[1] J[3] J[0] 2 2 2 3 2 - 2 J[0] J[5] J[6] J[7] - 2 J[8] J[2] J[4] - 2 J[0] J[6] 3 2 2 2 - J[8] J[0] J[2] - 8 J[1] J[6] J[2] - J[10] J[2] J[0] J[4] 3 3 + 2 J[6] J[0] J[8] J[3] J[1] - 2 J[0] J[5] J[1] + 4 J[7] J[2] J[3] 2 2 2 + 4 J[6] J[7] J[8] J[1] + 2 J[8] J[3] J[7] 3 2 2 + 14 J[7] J[0] J[2] J[3] J[4] + 2 J[9] J[6] J[3] + 6 J[8] J[2] J[3] 2 2 2 2 - J[9] J[6] J[1] - 2 J[10] J[0] J[1] J[3] + 2 J[7] J[9] J[5] J[2] 4 2 + 2 J[1] J[8] - 2 J[10] J[6] J[7] J[3] J[2] + 4 J[4] J[2] J[7] J[1] 2 2 3 - 8 J[9] J[2] J[3] J[4] + 8 J[3] J[1] J[2] J[5] + 4 J[9] J[1] J[6] 2 2 2 4 + 4 J[9] J[2] J[0] J[3] - 2 J[2] J[5] J[6] + 2 J[2] J[6] 3 2 2 + 4 J[6] J[7] J[5] + 4 J[3] J[7] J[6] J[4] + 4 J[2] J[5] J[6] J[1] 2 2 3 + 2 J[9] J[1] J[2] J[0] + 2 J[7] J[8] J[4] J[1] + J[10] J[0] J[2] 2 2 2 2 - 2 J[10] J[8] J[4] J[1] - J[6] J[0] J[2] - 4 J[2] J[8] J[4] J[6] 2 3 - 2 J[8] J[3] J[7] J[1] - 2 J[0] J[5] J[3] + 4 J[9] J[5] J[8] J[0] J[2] 2 3 2 - 4 J[9] J[2] J[7] J[3] + 2 J[10] J[3] J[5] - 2 J[9] J[1] J[5] J[0] 2 2 2 2 + 2 J[10] J[6] J[1] J[5] + 4 J[6] J[7] J[1] - 6 J[3] J[9] J[6] J[0] 2 2 2 + 6 J[6] J[7] J[1] J[0] + J[6] J[0] J[5] - 8 J[7] J[0] J[4] J[5] J[2] 2 2 2 2 + 2 J[10] J[3] J[1] J[5] - J[10] J[2] J[4] + 10 J[2] J[8] J[1] J[3] 2 2 3 - 7 J[3] J[7] J[2] - J[10] J[0] J[4] - 2 J[8] J[9] J[2] J[0] J[3] 2 2 2 + 2 J[9] J[5] J[0] J[4] + 4 J[8] J[1] J[7] J[2] - 2 J[3] J[7] J[4] J[6] 2 2 2 2 - 3 J[9] J[2] J[1] - 2 J[7] J[5] J[8] J[3] + 4 J[3] J[0] J[1] J[4] - 2 J[7] J[5] J[8] J[0] J[6] - 4 J[10] J[8] J[5] J[1] J[2] 2 + 8 J[3] J[9] J[1] J[7] J[2] + 8 J[3] J[1] J[7] J[0] + 2 J[10] J[7] J[1] J[4] J[0] - 4 J[10] J[7] J[5] J[2] J[0] 2 2 2 2 + 7 J[0] J[6] J[3] + 2 J[8] J[6] J[5] J[7] J[4] + 2 J[6] J[8] J[1] 3 2 2 2 2 - 2 J[7] J[9] J[4] + J[7] J[8] J[0] - J[0] J[2] J[5] 2 2 2 2 + 2 J[9] J[0] J[1] + 4 J[6] J[0] J[7] J[3] - 6 J[0] J[1] J[7] J[4] 3 2 2 - 2 J[10] J[6] J[7] J[1] J[4] + 4 J[7] J[1] J[4] + J[4] J[0] J[5] 2 + 2 J[8] J[2] J[1] J[4] J[5] + 2 J[9] J[1] J[3] J[2] 2 2 3 - 4 J[6] J[7] J[8] J[3] - 2 J[10] J[0] J[5] J[1] + 8 J[9] J[1] J[2] 2 2 2 2 + 2 J[3] J[5] J[6] J[4] + 2 J[8] J[2] J[6] + 2 J[7] J[9] J[4] J[0] 2 2 2 - 4 J[7] J[9] J[1] J[2] + 4 J[0] J[2] J[4] + 8 J[0] J[5] J[6] J[7] J[2] 3 3 + J[8] J[6] J[4] + 4 J[10] J[0] J[6] J[3] J[1] - 6 J[8] J[6] J[2] 2 2 3 + 2 J[0] J[5] J[2] J[4] - J[10] J[3] J[0] J[4] + 2 J[9] J[5] J[6] 2 2 - 3 J[10] J[6] J[2] - 10 J[6] J[7] J[1] J[0] J[2] 