A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calculation of the Taylor coefficients of the function with, 2, variables , x[1], x[2] 3 2 2 3 exp(-8 x[1] + 21 x[1] x[2] - 56 x[1] x[2] - 95 x[2] ) By Shalosh B. Ekhad m[1] m[2] Theorem: Let , F(m[1], m[2]), be the coefficient of, x[1] x[2] , in \ the Taylor expansion (around the origing) of the multivariable function \ (that may also be viewed as a formal power series) of, x[1], x[2] 3 2 2 3 exp(-8 x[1] + 21 x[1] x[2] - 56 x[1] x[2] - 95 x[2] ) The following PURE recurrence relations hold in each of the, 2, discrete variables, m[1], m[2] 3 2 F(m[1], m[2]) = -8/81225 (3875244660511 m[1] - 1529990456834 m[1] m[2] 2 3 2 + 175142627072 m[1] m[2] - 6190815232 m[2] - 34037460950809 m[1] 2 + 9300285044554 m[1] m[2] - 565122146176 m[2] + 88392357256131 m[1] - 12401946502218 m[2] - 66338907840459) F(m[1] - 3, m[2])/(m[1] 2 (-1 + m[1]) (-2 + m[1]) %1) - 4/27075 (77217009905609 m[1] 2 - 16789239540402 m[1] m[2] + 839214752320 m[2] - 516938176949231 m[1] + 56270993104690 m[2] + 596226863144550) F(m[1] - 6, m[2])/(m[1] (-1 + m[1]) (-2 + m[1]) %1) 259203488 (3621037 m[1] - 5883613 - 282016 m[2]) F(m[1] - 9, m[2]) - --------- -------------------------------------------------------- 9025 m[1] (-1 + m[1]) (-2 + m[1]) %1 %1 := 3621037 m[1] - 16746724 - 282016 m[2] 3 2 F(m[1], m[2]) = 3/64 (7944909 m[1] + 133424599 m[1] m[2] 2 3 2 - 251336120 m[1] m[2] - 1344272020 m[2] - 278200783 m[1] 2 + 1896437193 m[1] m[2] + 9509770090 m[2] - 3699327674 m[1] - 18729823346 m[2] + 6874570168) F(m[1], m[2] - 3)/(m[2] (m[2] - 1) 2 (m[2] - 2) (23163 m[1] + 58093 m[2] - 172444)) + 1/96 (243697598718 m[1] 2 - 270843710186 m[1] m[2] - 2212162153124 m[2] + 4115039186804 m[1] + 11296532394689 m[2] - 1864081457040) F(m[1], m[2] - 6)/(m[2] (m[2] - 1) (m[2] - 2) (23163 m[1] + 58093 m[2] - 172444)) 769510355 (23163 m[1] + 58093 m[2] + 1835) F(m[1], m[2] - 9) - --------- ------------------------------------------------------------- 16 m[2] (m[2] - 1) (m[2] - 2) (23163 m[1] + 58093 m[2] - 172444) and in Maple notation F(m[1],m[2]) = -8/81225*(3875244660511*m[1]^3-1529990456834*m[1]^2*m[2]+ 175142627072*m[1]*m[2]^2-6190815232*m[2]^3-34037460950809*m[1]^2+9300285044554* m[1]*m[2]-565122146176*m[2]^2+88392357256131*m[1]-12401946502218*m[2]-\ 66338907840459)/m[1]/(-1+m[1])/(-2+m[1])/(3621037*m[1]-16746724-282016*m[2])*F( m[1]-3,m[2])-4/27075*(77217009905609*m[1]^2-16789239540402*m[1]*m[2]+ 839214752320*m[2]^2-516938176949231*m[1]+56270993104690*m[2]+596226863144550)/m [1]/(-1+m[1])/(-2+m[1])/(3621037*m[1]-16746724-282016*m[2])*F(m[1]-6,m[2])-\ 259203488/9025*(3621037*m[1]-5883613-282016*m[2])/m[1]/(-1+m[1])/(-2+m[1])/( 3621037*m[1]-16746724-282016*m[2])*F(m[1]-9,m[2]) F(m[1],m[2]) = 3/64*(7944909*m[1]^3+133424599*m[1]^2*m[2]-251336120*m[1]*m[2]^2 -1344272020*m[2]^3-278200783*m[1]^2+1896437193*m[1]*m[2]+9509770090*m[2]^2-\ 3699327674*m[1]-18729823346*m[2]+6874570168)/m[2]/(m[2]-1)/(m[2]-2)/(23163*m[1] +58093*m[2]-172444)*F(m[1],m[2]-3)+1/96*(243697598718*m[1]^2-270843710186*m[1]* m[2]-2212162153124*m[2]^2+4115039186804*m[1]+11296532394689*m[2]-1864081457040) /m[2]/(m[2]-1)/(m[2]-2)/(23163*m[1]+58093*m[2]-172444)*F(m[1],m[2]-6)-769510355 /16*(23163*m[1]+58093*m[2]+1835)/m[2]/(m[2]-1)/(m[2]-2)/(23163*m[1]+58093*m[2]-\ 