Let M(x,q) be the bivariate weight enumerator for Motzkin paths of length n and area m. ---------------------------- M(x,1) is (1/2) / 2 \ x - 1 + \-3 x - 2 x + 1/ - ------------------------------ 2 2 x The Maclaurin polynomial of order 50 of M(x,1) is 50 49 2837208756709314025578 x + 973899740488107474693 x 48 47 + 334496473194459009429 x + 114956499435014161638 x 46 45 + 39532221379621112004 x + 13603677110519480289 x 44 43 + 4684478925507420069 x + 1614282136160911722 x 42 41 + 556704809728838604 x + 192137918101841817 x 40 39 + 66368199913921497 x + 22944749046030949 x 38 37 + 7939655757745265 x + 2750016719520991 x 36 35 + 953467954114363 x + 330931069469828 x 34 33 + 114988706524270 x + 40002464776083 x 32 31 30 + 13933569346707 x + 4859761676391 x + 1697385471211 x 29 28 27 + 593742784829 x + 208023278209 x + 73007772802 x 26 25 24 + 25669818476 x + 9043402501 x + 3192727797 x 23 22 21 + 1129760415 x + 400763223 x + 142547559 x 20 19 18 17 + 50852019 x + 18199284 x + 6536382 x + 2356779 x 16 15 14 13 12 + 853467 x + 310572 x + 113634 x + 41835 x + 15511 x 11 10 9 8 7 6 + 5798 x + 2188 x + 835 x + 323 x + 127 x + 51 x 5 4 3 2 + 21 x + 9 x + 4 x + 2 x + x + 1 This is the enumerating function for Motzkin paths of lengths 0 to 50 ---------------------------- The first derivative with respect to q of M(x,q) evaluated at q=1 is 2 / (1/2)\ | / 2 \ | \x - 1 + \-3 x - 2 x + 1/ / - --------------------------------- 2 / 2 \ 4 x \3 x + 2 x - 1/ The Maclaurin polynomial of order 50 is 50 49 393725844176714426273209 x + 130779509898557773457912 x 48 47 + 43434505121125890935500 x + 14423698796337447707216 x 46 45 + 4789189184897183557759 x + 1589965382829229180688 x 44 43 + 527776624989874521556 x + 175164311554822692452 x 42 41 + 58125935030294209435 x + 19284954722600733488 x 40 39 + 6397152897821915224 x + 2121626402922994240 x 38 37 + 703492213702678732 x + 233212294076450364 x 36 35 + 77292662170517910 x + 25610140323335972 x 34 33 + 8483281546400477 x + 2809211622529820 x 32 31 + 929957353197550 x + 307742673568448 x 30 29 + 101799399463446 x + 33660499478792 x 28 27 26 + 11124919273160 x + 3674980475284 x + 1213314272395 x 25 24 23 + 400337992056 x + 132003957436 x + 43493134160 x 22 21 20 + 14318240578 x + 4709218604 x + 1547195902 x 19 18 17 16 + 507710420 x + 166374109 x + 54433100 x + 17776102 x 15 14 13 12 + 5792528 x + 1882717 x + 610052 x + 196938 x 11 10 9 8 7 6 + 63284 x + 20219 x + 6412 x + 2014 x + 624 x + 190 x 5 4 3 2 + 56 x + 16 x + 4 x + x This is the enumerating function for the total area under Motzkin paths of lengths 0 to 50 ---------------------------- The second derivative with respect to q of M(x,q) evaluated at q=1 is // (1/2) 1 || / 2 \ 2 2 ------------------- \\6 \-3 x - 2 x + 1/ x + 9 x 3 / 2 \ 2 \3 x + 2 x - 1/ (1/2) (1/2) / 2 \ / 2 \ - \-3 x - 2 x + 1/ x + 6 x + 3 \-3 x - 2 x + 1/ \ / (1/2)\\ | | / 2 \ || - 3/ \x - 1 + \-3 x - 2 x + 1/ // The Maclaurin polynomial of order 50 is 50 49 60018342200338037677877348 x + 19296933101286518733609488 x 48 + 6199263698080227198142284 x 47 46 + 1989864878340188078554936 x + 638148525275823253485580 x 45 44 + 204463845700320595036566 x + 65446800460168367043524 x 43 42 + 20927463775993833063194 x + 6684649530995089298064 x 41 40 + 2132801351611949930800 x + 679682119677569514648 x 39 38 + 216330024731188111640 x + 68762676834303760938 x 37 36 + 21826327491660395718 x + 6917743669191139590 x 35 34 + 2189096467958236654 x + 691574611570532818 x 33 32 + 218092049172727284 x + 68646049633764662 x 31 30 + 21562999058936800 x + 6758613295772072 x 29 28 + 2113444502056238 x + 659220847569368 x 27 26 + 205064578941594 x + 63602189846300 x 25 24 23 + 19663675915232 x + 6058195939164 x + 1859367055168 x 22 21 20 + 568285389682 x + 172885496430 x + 52326531614 x 19 18 17 + 15747070550 x + 4708566186 x + 1397741884 x 16 15 14 13 + 411508510 x + 120008616 x + 34615478 x + 9856558 x 12 11 10 9 8 + 2763890 x + 760790 x + 204698 x + 53508 x + 13478 x 7 6 5 4 3 + 3224 x + 720 x + 142 x + 24 x + 2 x ---------------------------- The third derivative with respect to q of M(x,q) evaluated at q=1 is / / (1/2) 1 | | 5 / 2 \ 4 - ------------------- \3 \-9 x + 9 \-3 x - 2 x + 1/ x 4 / 2 \ 2 \3 x + 2 x - 1/ (1/2) 4 / 2 \ 3 3 + 51 x + 18 \-3 x - 2 x + 1/ x - 19 x (1/2) / 2 \ 2 2 - 23 \-3 x - 2 x + 1/ x + 29 x (1/2) (1/2) / 2 \ / 2 \ + 4 \-3 x - 2 x + 1/ x - 8 x - 4 \-3 x - 2 x + 1/ \ / (1/2)\\ | | / 2 \ || + 4/ \x - 1 + \-3 x - 2 x + 1/ // The Maclaurin polynomial of order 50 is 50 9979004468130168417175264776 x 49 + 3106319429243118326128674720 x 48 + 965500244434759228150013448 x 47 + 299624358508824414586000368 x 46 + 92830228609803801691576962 x 45 + 28711526293205226133117140 x 44 + 8864257109206515536392530 x 43 42 + 2731553417686681805399700 x + 840074129642896844202780 x 41 40 + 257823353303017341862260 x + 78954406223868119559108 x 39 38 + 24122867387456325143700 x + 7352324811989014139046 x 37 36 + 2235134820158962558236 x + 677642937030762691494 x 35 34 + 204854397770969777532 x + 61738962434758191720 x 33 32 + 18546335380787292696 x + 5551986136682617656 x 31 30 + 1655874305279387664 x + 491905326917738010 x 29 28 + 145507338980226060 x + 42844717681257498 x 27 26 + 12553418447343468 x + 3658501282256772 x 25 24 + 1060040930778276 x + 305207784810060 x 23 22 21 + 87270237544980 x + 24765147655230 x + 6969176674140 x 20 19 18 + 1943097572094 x + 536191372860 x + 146255771616 x 17 16 15 + 39375390936 x + 10444121904 x + 2723305584 x 14 13 12 11 + 696162930 x + 173866416 x + 42236346 x + 9921264 x 10 9 8 7 6 + 2235876 x + 477900 x + 95388 x + 17268 x + 2742 x 5 4 + 336 x + 30 x ---------------------------- The fourth derivative with respect to q of M(x,q) evaluated at q=1 is / / (1/2) 1 | | 9 / 2 \ 8 ----------------- \6 \-81 x + 27 \-3 x - 2 x + 1/ x 6 / 2 \ \3 x + 2 x - 1/ (1/2) 8 / 2 \ 7 7 + 54 x + 423 \-3 x - 2 x + 1/ x + 900 x (1/2) / 2 \ 6 6 - 567 \-3 x - 2 x + 1/ x - 477 x (1/2) / 2 \ 5 5 + 358 \-3 x - 2 x + 1/ x - 419 x (1/2) / 2 \ 4 4 - 533 \-3 x - 2 x + 1/ x + 342 x (1/2) / 2 \ 3 3 + 229 \-3 x - 2 x + 1/ x - 346 x (1/2) / 2 \ 2 2 - 107 \-3 x - 2 x + 1/ x + 110 x (1/2) / 2 \ + 15 \-3 x - 2 x + 1/ x - 20 x (1/2) \ / / 2 \ | | - 5 \-3 x - 2 x + 1/ + 5/ \x - 1 (1/2)\\ / 2 \ || + \-3 x - 2 x + 1/ // The Maclaurin polynomial of order 50 is 50 1795639905350278702398315176160 x 49 + 541242107922317104877461633872 x 48 + 162783949415947696581512835792 x 47 + 48846923923257197932181042880 x 46 + 14622566779296215235946962888 x 45 + 4366392606189850333291455096 x 44 + 1300419154507821041659623360 x 43 + 386233775204414743673888352 x 42 + 114383737160546428644603360 x 41 + 33772388816919545828797512 x 40 + 9939718893457655312569848 x 39 38 + 2915594381379157729128672 x + 852198065228595643684752 x 37 36 + 248157488791906501724160 x + 71976879968048955208584 x 35 34 + 20789035451921271462984 x + 5977763440180495850688 x 33 32 + 1710735223816974783528 x + 487114467464829365400 x 31 30 + 137953930187004727488 x + 38844259967134929912 x 29 28 + 10869916434550471800 x + 3021538989359636544 x 27 26 + 833882469783916848 x + 228349526112449952 x 25 24 + 62004782389977240 x + 16682170269390216 x 23 22 + 4443331800329088 x + 1170483585687792 x 21 20 + 304598289031296 x + 78202272710376 x 19 18 17 + 19777236483144 x + 4917749708160 x + 1199678979432 x 16 15 14 + 286354167096 x + 66659304288 x + 15071880384 x 13 12 11 + 3292868952 x + 690522096 x + 137744184 x 10 9 8 7 6 + 25823664 x + 4467888 x + 696144 x + 92880 x + 10056 x 5 4 + 672 x + 24 x ---------------------------- The fifth derivative with respect to q of M(x,q) evaluated at q=1 is / / 1 | | 11 10 - ----------------- \30 \-162 x - 1728 x 7 / 2 \ \3 x + 2 x - 1/ (1/2) / 2 \ 9 9 + 1026 \-3 x - 2 x + 1/ x + 8532 x (1/2) / 2 \ 8 8 + 1224 \-3 x - 2 x + 1/ x - 11532 x (1/2) / 2 \ 7 7 - 5718 \-3 x - 2 x + 1/ x + 10291 x (1/2) / 2 \ 6 6 + 6609 \-3 x - 2 x + 1/ x - 11307 x (1/2) / 2 \ 5 5 - 4902 \-3 x - 2 x + 1/ x + 6189 x (1/2) / 2 \ 4 4 + 3031 \-3 x - 2 x + 1/ x - 3519 x (1/2) / 2 \ 3 3 - 732 \-3 x - 2 x + 1/ x + 984 x (1/2) / 2 \ 2 2 + 288 \-3 x - 2 x + 1/ x - 292 x (1/2) / 2 \ - 16 \-3 x - 2 x + 1/ x + 22 x (1/2) \ / / 2 \ | | + 6 \-3 x - 2 x + 1/ - 6/ \x - 1 (1/2)\\ / 2 \ || + \-3 x - 2 x + 1/ // The Maclaurin polynomial of order 50 is 50 346966089354179363182313163390240 x 49 + 101274567252871177823792282406720 x 48 + 29475496915214913503117426866080 x 47 + 8552911481426674496433485113920 x 46 + 2474001244702020161810106925920 