2 2 2 3 + 4 J[4] J[9] J[5] J[3] - 4 J[9] J[2] J[5] J[0] + 2 J[4] J[2] 2 3 - 8 J[8] J[1] J[4] J[6] J[5] - 2 J[8] J[9] J[1] J[5] + 4 J[3] J[8] J[1] 3 3 2 + 2 J[7] J[4] J[1] - 2 J[3] J[9] J[0] - J[10] J[1] J[4] J[2] 2 2 + J[0] J[5] J[6] - 2 J[8] J[9] J[7] J[4] J[2] 2 - 8 J[0] J[2] J[6] J[1] J[5] - 4 J[0] J[8] J[3] J[5] 2 2 2 3 2 + 8 J[6] J[7] J[1] J[3] - J[9] J[6] J[3] + 2 J[4] J[0] 3 2 2 - 2 J[3] J[7] J[4] - J[10] J[0] J[5] J[2] - 7 J[4] J[1] J[0] J[6] 2 3 2 2 + 2 J[10] J[0] J[2] J[3] + 2 J[3] J[6] J[7] + 4 J[1] J[0] J[3] 3 + 2 J[6] J[7] J[3] - 6 J[7] J[8] J[4] J[1] J[6] 2 2 2 - 2 J[10] J[6] J[7] J[1] J[0] - J[8] J[6] J[2] + 2 J[10] J[0] J[4] J[2] 2 2 2 + 6 J[0] J[5] J[6] J[1] + 2 J[9] J[6] J[0] J[1] + J[10] J[8] J[6] J[3] 4 2 3 - J[10] J[2] - 2 J[7] J[5] J[0] J[4] + 2 J[2] J[5] J[7] 2 2 3 2 + J[10] J[7] J[0] - 2 J[10] J[6] J[7] J[5] J[2] + 2 J[6] J[0] 2 - 2 J[4] J[6] J[1] J[3] + 2 J[10] J[8] J[0] J[1] J[3] 2 2 2 2 + 2 J[6] J[8] J[0] J[4] - 2 J[9] J[2] J[0] J[3] + 2 J[8] J[3] J[6] 2 2 3 + 4 J[10] J[7] J[1] J[3] + 2 J[7] J[9] J[1] J[2] + 4 J[9] J[2] J[3] 2 4 2 2 + J[0] J[6] J[8] J[4] - 2 J[4] J[2] + 4 J[4] J[6] J[2] 2 2 2 + 4 J[4] J[0] J[7] J[3] + J[7] J[0] J[6] + 2 J[7] J[8] J[4] J[0] J[1] 3 - 2 J[3] J[5] J[0] - 2 J[10] J[8] J[2] J[1] J[3] 2 2 2 - 4 J[8] J[9] J[7] J[3] J[1] - 6 J[3] J[7] J[4] J[5] + J[2] J[5] J[1] 2 2 2 2 3 + 2 J[7] J[5] J[4] + J[7] J[5] J[0] + 2 J[3] J[6] J[5] 2 2 3 3 2 - J[10] J[3] J[5] + 2 J[10] J[2] J[4] + J[8] J[0] 2 3 3 - 10 J[8] J[2] J[3] J[1] + 2 J[7] J[1] J[2] - 3 J[6] J[0] J[4] 2 4 4 - 2 J[9] J[6] J[0] J[7] - J[0] J[8] - J[0] J[6] 2 2 2 4 + 3 J[10] J[6] J[4] J[5] + 4 J[0] J[8] J[4] + 4 J[2] J[6] 2 2 2 2 - 6 J[1] J[5] J[3] J[6] - 4 J[10] J[3] J[1] - 2 J[9] J[1] J[3] J[4] + 8 J[8] J[1] J[4] J[6] J[3] + 2 J[3] J[9] J[6] J[0] J[4] 2 2 2 2 2 - J[10] J[0] J[5] J[6] + 2 J[1] J[8] J[3] + J[3] J[0] J[4] 2 2 2 3 - 3 J[2] J[3] J[4] + 10 J[3] J[9] J[1] J[4] - 2 J[3] J[1] J[6] 2 2 2 - 4 J[3] J[9] J[1] J[7] J[4] + 7 J[8] J[3] J[6] J[2] + 5 J[2] J[6] J[0] 2 4 2 2 - 4 J[0] J[3] J[4] J[5] + J[6] J[0] - J[8] J[5] J[4] 2 4 2 + 4 J[4] J[8] J[1] J[3] - J[4] J[0] + 4 J[3] J[5] J[8] J[7] 2 - 4 J[0] J[8] J[6] J[3] - 4 J[10] J[5] J[0] J[1] J[4] 2 2 2 - 3 J[0] J[8] J[5] J[2] + J[0] J[8] J[5] + 2 J[10] J[2] J[4] J[5] J[3] 2 2 + 2 J[10] J[2] J[5] + 4 J[7] J[9] J[4] J[5] J[3] 2 - 2 J[0] J[8] J[2] J[4] - 2 J[9] J[6] J[8] J[5] J[0] 2 3 2 + 2 