172444)*F(m[1],m[2]-9) Subject to the initial conditions F(0, 0) = 1, F(0, 1) = 0, F(0, 2) = 0, F(0, 3) = -95, F(0, 4) = 0, F(0, 5) = 0, F(0, 6) = 9025/2, F(0, 7) = 0, F(0, 8) = 0, F(1, 0) = 0, F(1, 1) = 0, F(1, 2) = -56, F(1, 3) = 0, F(1, 4) = 0, F(1, 5) = 5320, F(1, 6) = 0, F(1, 7) = 0, F(1, 8) = -252700, F(2, 0) = 0, F(2, 1) = 21, F(2, 2) = 0, F(2, 3) = 0, F(2, 4) = -427, F(2, 5) = 0, F(2, 6) = 0, F(2, 7) = -108395/2, F(2, 8) = 0, F(3, 0) = -8, F(3, 1) = 0, F(3, 2) = 0, F(3, 3) = -416, F(3, 4) = 0, F(3, 5) = 0, F(3, 6) = 139052/3, F(3, 7) = 0, F(3, 8) = 0, F(4, 0) = 0, F(4, 1) = 0, F(4, 2) = 1337/2, F(4, 3) = 0, F(4, 4) = 0, 3578603 F(4, 5) = -61159/2, F(4, 6) = 0, F(4, 7) = 0, F(4, 8) = -------, 12 F(5, 0) = 0, F(5, 1) = -168, F(5, 2) = 0, F(5, 3) = 0, F(5, 4) = -8932, F(5, 5) = 0, F(5, 6) = 0, F(5, 7) = 991984, F(5, 8) = 0, F(6, 0) = 32, F(6, 1) = 0, F(6, 2) = 0, F(6, 3) = 15823/2, F(6, 4) = 0, F(6, 5) = 0, F(6, 6) = -1896563/6, F(6, 7) = 0, F(6, 8) = 0, F(7, 0) = 0, F(7, 1) = 0, F(7, 2) = -3556, F(7, 3) = 0, F(7, 4) = 0, F(7, 5) = -12040, F(7, 6) = 0, F(7, 7) = 0, F(7, 8) = 22374590/3, F(8, 0) = 0, F(8, 1) = 672, F(8, 2) = 0, F(8, 3) = 0, F(8, 4) = 745787/8, F(8, 5) = 0, F(8, 6) = 0, F(8, 7) = -36409317/8, F(8, 8) = 0 -------------------------------------------- By the way, it took, 1.035, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000, 2000) = -10846563283160044429938460593559211629913436952203959304511\ 315252659114194395813755355629921090542774282561590034293480900358676126\ 269296062360899722939181234810301263682225980273350265108722714289062010\ 269853915454589269286175537810099777395657328713730775866126848184858692\ 387395393522420843962312224647623145466294766186908120690683342877162557\ 081692847157882227589818607008528075680487285155931094861499204022267105\ 920811716225368308045826396980859122735142985392406959743557909101304697\ 493735471362661573401125365223680955730122277188733669441071232396740537\ 303036372537893214851563094792139494055122795210731936056330389609477262\ 062152181929890391736115518741790657893897969043150344874391666974416102\ 446030199867634421824991959646244918340985649564051003522465197075547131\ 804824371567507586949747006472538830051421343282055837678803758376684931\ 065445234400338799358197453480175946247228860787227110088338650324055269\ 818283935221311849978505850245744873360230883103595685721200678043770203\ 600500127442195900702583503043536174873821744850979839076047755641163220\ 268171330757858720548420909610453587468979663496489504189027391770878742\ 376765051094389298021645902457742262900430142234099839145781710110411166\ 051885394301435206250762402202047895171111489920735808648990161885370055\ 708959541062447022980595736869409323724958145058222181020387668268234998\ 926278624942384339899176009392885421621088819650454791531633674605386946\ 221138824519223885574031776595318289543269914213586609969127503092571702\ 119297121206504858525337377939484290510876354440230514189894149810957836\ 088954848032667550810666111512313369946517274108306963519919437873830926\ 972644311688692294211894729873980096798967398137679101006693123603846058\ 072842104926251891477546651194206251436103500403275085012799633571777359\ 198792006132954759492308626968726885256972311402780741814888502253692091\ 