x 45 + 713273117107440055905862036320 x 44 + 204934109407025992504936319400 x 43 + 58668183075170138664377478960 x 42 + 16731820149133306059686895120 x 41 + 4752828317539417358826096480 x 40 + 1344432913997269578850785000 x 39 + 378622695088936874913650640 x 38 + 106132653125976454789641000 x 37 + 29604087692493142147648320 x 36 + 8214668189909158992126360 x 35 34 + 2266878479028381745175040 x + 621896203304665648689360 x 33 32 + 169549451772069154858560 x + 45918211173459407248080 x 31 30 + 12347714696922644944320 x + 3295211820565489521360 x 29 28 + 872234607753224652960 x + 228858468158944442040 x 27 26 + 59481621014447192880 x + 15301776243356978640 x 25 24 + 3892794257745459360 x + 978379152842390760 x 23 22 + 242651565718941840 x + 59308831918817640 x 21 20 + 14264509073297280 x + 3369977772440760 x 19 18 + 780416579975040 x + 176718770242800 x 17 16 15 + 39012927202560 x + 8366438648880 x + 1735208403840 x 14 13 12 + 346122132360 x + 65930273760 x + 11882349600 x 11 10 9 8 + 2000795280 x + 309448560 x + 42800160 x + 5110200 x 7 6 5 + 485040 x + 33840 x + 960 x ---------------------------- The sixth derivative with respect to q of M(x,q) evaluated at q=1 is / / 1 | | 2 - ----------------- \180 \7 - 35 x + 665 x 9 / 2 \ \3 x + 2 x - 1/ (1/2) / 2 \ 14 + 243 \-3 x - 2 x + 1/ x (1/2) / 2 \ 13 + 1458 \-3 x - 2 x + 1/ x (1/2) / 2 \ 12 - 91584 \-3 x - 2 x + 1/ x (1/2) / 2 \ 11 + 227934 \-3 x - 2 x + 1/ x (1/2) / 2 \ 10 3 4 - 310128 \-3 x - 2 x + 1/ x - 3437 x + 17070 x (1/2) 5 6 / 2 \ 9 - 58708 x + 126404 x + 323482 \-3 x - 2 x + 1/ x (1/2) / 2 \ 8 - 315669 \-3 x - 2 x + 1/ x (1/2) / 2 \ 7 + 199216 \-3 x - 2 x + 1/ x (1/2) / 2 \ 6 - 122282 \-3 x - 2 x + 1/ x (1/2) / 2 \ 5 15 + 45722 \-3 x - 2 x + 1/ x + 729 x (1/2) / 2 \ 4 - 15756 \-3 x - 2 x + 1/ x (1/2) / 2 \ 3 14 13 + 2826 \-3 x - 2 x + 1/ x + 23085 x - 14256 x (1/2) 12 / 2 \ 2 - 193050 x - 651 \-3 x - 2 x + 1/ x (1/2) / 2 \ 10 11 + 28 \-3 x - 2 x + 1/ x - 203850 x + 362916 x (1/2)\ 8 9 7 / 2 \ | + 209209 x - 64219 x - 243450 x - 7 \-3 x - 2 x + 1/ / / (1/2)\\ | / 2 \ || \x - 1 + \-3 x - 2 x + 1/ // The Maclaurin polynomial of order 50 is 50 71465175792898680290246330377605120 x 49 + 20200043578034820420907091039400480 x 48 + 5689260961350901981417718342097120 x 47 + 1596380651689297024342077324328800 x 46 + 446190992666465169382719462342000 x 45 + 124202544409072050907283994290400 x 44 + 34425604897344299477040429270960 x 43 + 9499133631084820438805138877120 x 42 + 2608801092458422025827636776960 x 41 + 712932339052133390758599598320 x 40 + 193817812527508910139847684800 x 39 + 52402943004870245649602535360 x 38 + 14086490921686017173672892000 x 37 + 3763504511992345690935955440 x 36 + 999008885146971052018207440 x 35 + 263368799420900619029114160 x 34 + 68927471979936562739463840 x 33 + 17899899830040241726352640 x 32 + 4610160568906477430531760 x 31 30 + 1176908507529878117606400 x + 297617377725001780092720 x 29 28 + 74500703718983697870720 x + 18446404508309472518160 x 27 26 + 4513720855263424992000 x + 1090446746421253556160 x 25 24 + 259800967911432937680 x + 60967411751561703840 x 23 22 + 14071855675127415840 x + 3189212634796028640 x 21 20 + 708372623473529040 x + 153855824628834000 x 19 18 + 32590721974669200 x + 6711778747104480 x 17 16 + 1338758144252640 x + 257444802065520 x 15 14 13 + 47457150893280 x + 8325905515920 x + 1377338038560 x 12 11 10 + 212251779360 x + 29959376400 x + 3785919840 x 9 8 7 6 5 + 412457040 x + 36920880 x + 2380320 x + 99360 x + 720 x ---------------------------- The seventh derivative with respect to q of M(x,q) evaluated at q=1 is / / 1 | | 2 ------------------ \1260 \-8 + 32 x - 1284 x 10 / 2 \ \3 x + 2 x - 1/ (1/2) / 2 \ 14 - 289656 \-3 x - 2 x + 1/ x (1/2) / 2 \ 13 - 188136 \-3 x - 2 x + 1/ x (1/2) / 2 \ 12 + 2386971 \-3 x - 2 x + 1/ x (1/2) / 2 \ 11 - 5245638 \-3 x - 2 x + 1/ x (1/2) / 2 \ 10 3 4 + 6524128 \-3 x - 2 x + 1/ x + 6518 x - 54481 x (1/2) 5 6 / 2 \ 16 + 215061 x - 828936 x + 729 \-3 x - 2 x + 1/ x (1/2) / 2 \ 15 + 33534 \-3 x - 2 x + 1/ x (1/2) / 2 \ 9 - 5688308 \-3 x - 2 x + 1/ x (1/2) / 2 \ 8 + 3721651 \-3 x - 2 x + 1/ x (1/2) / 2 \ 7 - 1712564 \-3 x - 2 x + 1/ x (1/2) / 2 \ 6 + 723360 \-3 x - 2 x + 1/ x (1/2) / 2 \ 5 17 16 - 173674 \-3 