J[6] J[7] J[2] J[3] + 2 J[8] J[3] J[1] - 2 J[4] J[7] J[5] J[3] 2 2 2 2 + J[10] J[8] J[6] J[1] - 2 J[3] J[6] J[4] - 4 J[6] J[7] J[2] J[5] 2 2 2 3 - 2 J[4] J[8] J[0] + 2 J[0] J[2] J[5] J[3] - 2 J[7] J[2] J[5] 4 2 2 - J[10] J[1] + 2 J[4] J[9] J[0] J[3] - J[10] J[8] J[6] J[0] 2 2 2 3 + 2 J[7] J[9] J[4] J[1] - J[0] J[6] J[4] - 2 J[1] J[5] J[6] 2 2 2 - 2 J[7] J[5] J[8] J[0] + 2 J[9] J[2] J[6] J[5] J[0] + 3 J[7] J[1] J[2] 2 - 2 J[9] J[1] J[5] J[4] - 4 J[9] J[6] J[8] J[1] J[2] 2 3 + 2 J[8] J[2] J[4] J[6] + 2 J[8] J[9] J[4] J[6] J[1] - 3 J[2] J[8] J[0] 3 2 2 3 - 3 J[0] J[2] + 4 J[7] J[8] J[0] J[1] - 2 J[6] J[7] J[3] 2 4 3 3 - J[10] J[8] J[0] J[3] + J[3] J[0] - 2 J[3] J[5] J[6] + J[8] J[4] J[2] 3 - 2 J[2] J[3] J[5] J[6] J[4] - 6 J[3] J[1] J[2] 2 3 - 2 J[10] J[3] J[7] J[0] J[4] + J[10] J[0] J[2] J[6] - 4 J[3] J[7] J[2] 3 2 - 3 J[6] J[0] J[2] + J[7] J[2] J[6] J[0] + 4 J[8] J[3] J[7] J[6] J[4] 2 2 2 2 + 4 J[8] J[2] J[1] J[6] + J[0] J[5] J[2] - 2 J[7] J[9] J[4] J[5] 3 2 2 + 2 J[2] J[9] J[4] J[0] J[5] + 4 J[3] J[7] J[2] + 2 J[10] J[3] J[0] 2 3 3 2 + 4 J[1] J[6] - 2 J[7] J[5] J[2] - 2 J[4] J[9] J[0] J[5] 2 2 2 2 + 2 J[7] J[9] J[2] J[5] - 3 J[7] J[5] J[6] + 2 J[8] J[1] J[4] J[5] 2 3 2 2 - 2 J[1] J[6] J[5] J[2] + 2 J[0] J[3] J[5] - 9 J[4] J[6] J[1] 2 2 + 2 J[10] J[1] J[2] J[4] J[3] - 4 J[2] J[6] J[3] 2 2 2 2 + 4 J[3] J[1] J[4] J[7] - 3 J[10] J[6] J[0] J[2] + J[9] J[2] J[0] 2 + 12 J[0] J[8] J[2] J[1] J[3] - 2 J[3] J[9] J[5] J[6] 2 2 2 2 + 4 J[3] J[9] J[6] J[0] + 3 J[2] J[1] J[4] - 6 J[3] J[7] J[2] J[5] 4 3 2 4 - J[0] J[2] - 2 J[3] J[5] J[8] - 4 J[9] J[1] J[6] J[4] - J[3] J[6] 2 2 3 2 + J[9] J[0] J[2] + 2 J[3] J[1] J[4] + 6 J[0] J[8] J[4] J[1] 2 2 2 2 + 4 J[9] J[1] J[2] J[3] - 2 J[7] J[8] J[2] J[1] + J[10] J[1] J[5] 3 2 2 + 2 J[10] J[5] J[1] - 4 J[7] J[5] J[8] J[4] - 12 J[4] J[6] J[7] J[1] 3 - 2 J[10] J[5] J[1] J[6] J[4] + J[8] J[0] J[4] 2 - 2 J[0] J[6] J[2] J[7] J[3] - 4 J[6] J[7] J[1] J[3] 2 2 + 4 J[7] J[5] J[2] J[3] + 2 J[9] J[5] J[0] J[3] 2 - 6 J[7] J[8] J[2] J[1] J[0] + 2 J[9] J[6] J[7] J[0] 2 2 2 2 2 + 2 J[3] J[1] J[6] J[2] + J[0] J[8] J[1] - J[9] J[2] J[4] 2 2 2 - 2 J[3] J[9] J[6] J[7] + 4 J[4] J[1] J[5] J[2] - 4 J[8] J[3] J[0] J[1] 3 2 2 - 2 J[10] J[5] J[1] - 6 J[7] J[5] J[1] J[0] + 4 J[8] J[9] J[2] J[1] 2 2 2 + 3 J[3] J[0] J[6] J[4] + 6 J[8] J[9] J[1] J[3] + 2 J[6] J[0] J[2] J[4] 2 3 2 + 4 J[9] J[6] J[1] J[4] + J[10] J[0] J[4] + 3 J[0] J[6] J[8] J[4] 2 2 4 + 2 J[7] J[1] J[2] J[6] + 4 J[10] J[3] J[5] J[1] - J[7] J[2] 2 3 - 4 J[9] J[2] J[5] J[6] - 2 J[0] J[6] J[8] J[5] J[3] - 4 J[8] J[0] J[4] 3 2 2 2 + 4 J[2] J[1] + 2 J[10] J[1] J[5] J[4] + 8 J[2] J[8] J[7] J[3] 2 2 2 2 2 2 - 5 J[2] J[8] J[1] + J[9] J[0] J[4] - 3 J[6] J[7] J[4] 3 - 6 J[3] J[1] J[8] J[2] J[4] - J[6] J[8] J[0] 2 2 2 - 8 J[8] J[6] J[2] J[3] J[5] - 2 J[9] J[6] J[7] J[1] + J[6] J[0] J[8] 2 2 5 + J[10] J[0] J[6] J[2] + 2 J[8] J[9] J[1] J[0] + J[0] 3 2 2 + 3 J[6] J[8] J[4] + J[0] J[4] J[5] + 4 J[10] J[7] J[1] J[0] J[2] 2 2 2 + 8 J[1] J[4] J[6] + 2 J[7] J[8] J[4] J[2] 3 2 - 4 J[10] J[3] J[6] J[4] J[5] + 2 J[0] J[5] J[1] - 2 J[8] J[9] J[7] J[0] 2 3 2 + 2 J[10] J[6] J[4] J[2] + 2 J[0] J[3] J[1] - 4 J[9] J[2] J[4] J[5] 3 + 2 J[10] J[7] J[4] J[0] J[5] + 2 J[10] J[7] J[1] 2 3 2 - 2 J[3] J[9] J[6] J[5] - 2 J[0] J[5] - 10 J[4] J[8] J[2] J[7] J[3] 2 3 2 2 - 2 J[8] J[9] J[1] J[4] J[0] - 2 J[4] J[0] - J[10] J[6] J[4] 2 2 2 2 2 2 + 4 J[10] J[1] J[2] - J[1] J[8] J[4] + 4 J[6] J[1] J[5] 2 3 2 - 6 J[2] J[3] J[7] J[1] - 2 J[8] J[3] J[5] - 4 J[0] J[6] J[7] J[3] 2 2 2 2 - 8 J[3] J[1] J[7] J[0] - 2 J[10] J[7] J[1] J[0] - J[10] J[7] J[3] 2 2 + 2 J[8] J[3] J[4] J[5] - 2 J[10] J[8] J[1] J[0] 2 3 - 5 J[0] J[8] J[1] J[6] + J[10] J[8] J[0] + 2 J[9] J[6] J[8] J[5] J[2] 2 + 2 J[10] J[2] J[7] J[4] J[5] + J[10] J[3] J[6] J[4] 2 2 3 2 + 2 J[6] J[8] J[5] J[3] - J[4] J[6] + 2 J[8] J[5] J[7] J[1] - 4 J[9] J[2] J[7] J[5] J[3] + J[10] J[6] J[0] J[4] J[2] 2 2 2 3 - J[10] J[7] J[1] - 4 J[6] J[7] J[1] J[3] - 2 J[7] J[1] J[0] 2 2 2 3 2 - 3 J[6] J[8] J[5] - 4 J[1] J[4] - 12 J[3] J[1] J[4] J[7] 3 2 2 + 3 J[4] J[8] J[2] + 6 J[8] J[6] J[2] J[0] - 2 J[9] J[5] J[1] J[0] 3 3 2 + 2 J[0] J[8] J[5] J[3] J[2] - J[10] J[6] J[0] + 4 J[2] J[3] 2 + 2 J[9] J[6] J[3] J[1] + 2 J[8] J[9] J[2] J[6] J[3] 2 2 - 10 J[4] J[2] J[7] J[1] - 4 J[7] J[1] J[4] J[5] 2 2 + 4 J[0] J[8] J[4] J[3] + 6 J[3] J[0] J[1] J[2] 3 2 2 + 4 J[7] J[3] J[1] J[5] J[4] + 4 J[9] J[1] J[4] - 2 J[8] J[2] J[4] 2 2 2 3 + 4 J[3] J[1] J[4] + 2 J[0] J[5] J[6] J[2] + 2 J[1] J[7] J[0] 3 3 3 - 4 J[9] J[1] J[0] + 3 J[0] J[6] J[2] - 2 J[4] J[9] J[5] 2 2 5 - 2 J[1] J[5] J[3] J[4] + 6 J[0] J[8] J[2] J[4] - J[6] 2 2 2 - 2 J[8] J[9] J[4] J[3] + J[9] J[6] J[0] + 4 J[8] J[9] J[4] J[2] J[3] 2 2 2 + 6 J[7] J[5] J[2] J[3] + 2 J[9] J[6] J[8] J[1] J[0] - 2 J[8] J[1] J[5] 2 2 2 2 2 - 4 J[8] J[5] J[6] J[1] - J[3] J[0] J[2] - 2 J[8] J[1] J[4] 2 2 2 2 + 2 J[10] J[0] J[2] - 2 J[9] J[5] J[8] J[4] J[2] + J[4] J[2] J[5] 3 2 2 - 5 J[0] J[2] J[6] + 2 J[8] J[9] J[7] J[3] - 2 J[1] J[6] J[8] J[3] 3 2 2 + 2 J[3] J[0] J[7] - 2 J[7] J[8] J[5] J[2] + 6 J[4] J[6] J[1] J[3] 2 2 - 3 J[7] J[0] J[2] J[4] + 2 J[10] J[7] J[3] J[1] 2 2 - 2 J[4] J[9] J[0] J[3] - J[10] J[8] J[2] J[0] 2 + 4 J[10] J[6] J[2] J[5] J[3] + 2 J[1] J[6] J[8] J[4] 2 2 - J[9] J[6] J[0] J[2] - 4 J[9] J[1] J[4] J[0] 2 2 2 - 2 J[10] J[8] J[5] J[3] J[4] - 3 J[3] J[1] J[2] - J[10] J[8] J[0] J[2] 3 3 3 - 4 J[9] J[1] J[2] - 2 J[4] J[8] J[2] - 6 J[4] J[6] J[2] 2 2 2 2 + J[10] J[0] J[6] - 2 J[7] J[5] J[1] J[2] - J[10] J[7] J[0] J[2] 2 2 2 - 3 J[7] J[8] J[6] J[0] + 2 J[8] J[9] J[4] J[5] + 3 J[4] J[8] J[0] J[2] 2 2 2 2 2 + 3 J[8] J[2] J[0] - J[4] J[8] J[5] + 2 J[8] J[5] J[1] J[4] 3 4 + 2 J[9] J[1] J[0] - 4 J[8] J[9] J[2] J[1] J[0] + 3 J[1] J[0] 3 3 - 2 J[8] J[3] J[7] + 2 J[1] J[8] J[5] + J[10] J[8] J[6] J[4] J[0] 2 + 6 J[7] J[5] J[0] J[3] - 3 J[8] J[6] J[2] J[0] J[4] 2 2 2 2 - 4 J[8] J[3] J[6] J[4] - 6 J[10] J[7] J[1] J[3] + J[10] J[0] J[4] 3 2 3 2 + 2 J[4] J[8] + 3 J[4] J[0] J[6] + 4 J[9] J[1] J[5] J[4] 2 - 2 J[7] J[9] J[0] J[6] J[4] - 2 J[3] J[0] J[4] J[5] 2 + 4 J[10] J[6] J[7] J[1] J[2] - 2 J[8] J[3] J[7] J[0] 2 2 - 3 J[1] J[6] J[8] + 4 J[3] J[9] J[1] J[7] J[6] 2 2 3 + 2 J[10] J[8] J[5] J[1] J[0] + 2 J[6] J[2] J[5] - 4 J[2] J[3] J[5] 4 - 2 J[10] J[8] J[6] J[1] J[3] + J[7] J[0] + 2 J[6] J[9] J[0] J[3] J[2] 2 2 2 2 3 + 3 J[3] J[0] J[7] - 3 J[8] J[3] J[4] - 2 J[1] J[5] J[4] 2 2 + 2 J[10] J[1] J[4] J[6] + 6 J[7] J[1] J[8] J[3] 2 2 2 2 - 2 J[7] J[5] J[8] J[1] - J[10] J[6] J[0] J[4] - J[3] J[5] J[6] 2 2 2 - 3 J[3] J[1] J[6] + 4 J[3] J[7] J[6] J[5] + 2 J[8] J[9] J[7] J[0] J[2] 3 2 2 2 + J[0] J[6] J[8] + 2 J[3] J[5] J[0] - 2 J[7] J[5] J[1] J[6] 4 2 + 2 J[2] J[8] - 4 J[9] J[2] J[4] J[0] J[3] + 2 J[8] J[6] J[2] J[5] 2 2 2 2 - J[7] J[2] J[0] - 2 J[4] J[2] J[6] J[5] - 6 J[2] J[1] J[4] J[5] 3 2 + 2 J[4] J[2] J[6] + 2 J[10] J[8] J[5] J[1] J[4] + 4 J[9] J[2] J[6] J[7] 2 2 + 6 J[1] J[5] J[7] J[2] + 2 J[7] J[1] J[4] J[5] 2 2 2 3 + 11 J[0] J[2] J[1] J[6] - 10 J[8] J[2] J[3] - 6 J[9] J[1] J[4] 2 2 2 2 + J[10] J[8] J[2] J[4] - J[9] J[6] J[4] J[0] + 2 J[7] J[0] J[4] 4 2 2 - J[10] J[0] - 2 J[10] J[3] J[6] J[2] + J[0] J[2] J[4] J[6] 2 3 2 - 2 J[7] J[9] J[5] J[0] - 2 J[1] J[6] J[5] - 4 J[8] J[9] J[1] J[3] 2 2 2 + 2 J[3] J[0] J[8] J[5] + 2 J[7] J[8] J[4] J[0] J[5] - 2 