x - 2 x + 1/ x + 729 x + 32805 x (1/2) 15 / 2 \ 4 + 701784 x + 51649 \-3 x - 2 x + 1/ x (1/2) / 2 \ 3 14 13 - 5274 \-3 x - 2 x + 1/ x - 3671298 x + 7714953 x (1/2) 12 / 2 \ 2 - 10648143 x + 1276 \-3 x - 2 x + 1/ x (1/2) / 2 \ 10 11 - 24 \-3 x - 2 x + 1/ x - 10754048 x + 11438146 x 8 9 7 - 4792319 x + 7540939 x + 2105726 x (1/2)\ / (1/2)\\ / 2 \ | | / 2 \ || + 8 \-3 x - 2 x + 1/ / \x - 1 + \-3 x - 2 x + 1/ // The Maclaurin polynomial of order 50 is 50 15586598131206135465994466767093667520 x 49 + 4266181792839959234747171190407399040 x 48 + 1162704955216479081552404420673869280 x 47 + 315471123087205012921328675639473440 x 46 + 85196511405651364742478443594119920 x 45 + 22896136344646460872089845508349440 x 44 + 6121814609413052710477541641110960 x 43 + 1628050118961930624848758454166240 x 42 + 430535494415418589853493287778720 x 41 + 113182376450229450517386741137280 x 40 + 29569400970024545082209203871760 x 39 + 7674576497851251368219943458880 x 38 + 1978138353314899282423805128800 x 37 + 506148457429255688400091848480 x 36 + 128508462546511748292890403840 x 35 + 32360460016901510871400272480 x 34 + 8078008461394032619683019200 x 33 + 1997820789937275454507480320 x 32 + 489218479850348806660200000 x 31 + 118535085683064911042663040 x 30 + 28396216892182681494504720 x 29 + 6720145420881140743461120 x 28 27 + 1569618970725545826040080 x + 361452074279829876443520 x 26 25 + 81966064454190494272800 x + 18279424700187916305600 x 24 23 + 4002892163850099406320 x + 859231138057853611680 x 22 21 + 180425089927209748320 x + 36975989167609897440 x 20 19 + 7375481808958740960 x + 1427264900435142720 x 18 17 + 266922907469789760 x + 48018453138936000 x 16 15 + 8262105478774560 x + 1350030661791840 x 14 13 12 + 207614312347680 x + 29698610014080 x + 3891115453680 x 11 10 9 + 456826970880 x + 46660239360 x + 3935725920 x 8 7 6 + 257518800 x + 10604160 x + 236880 x ---------------------------- The eigth derivative with respect to q of M(x,q) evaluated at q=1 is / / 1 | | 2 - ------------------ \10080 \-9 + 42 x - 2349 x 12 / 2 \ \3 x + 2 x - 1/ (1/2) / 2 \ 14 + 428622645 \-3 x - 2 x + 1/ x (1/2) / 2 \ 13 - 466714234 \-3 x - 2 x + 1/ x (1/2) / 2 \ 12 + 434807031 \-3 x - 2 x + 1/ x (1/2) / 2 \ 11 - 323932919 \-3 x - 2 x + 1/ x (1/2) / 2 \ 10 3 4 + 211728140 \-3 x - 2 x + 1/ x + 15251 x - 166960 x 5 6 + 910237 x - 4563630 x (1/2) / 2 \ 16 + 164342709 \-3 x - 2 x + 1/ x (1/2) / 2 \ 15 - 312654384 \-3 x - 2 x + 1/ x (1/2) / 2 \ 20 + 4374 \-3 x - 2 x + 1/ x (1/2) / 2 \ 19 + 292329 \-3 x - 2 x + 1/ x (1/2) / 2 \ 18 + 3568455 \-3 x - 2 x + 1/ x (1/2) / 2 \ 17 - 48756573 \-3 x - 2 x + 1/ x (1/2) / 2 \ 9 - 104443775 \-3 x - 2 x + 1/ x (1/2) / 2 \ 8 + 45322685 \-3 x - 2 x + 1/ x (1/2) / 2 \ 7 - 14380841 \-3 x - 2 x + 1/ x (1/2) / 2 \ 6 + 4131258 \-3 x - 2 x + 1/ x (1/2) / 2 \ 5 20 19 - 771362 \-3 x - 2 x + 1/ x - 269001 x + 10825650 x 18 17 16 15 - 12621906 x - 88550496 x + 287755119 x - 395011575 x (1/2) / 2 \ 4 + 158653 \-3 x - 2 x + 1/ x (1/2) / 2 \ 3 14 - 12965 \-3 x - 2 x + 1/ x + 252475941 x 13 12 + 30554217 x - 247033274 x (1/2) (1/2) / 2 \ 2 / 2 \ + 2334 \-3 x - 2 x + 1/ x - 33 \-3 x - 2 x + 1/ x 10 11 8 9 - 217822011 x + 305234235 x - 50635834 x + 125688554 x (1/2)\ / 7 / 2 \ | | + 17449971 x + 9 \-3 x - 2 x + 1/ / \x - 1 (1/2)\\ / 2 \ || + \-3 x - 2 x + 1/ // The Maclaurin polynomial of order 50 is 50 3578565302003488494700842999067636318080 x 49 + 948418985863613045856432520042707536640 x 48 + 250108158504119483350796545499582522880 x 47 + 65613894689867633669842246586596584960 x 46 + 17119914064892393262167097205370426880 x 45 + 4441559192210843464398372899908859520 x 44 + 1145459930779349837429458116281852160 x 43 + 293568039928107115597313825785182720 x 42 + 74745739079780874658138887475305600 x 41 + 18900218263251723985507276960168320 x 40 + 4744519780871734210200730779060480 x 39 + 1181931465362281098717406815521280 x 38 + 292066795519726084149059396173440 x 37 + 71558511387316020425671354500480 x 36 + 17374455268076396837433530062080 x 35 + 4178240905186316286559430824320 x 34 + 994597178270365918253946044160 x 33 + 234199560902419129159938902400 x 32 + 54512335213793389175781605760 x 31 + 12532137767379564079696634880 x 30 + 2843094326189248880907701760 x 29 + 