J[8] J[6] J[4] 2 2 2 + 14 J[0] J[2] J[1] J[7] - J[10] J[8] J[0] J[4] - 2 J[9] J[1] J[6] J[0] 2 2 2 - 4 J[10] J[0] J[6] J[3] + 2 J[10] J[7] J[5] J[6] J[0] - J[0] J[3] J[4] 2 3 2 2 - 8 J[3] J[1] J[2] J[5] - 3 J[0] J[2] J[4] + 2 J[0] J[8] J[3] 4 2 2 + 2 J[4] J[6] - 4 J[0] J[5] J[6] J[3] + J[0] J[6] J[8] J[2] 2 2 - 2 J[8] J[2] J[7] J[3] + 2 J[10] J[7] J[5] J[2] 2 3 - 2 J[6] J[2] J[7] J[5] + 6 J[0] J[5] J[1] J[4] J[6] + 2 J[9] J[2] J[5] 2 2 3 2 - 2 J[6] J[7] J[0] J[4] - J[9] J[0] + 8 J[0] J[2] J[1] J[5] 3 2 2 2 + 2 J[7] J[1] J[4] + 2 J[9] J[4] J[1] - 2 J[8] J[0] J[5] J[1] 2 3 + 4 J[6] J[8] J[1] J[5] + 2 J[3] J[0] J[1] + 3 J[10] J[8] J[2] J[0] J[4] 2 2 2 3 + 2 J[4] J[8] J[7] J[3] - 6 J[7] J[1] J[4] + 2 J[8] J[5] J[7] 2 4 3 2 3 + J[10] J[7] J[4] J[2] - J[10] J[5] - J[8] J[3] + J[10] J[6] J[2] 2 2 - 2 J[8] J[1] J[7] J[0] - 4 J[10] J[2] J[7] J[1] 2 2 2 - 2 J[3] J[1] J[2] J[5] - 4 J[9] J[1] J[4] J[2] - J[10] J[8] J[5] J[0] 4 4 + 4 J[1] J[4] + J[0] J[5] - 6 J[0] J[6] J[2] J[5] J[3] 2 2 3 2 - J[7] J[8] J[4] - 2 J[8] J[9] J[1] + 8 J[3] J[1] J[4] J[5] 2 2 - 2 J[9] J[1] J[4] J[0] - 3 J[10] J[0] J[4] J[2] 2 2 3 + 4 J[7] J[9] J[4] J[6] + 6 J[7] J[0] J[2] J[5] - 8 J[1] J[3] J[0] 2 2 4 + 2 J[7] J[2] J[4] + 2 J[0] J[2] - 2 J[6] J[7] J[8] J[2] J[1] + 2 J[7] J[8] J[2] J[3] J[0] - J[10] J[8] J[6] J[4] J[2] 2 2 3 2 3 + 4 J[3] J[5] J[8] - 6 J[1] J[8] J[3] + 2 J[7] J[2] 3 2 + 8 J[9] J[1] J[5] J[3] J[2] - 6 J[8] J[3] J[7] J[0] J[4] - J[8] J[1] 2 2 2 3 + 2 J[8] J[2] J[1] - J[8] J[2] + 4 J[7] J[5] J[8] J[1] J[3] 2 2 3 2 - 2 J[10] J[5] J[2] J[4] - 2 J[8] J[2] J[5] J[1] - 5 J[2] J[6] 2 2 3 2 + J[9] J[4] J[0] - 4 J[3] J[1] J[2] - 2 J[3] J[1] J[4] J[5] 2 2 2 2 + 2 J[8] J[5] J[7] J[0] - 2 J[6] J[0] J[7] J[5] + 4 J[2] J[6] J[4] 3 2 2 + 8 J[9] J[2] J[6] J[3] J[4] + J[10] J[6] J[0] + J[9] J[0] J[3] 2 3 2 - 4 J[10] J[1] J[6] J[2] + 2 J[2] J[5] J[3] + 2 J[8] J[2] J[1] J[5] 2 2 + 8 J[4] J[8] J[2] J[3] - 2 J[10] J[3] J[0] J[5] 3 2 + 2 J[10] J[8] J[1] J[3] J[4] + 2 J[0] J[2] J[4] + 4 J[9] J[2] J[5] J[1] 2 2 2 - 2 J[8] J[9] J[2] J[3] + 2 J[10] J[6] J[3] 2 2 2 - 4 J[3] J[9] J[1] J[7] J[0] - 4 J[2] J[3] J[5] - 2 J[9] J[0] J[3] J[1] 2 2 3 + 4 J[0] J[1] J[7] J[4] J[2] + 5 J[1] J[0] J[4] + 3 J[0] J[4] J[2] 2 2 2 2 + 6 J[3] J[9] J[1] J[0] + J[7] J[0] J[4] + 4 J[3] J[1] J[6] J[5] 2 2 2 2 + J[10] J[1] J[6] J[0] - 4 J[6] J[8] J[4] - 6 J[6] J[1] J[7] J[0] 2 2 2 - 2 J[4] J[1] J[8] J[3] - 2 