635863275288307970515946880 x 28 + 140043494878016303321717760 x 27 + 30335468025705580509125760 x 26 + 6453871520362315587740160 x 25 24 + 1346431452245676172414080 x + 274953289527358684542720 x 23 22 + 54846095493262577736960 x + 10661221853837710001280 x 21 20 + 2013888036614960822400 x + 368479928973864879360 x 19 18 + 65052454164309480960 x + 11029966099686737280 x 17 16 + 1786102754573934720 x + 274322657569322880 x 15 14 + 39617190744284160 x + 5321013605308800 x 13 12 11 + 655089821043840 x + 72524725983360 x + 7021458702720 x 10 9 8 + 572780799360 x + 36663177600 x + 1702713600 x 7 6 + 41207040 x + 403200 x ---------------------------- The ninth derivative with respect to q of M(x,q) evaluated at q=1 is / / 1 | | 2 ------------------ \90720 \10 - 30 x + 3997 x 13 / 2 \ \3 x + 2 x - 1/ (1/2) / 2 \ 14 - 12080285819 \-3 x - 2 x + 1/ x (1/2) / 2 \ 13 + 11076904832 \-3 x - 2 x + 1/ x (1/2) / 2 \ 12 - 7963849973 \-3 x - 2 x + 1/ x (1/2) / 2 \ 11 + 4538599106 \-3 x - 2 x + 1/ x (1/2) / 2 \ 10 3 4 - 2172041153 \-3 x - 2 x + 1/ x - 22985 x + 410238 x 5 6 - 2301628 x + 16719791 x (1/2) / 2 \ 16 - 5574479094 \-3 x - 2 x + 1/ x (1/2) / 2 \ 15 + 9906400668 \-3 x - 2 x + 1/ x (1/2) / 2 \ 20 + 33840180 \-3 x - 2 x + 1/ x (1/2) / 2 \ 19 - 190571292 \-3 x - 2 x + 1/ x (1/2) / 2 \ 18 + 19423800 \-3 x - 2 x + 1/ x (1/2) / 2 \ 17 + 1732653936 \-3 x - 2 x + 1/ x (1/2) / 2 \ 22 + 6561 \-3 x - 2 x + 1/ x (1/2) / 2 \ 21 + 39366 \-3 x - 2 x + 1/ x (1/2) / 2 \ 9 23 22 + 797104318 \-3 x - 2 x + 1/ x - 6561 x - 1642437 x (1/2) 21 / 2 \ 8 - 4875552 x - 269964712 \-3 x - 2 x + 1/ x (1/2) / 2 \ 7 + 61492842 \-3 x - 2 x + 1/ x (1/2) / 2 \ 6 - 15524551 \-3 x - 2 x + 1/ x (1/2) / 2 \ 5 20 + 1933106 \-3 x - 2 x + 1/ x + 532523322 x 19 18 17 - 3019887198 x + 8229865410 x - 14674954392 x 16 15 + 19805836524 x - 21688057201 x (1/2) / 2 \ 4 - 399222 \-3 x - 2 x + 1/ x (1/2) / 2 \ 3 14 + 19008 \-3 x - 2 x + 1/ x + 20246564133 x 13 12 - 15664939185 x + 10506800295 x (1/2) (1/2) / 2 \ 2 / 2 \ - 3997 \-3 x - 2 x + 1/ x + 20 \-3 x - 2 x + 1/ x 10 11 8 + 2632451021 x - 5634232465 x + 308057296 x (1/2)\ / 9 7 / 2 \ | | - 957789668 x - 73436335 x - 10 \-3 x - 2 x + 1/ / \x (1/2)\\ / 2 \ || - 1 + \-3 x - 2 x + 1/ // The Maclaurin polynomial of order 50 is 50 860499666989177300974511236735625777616000 x 49 + 220805314056333979811175276018664357409280 x 48 + 56337341227369732668632849079311029927680 x 47 + 14288979788228635978352877217711034403840 x 46 + 3601692693668882019338666784935044974720 x 45 + 901959258241133488642950311583773790720 x 44 + 224340285777841883129775676481614631040 x 43 + 55401692585808059281954242889513159680 x 42 + 13579339035874723233354337102004246400 x 41 + 3302223263136912629562063404255481600 x 40 + 796389402288699971680437899587975680 x 39 + 190388184705384744969848382421328640 x 38 + 45095974630910051770299878215793280 x 37 + 10577626731442651817855776933716480 x 36 + 2455501571174149842795033909957120 x 35 + 563792704760334964166097545871360 x 34 + 127945681048371637428471984470400 x 33 + 28676685476432257823239495395840 x 32 + 6342582347380795178352038872320 x 31 + 1383042620450101517089006602240 x 30 + 297024237991766985649600694400 x 29 + 62753670325065972624858823680 x 28 + 13026349929332690507378102400 x 27 + 2652893851732954499342292480 x 26 + 529209411189181366201689600 x 25 + 103216269259311266811513600 x 24 + 19641349470370302439107840 x 23 22 + 3637905339566666441952000 x + 654001497345131930252160 x 21 20 + 113747791185930058053120 x + 19067118472203167036160 x 19 18 + 3066436147124875553280 x + 470566975984370496000 x 17 16 + 68448539680041185280 x + 9360560574776759040 x 15 14 + 1191099546713180160 x + 139176869849216640 x 13 12 + 14672837784890880 x + 1363530790298880 x 11 10 9 + 107838361774080 x + 6934206424320 x + 329002248960 x 8 7 6 + 10475982720 x + 132814080 x + 362880 x ---------------------------- ---------------------------- ---------------------------- So... 1. the generating function for the number of Motzkin paths of lengths n is (1/2) / 2 \ x - 1 + \-3 x - 2 x + 1/ - ------------------------------ 2 2 x and the enumerating function for the number Motzkin paths of lengths 0 to 50 is 50 49 2837208756709314025578 x + 973899740488107474693 