J[9] J[6] J[7] J[2] + 2 J[7] J[9] J[1] J[0] 2 2 - J[10] J[7] J[4] J[0] + 2 J[0] J[5] J[1] J[3] 3 2 + 4 J[10] J[0] J[4] J[5] J[3] - 2 J[7] J[9] J[2] + 5 J[1] J[2] J[6] J[4] 2 2 2 2 - 2 J[9] J[5] J[8] J[3] - 10 J[0] J[6] J[1] J[3] - 2 J[7] J[4] J[2] 2 2 2 2 + 4 J[8] J[6] J[4] J[5] + 2 J[7] J[9] J[0] J[3] - J[0] J[6] J[4] 2 2 2 - 4 J[8] J[6] J[2] J[1] J[3] - 3 J[3] J[7] J[6] + 2 J[9] J[6] J[0] J[5] 3 5 2 - 2 J[3] J[0] J[7] - J[2] - J[10] J[0] J[4] J[5] 2 2 2 2 2 + 2 J[9] J[5] J[3] J[4] + 2 J[7] J[0] J[2] + J[8] J[6] J[0] 3 2 - 4 J[7] J[9] J[4] J[2] J[0] + 2 J[2] J[3] J[5] - 2 J[3] J[9] J[4] J[5] 2 2 2 2 - 2 J[8] J[6] J[2] J[0] + 4 J[9] J[1] J[4] J[5] - 6 J[3] J[0] J[6] 2 + 2 J[8] J[6] J[2] J[5] J[1] - 4 J[10] J[2] J[6] J[4] 2 - 2 J[7] J[9] J[1] J[0] + 8 J[9] J[1] J[6] J[5] J[3] 2 2 + 2 J[10] J[7] J[4] J[3] + 6 J[7] J[9] J[4] J[2] 2 2 - 2 J[8] J[9] J[5] J[0] J[4] - 2 J[3] J[6] J[2] 2 2 - 2 J[10] J[3] J[5] J[2] - 2 J[10] J[6] J[3] J[5] 2 2 3 - 2 J[10] J[5] J[7] J[1] + 4 J[8] J[6] J[5] J[3] + J[10] J[8] J[4] 2 2 2 2 2 + 2 J[9] J[5] J[8] J[1] + 6 J[3] J[7] J[4] - 2 J[6] J[8] J[3] 2 3 2 2 + 2 J[3] J[0] J[2] J[5] - 2 J[7] J[1] J[8] + 2 J[10] J[0] J[5] 2 2 3 3 - 2 J[9] J[5] J[8] J[2] - J[8] J[0] + 4 J[10] J[3] J[1] + 2 J[10] J[6] J[1] J[5] J[2] - 4 J[7] J[3] J[6] J[2] J[4] 2 2 2 2 2 - 6 J[10] J[1] J[0] J[2] + 2 J[8] J[3] J[0] + 6 J[6] J[8] J[2] 2 2 2 2 - 6 J[0] J[2] J[1] + 2 J[7] J[1] J[5] J[0] + 4 J[2] J[1] J[3] J[6] 2 + 4 J[7] J[5] J[8] J[2] J[4] - 5 J[0] J[5] J[6] J[4] 2 + 4 J[7] J[0] J[4] J[5] + 2 J[0] J[2] J[1] J[3] J[4] 2 2 - 2 J[10] J[3] J[1] J[4] - 2 J[7] J[5] J[0] J[3] 2 3 2 + 10 J[10] J[5] J[0] J[2] J[1] - 4 J[3] J[9] J[4] J[6] - 3 J[0] J[3] 2 2 2 + 2 J[7] J[9] J[4] J[3] + 4 J[8] J[1] J[5] J[3] - 4 J[8] J[2] J[6] J[0] where J[a](t) is: infinity ----- (2 n + a) \ t ) ----------- / n! (n + a)! ----- n = 0 The sequence of the number of such walks of length 2n for n=1.., 50, is: [1, 3, 15, 105, 945, 10394, 135057, 2023020, 34284920, 647659574, 13471248273, 305364739239, 7477837120725, 196339572677700, 5491374604824840, 162677781287102790, 5079143986721679870, 166406785107656869650, 5698966984020859624950, 203321476416593182445694, 7533794159176327055809470, 289141399514122275996989900, 11466150928975939345856278380, 468799060032886468347672828620, 19722446257347790665717402050580, 852243935053102928688060969824340, 37765155125166681537920579059052845, 1713564952829203213189675467051105625, 79506920915068919834682409461449354175, 3767616335149216720732158818893539776160, 182134370697342662421085149434603757359910, 8972757853336159750829453571589014833341230, 450038014909090386653899641951856443075067550, 22960093831546540747523361065181023424356491050, 1190528467155931357824012274908338611545182537610, 62692238883186404544156475119327141057433305921470, 3350311399398847739478249959197457940804003114969710, 181578613251314383099059136733233502320649028563047500, 9974298558522095221171525865448541780890735549148164750, 554990842673740116989211855581937735595019304753903009000, 31263586490398087965432957696329299838267544542184773392080, 1782049671736735500648714411451129430739005946631593084321540, 102735582441452048964871854361730713599876850610090715347479650, 5987514165608340441310824868846011043577464245276600310205612550, 352624752427972045868919778491745323003210105361508517424420791138, 20977135976985612543363018341824820825866757445882312148364934460920, 1260036512469674700998572512374337001250853595950626401980979808049240, 76395723788098547510501660664625455696086883266603680413188823533353260, 4673667428647253256112763656188092817737739249131545093115101538140726140, 288410491500694910751500881900969349109956226391099393569679767086527301700 ] The sequence is annihilated by the recurrence operator 1800 (2 n + 5) (2 n + 3) (1 + 2 n) (n + 2) (1 + n) - -------------------------------------------------- (n + 8) (n + 18) (n + 17) (n + 15) (n + 12) 2 4 (2 n + 5) (2 n + 3) (n + 2) (259 n + 4611 n + 18962) N + --------------------------------------------------------- (n + 8) (n + 18) (n + 17) (n + 15) (n + 12) 4 3 2 2 2 (2 n + 5) (35 n + 1484 n + 22363 n + 139454 n + 294368) N 3 - --------------------------------------------------------------- + N (n + 8) (n + 18) (n + 17) (n + 15) (n + 12) The asymptotic behaviour is 21 n / 1705 3041665 1889861875\ 0.1783005393 10 100 |1 - ---- + ------- - ----------| | 8 n 2 3 | \ 128 n 1024 n / - -------------------------------------------------------- 55/2 n The expected number of visits to the origin is , 1.010316156 The relative Polya constant, i.e. the probablity of returning to the origin, while staying in [x[2] <= x[1], x[3] <= x[2], x[4] <= x[3], x[5] <= x[4], 0 <= x[5]] is equal to, 0.0102108196 The whole thing took, 257.729, seconds of CPU time