x 48 47 + 334496473194459009429 x + 114956499435014161638 x 46 45 + 39532221379621112004 x + 13603677110519480289 x 44 43 + 4684478925507420069 x + 1614282136160911722 x 42 41 + 556704809728838604 x + 192137918101841817 x 40 39 + 66368199913921497 x + 22944749046030949 x 38 37 + 7939655757745265 x + 2750016719520991 x 36 35 + 953467954114363 x + 330931069469828 x 34 33 + 114988706524270 x + 40002464776083 x 32 31 30 + 13933569346707 x + 4859761676391 x + 1697385471211 x 29 28 27 + 593742784829 x + 208023278209 x + 73007772802 x 26 25 24 + 25669818476 x + 9043402501 x + 3192727797 x 23 22 21 + 1129760415 x + 400763223 x + 142547559 x 20 19 18 17 + 50852019 x + 18199284 x + 6536382 x + 2356779 x 16 15 14 13 12 + 853467 x + 310572 x + 113634 x + 41835 x + 15511 x 11 10 9 8 7 6 + 5798 x + 2188 x + 835 x + 323 x + 127 x + 51 x 5 4 3 2 + 21 x + 9 x + 4 x + 2 x + x + 1 ---------------------------- 2. the generating function for the total area under Motzkin paths of lengths n is 2 / (1/2)\ | / 2 \ | \x - 1 + \-3 x - 2 x + 1/ / - --------------------------------- 2 / 2 \ 4 x \3 x + 2 x - 1/ and the enumerating function for the total area under Motzkin paths of lengths 0 to 50 is 50 49 393725844176714426273209 x + 130779509898557773457912 x 48 47 + 43434505121125890935500 x + 14423698796337447707216 x 46 45 + 4789189184897183557759 x + 1589965382829229180688 x 44 43 + 527776624989874521556 x + 175164311554822692452 x 42 41 + 58125935030294209435 x + 19284954722600733488 x 40 39 + 6397152897821915224 x + 2121626402922994240 x 38 37 + 703492213702678732 x + 233212294076450364 x 36 35 + 77292662170517910 x + 25610140323335972 x 34 33 + 8483281546400477 x + 2809211622529820 x 32 31 + 929957353197550 x + 307742673568448 x 30 29 + 101799399463446 x + 33660499478792 x 28 27 26 + 11124919273160 x + 3674980475284 x + 1213314272395 x 25 24 23 + 400337992056 x + 132003957436 x + 43493134160 x 22 21 20 + 14318240578 x + 4709218604 x + 1547195902 x 19 18 17 16 + 507710420 x + 166374109 x + 54433100 x + 17776102 x 15 14 13 12 + 5792528 x + 1882717 x + 610052 x + 196938 x 11 10 9 8 7 6 + 63284 x + 20219 x + 6412 x + 2014 x + 624 x + 190 x 5 4 3 2 + 56 x + 16 x + 4 x + x ---------------------------- 3. the generating function for the sum of the squares of the areas under Motzkin paths of lengths n is // (1/2)\ / 1 || / 2 \ | | 5 - ---------------------- \\x - 1 + \-3 x - 2 x + 1/ / \9 x 3 / 2 \ 2 4 \3 x + 2 x - 1/ x (1/2) / 2 \ 4 4 - 3 \-3 x - 2 x + 1/ x - 15 x (1/2) / 2 \ 3 3 + 14 \-3 x - 2 x + 1/ x - 26 x (1/2) / 2 \ 2 2 - 8 \-3 x - 2 x + 1/ x + 4 x (1/2) (1/2) / 2 \ / 2 \ - 4 \-3 x - 2 x + 1/ x + 5 x + \-3 x - 2 x + 1/ \\ || - 1// and the enumerating function for the sum of the squares of the areas under Motzkin paths of lengths 0 to 50 is 50 49 60412068044514752104150557 x + 19427712611185076507067400 x 48 + 6242698203201353089077784 x 47 46 + 2004288577136525526262152 x + 642937714460720437043339 x 45 44 + 206053811083149824217254 x + 65974577085158241565080 x 43 42 + 21102628087548655755646 x + 6742775466025383507499 x 41 40 + 2152086306334550664288 x + 686079272575391429872 x 39 38 + 218451651134111105880 x + 69466169048006439670 x 37 36 + 22059539785736846082 x + 6995036331361657500 x 35 34 + 2214706608281572626 x + 700057893116933295 x 33 32 + 220901260795257104 x + 69576006986962212 x 31 30 + 21870741732505248 x + 6860412695235518 x 29 28 + 2147105001535030 x + 670345766842528 x 27 26 + 208739559416878 x + 64815504118695 x 25 24 23 + 20064013907288 x + 6190199896600 x + 1902860189328 x 22 21 20 + 582603630260 x + 177594715034 x + 53873727516 x 19 18 17 + 16254780970 x + 4874940295 x + 1452174984 x 16 15 14 13 + 429284612 x + 125801144 x + 36498195 x + 10466610 x 12 11 10 9 8 + 2960828 x + 824074 x + 224917 x + 59920 x + 15492 x 7 6 5 4 3 2 + 3848 x + 910 x + 198 x + 40 x + 6 x + x ---------------------------- 4. the generating function for the sum of the cubes of the areas under Motzkin paths of lengths n is // (1/2)\ / 1 || / 2 \ | | ---------------------- \\x - 1 + \-3 x - 2 x + 1/ / \27 4 / 2 \ 2 4 \3 x + 2 x - 1/ x (1/2) / 2 \ 6 7 \-3 x - 2 x + 1/ x + 27 x (1/2) / 2 \ 5 6 - 108 \-3 x - 2 x + 1/ x - 171 x (1/2) / 2 \ 4 5 + 135 \-3 x - 2 x + 1/ x + 375 x (1/2) / 2 \ 3 4 + 46 \-3 x - 2 x + 1/ x - 173 x (1/2) / 2 \ 2 3 + 9 \-3 x - 2 x + 1/ x - 49 x (1/2) (1/2) / 2 \ 2 / 2 \ - 6 \-3 x - 2 x + 1/ x - 15 x + \-3 x - 2 x + 1/ \\ || + 7 x - 1// and the enumerating function for the sum of the cubes of the areas under Motzkin of lengths 0 to 50 is 50 10159453220575359244635170029 x 49 + 3164341008056876440102961096 x 48 + 984141470034121035635375800 x 47 + 305608376842641316269372392 x 46 + 94749463374816168635591461 x 45 + 29326507795689017147407526 x 44 + 9061125287212010512044658 x 43 42 + 2794510973326218127281734 x + 860186204170912406306407 x 41 40 + 264241042312575792388148 x + 80999849735798650018276 x 39 38 + 24773979088052812472860 x + 7559316334705628100592 x 37 36 + 2300847014928020195754 x + 698473460700506628174 x 35 34 + 211447297315167823466 x + 63822169551016190651 x 33 32 + 19203420739928004368 x + 5758854242937109192 x 31 30 + 1720871045129766512 x + 512282966204517672 x 29 28 + 151881332985873566 x + 44833505143238762 x 27 26 + 13172287164643534 x + 3850521166068067 x 25 24 + 1119432296516028 x + 323514376584988 x 23 22 21 + 92891831844644 x + 26484322064854 x + 7492542382034 x 20 19 18 + 2101624362838 x + 583940294930 x + 160547844283 x 17 16 15 + 43623049688 x + 11696423536 x + 3089123960 x 14 13 12 11 + 801892081 x + 204046142 x + 50724954 x + 12266918 x 10 9 8 7 6 + 2870189 x + 644836 x + 137836 x + 27564 x + 5092 x 5 4 3 2 + 818 x + 118 x + 10 x + x ---------------------------- ---------------------------- The k-th entry of the following list is the average area under Motzkin paths of length k 0., 0.5000000000, 1., 1.777777778, 2.666666667, 3.725490196, 4.913385827, 6.235294118, 7.679041916, 9.240859232, 10.91479821, 12.69666688, 14.58233537, 16.56825422, 18.65115980, 20.82810700, 23.09639555, 25.45354739, 27.89727442, 30.42545670, 33.03612238, 35.72743145, 38.49766161, 41.34519628, 44.26851420, 47.26618046, 50.33683859, 53.47920372, 56.69205646, 59.97423755, 63.32464307, 66.74222018, 70.22596328, 73.77491062, 77.38814118, 81.06477185, 84.80395498, 88.60487598, 92.46675127, 96.38882637, 100.3703741, 104.4106931, 108.5091061, 112.6649588, 116.8776184, 121.1464729, 125.4709292, 129.8504128, 134.2843667, 138.7722505, 143.3135393, 147.9077235, 152.5543078, 157.2528104, 162.0027626, 166.8037083, 171.6552031, 176.5568142, 181.5081196, 186.5087080, 191.5581777, 196.6561372, 201.8022040, 206.9960043, 212.2371733, 217.5253542, 222.8601981, 228.2413638, 233.6685177, 239.1413329, 244.6594896, 250.2226748, 255.8305815, 261.4829091, 267.1793631, 272.9196546, 278.7035003, 284.5306224, 290.4007484, 296.3136108, 302.2689469, 308.2664991, 314.3060141, 320.3872434, 326.5099426, 332.6738718, 338.8787949, 345.1244802, 351.4106995, 357.7372287, 364.1038472, 370.5103380, 376.9564876, 383.4420860, 389.9669263, 396.5308050, 403.1335216, 409.7748787, 416.4546820, 423.1727400, 429.9288639, 436.7228679, 443.5545688, 450.4237860, 457.3303415, 464.2740598, 471.2547679, 478.2722951, 485.3264731, 492.4171359, 499.5441197, 506.7072630, 513.9064062, 521.1413920, 528.4120651, 535.7182721, 543.0598618, 550.4366845, 557.8485929, 565.2954411, 572.7770852, 580.2933831, 587.8441943, 595.4293801, 603.0488035, 610.7023288, 618.3898224, 626.1111519, 633.8661865, 641.6547970, 649.4768556, 657.3322360, 665.2208132, 673.1424638, 681.0970655, 689.0844977, 697.1046407, 705.1573764, 713.2425880, 721.3601596, 729.5099769, 737.6919266, 745.9058968, 754.1517764, 762.4294557, 770.7388262, 779.0797802, 787.4522115, 795.8560146, 804.2910852, 812.7573200, 821.2546168, 829.7828743, 838.3419923, 846.9318714, 855.5524133, 864.2035205, 872.8850966, 881.5970461, 890.3392741, 899.1116869, 907.9141915, 916.7466959, 925.6091088, 934.5013397, 943.4232991, 952.3748982, 961.3560489, 970.3666640, 979.4066571, 988.4759424, 997.5744349, 1006.702051, 1015.858706, 1025.044318, 1034.258804, 1043.502084, 1052.774076, 1062.074701, 1071.403879, 1080.761531, 1090.147580, 1099.561948, 1109.004558, 1118.475334, 1127.974200, 1137.501082, 1147.055905, 1156.638595, 1166.249079, 1175.887285, 1185.553141, 1195.246574, 1204.967515, 1214.715892, 1224.491636, 1234.294677, 1244.124947, 1253.982377, 1263.866900, 1273.778449, 1283.716956, 1293.682355, 1303.674581, 1313.693568, 1323.739251, 1333.811567, 1343.910450, 1354.035838, 1364.187668, 1374.365877, 1384.570403, 1394.801184, 1405.058159, 1415.341268, 1425.650450, 1435.985644, 1446.346792, 1456.733834, 1467.146712, 1477.585366, 1488.049740, 1498.539776, 1509.055416, 1519.596603, 1530.163282, 1540.755395, 1551.372889, 1562.015706, 1572.683793, 1583.377094, 1594.095555, 1604.839123, 1615.607744, 1626.401364, 1637.219931, 1648.063392, 1658.931695, 1669.824788, 1680.742620, 1691.685139, 1702.652295, 1713.644036, 1724.660312, 1735.701074, 1746.766272, 1757.855856, 1768.969777, 1780.107987, 1791.270437 The variances are: 0., 0.2500000000, 0.5000000000, 1.283950617, 2.317460317, 3.963860054, 6.157852316, 9.083955564, 12.79279429, 17.40222449, 22.99791485, 29.68034412, 37.54337626, 46.68374888, 57.19697414, 69.17907481, 82.72586253, 97.93323386, 114.8970507, 133.7131884, 154.4775167, 177.2859071, 202.2342303, 229.4183574, 258.9341589, 290.8775056, 325.3442682, 362.4303172, 402.2315231, 444.8437566, 490.3628881, 538.8847880, 590.5053270, 645.3203753, 703.4258034, 764.9174819, 829.8912810, 898.4430712, 970.6687229, 1046.664106, 1126.525092, 1210.347551, 1298.227352, 1390.260367, 1486.542466, 1587.169519, 1692.237397, 1801.841969, 1916.079107, 2035.044680, 2158.834559, 2287.544615, 2421.270717, 2560.108736, 2704.154543, 2853.504008, 3008.253000, 3168.497391, 3334.333051, 3505.855850, 3683.161659, 3866.346347, 4055.505786, 4250.735845, 4452.132395, 4659.791307, 4873.808450, 5094.279694, 5321.300912, 5554.967972, 5795.376744, 6042.623100, 6296.802910, 6558.012044, 6826.346372, 7101.901765, 7384.774093, 7675.059226, 7972.853036, 8278.251391, 8591.350162, 8912.245220, 9241.032436, 9577.807679, 9922.666819, 10275.70573, 10637.02028, 11006.70633, 11384.85977, 11771.57645, 12166.95226, 12571.08305, 12984.06471, 13405.99309, 13836.96408, 14277.07354, 14726.41734, 15185.09135, 15653.19145, 16130.81350, 16618.05337, 17115.00693, 17621.77006, 18138.43863, 18665.10849, 19201.87554, 19748.83562, 20306.08463, 20873.71842, 21451.83287, 22040.52384, 22639.88721, 23250.01885, 23871.01462, 24502.97040, 25145.98206, 25800.14547, 26465.55650, 27142.31102, 27830.50490, 28530.23401, 29241.59422, 29964.68140, 30699.59142, 31446.42015, 32205.26346, 32976.21723, 33759.37732, 34554.83960, 35362.69994, 36183.05422, 37015.99830, 37861.62806, 38720.03936, 39591.32807, 40475.59007, 41372.92123, 42283.41741, 43207.17449, 44144.28833, 45094.85481, 46058.96980, 47036.72917, 48028.22878, 49033.56451, 50052.83223, 51086.12781, 52133.54712, 53195.18603, 54271.14041, 55361.50613, 56466.37905, 57585.85506, 58720.03003, 59868.99981, 61032.86028, 62211.70732, 63405.63679, 64614.74456, 65839.12651, 67078.87850, 68334.09640, 69604.87609, 70891.31343, 72193.50430, 73511.54457, 74845.53010, 76195.55676, 77561.72043, 78944.11698, 80342.84228, 81757.99219, 83189.66259, 84637.94935, 86102.94833, 87584.75542, 89083.46647, 90599.17737, 92131.98397, 93681.98216, 95249.26779, 5 5 96833.93675, 98436.08489, 1.000558081 10 , 1.016932022 10 , 5 5 5 1.033483632 10 , 1.050213868 10 , 1.067123690 10 , 5 5 5 1.084214055 10 , 1.101485924 10 , 1.118940254 10 , 5 5 5 1.136578004 10 , 1.154400133 10 , 1.172407600 10 , 5 5 5 1.190601364 10 , 1.208982382 10 , 1.227551614 10 , 5 5 5 1.246310019 10 , 1.265258554 10 , 1.284398180 10 , 5 5 5 1.303729854 10 , 1.323254536 10 , 1.342973183 10 , 5 5 5 1.362886755 10 , 1.382996211 10 , 1.403302508 10 , 5 5 5 1.423806606 10 , 1.444509464 10 , 1.465412040 10 , 5 5 5 1.486515293 10 , 1.507820182 10 , 1.529327665 10 , 5 5 5 1.551038701 10 , 1.572954248 10 , 1.595075266 10 , 5 5 5 1.617402713 10 , 1.639937548 10 , 1.662680730 10 , 5 5 5 1.685633217 10 , 1.708795967 10 , 1.732169941 10 , 5 5 5 1.755756095 10 , 1.779555390 10 , 1.803568784 10 , 5 5 5 1.827797234 10 , 1.852241702 10 , 1.876903144 10 , 5 5 5 1.901782519 10 , 1.926880787 10 , 1.952198906 10 , 5 5 5 1.977737834 10 , 2.003498531 10 , 2.029481955 10 , 5 5 5 2.055689064 10 , 2.082120818 10 , 2.108778176 10 , 5 5 5 2.135662095 10 , 2.162773535 10 , 2.190113454 10 , 5 5 5 2.217682811 10 , 2.245482565 10 , 2.273513674 10 , 5 5 5 2.301777098 10 , 2.330273794 10 , 2.359004722 10 , 5 5 5 2.387970839 10 , 2.417173106 10 , 2.446612481 10 , 5 5 2.476289922 10 , 2.506206388 10