755590018258046463399939964364740434170412619226065568537476972877867353\ 468841531218513867771894739132469934138408544675929070481963757745141681\ / 8524003848119 / 796970584193762227357123910514252411822186874545876046\ / 836972189183393238208718271865976772418417663836671291929817618001789807\ 946576737702222485085383810905607143855540693053458184710766551393791552\ 133411623930712215932513818773770079933480650463246007878627356696205430\ 661326383702353846831777262003725301519433181155155091987178474528531438\ 383869980103524978517216707156744493514301299317663850335799558165618760\ 711368020024667574573270804453549908345919925278080888125234300347856574\ 115071779597300456008716253163630165754427861729685615081176080993552666\ 626656674693869888873171465426543806413897173560154918448125229489182256\ 078527367401203973223812850750969001331680046395921070825318301872507539\ 835079067414543165556493918778989231414086681169197024635841077124492426\ 998608980158775163916460723412740250240529709058815980556593285261829371\ 077311762692278707175017459649368903550135145361866998383604811975796226\ 751853784097955838247456940425707698604356654090873252038168042899845163\ 833683523118634880971461739991399343451392133110993472227510001849815641\ 762350408038016104641173191928444182626272781503493799110003672601906506\ 214638247565749345898221223688780063665238292998196084868630444263583227\ 556088349919095251862216258922967271089939599391333924336280933040631318\ 381326703258149119950498260946120948220637667522404333127791965490248186\ 447953624639615619177369200019076920372484928142559963103602627147162956\ 689322431864384549364016603863382448329551948509559539199736349712480270\ 126238363968822929643436412518995240463428382267384763526575688301273240\ 882775411196391627112742867532843154605182513792673518856283130355191762\ 542716342467835474829450465485497814084290400436063122870971050874548805\ 886792250366039520475342072468728668569173932693205896634399669726290089\ 045512161948791076523459569128770395056192014048485602281169080305919295\ 324040297160875148427981765421232333308273631292044332003818002697889234\ 768436606942960832395737514794857956469974650500466445004703944173433466\ 603572214362605860924995970957076744928804408372216902268920585947905804\ 093114047169766211172442346532497190004508772619270165160985409048481327\ 858886151117060656366471395837457611259060761371812018838884452631317707\ 331468718578396505102232927914905704463604195122607965290707212978818786\ 652962162179494270921080832000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000 This took, 1.956, seconds. --------------------------- A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 2, variables , x[1], x[2] 3 2 2 3 exp(-83 x[1] + 70 x[1] x[2] + 15 x[1] x[2] + 82 x[2] ) By Shalosh B. Ekhad m[1] m[2] Theorem: Let , F(m[1], m[2]), be the coefficient of, x[1] x[2] , in \ the Taylor expansion (around the origing) of the multivariable function \ (that may also be viewed as a formal power series) of, x[1], x[2] 3 2 2 3 exp(-83 x[1] + 70 x[1] x[2] + 15 x[1] x[2] + 82 x[2] ) The following PURE recurrence relations hold in each of the, 2, discrete variables, m[1], m[2] 3 2 F(m[1], m[2]) = -3/6724 (28377324465560 m[1] - 708704238860 m[1] m[2] 2 3 2 - 9411282875 m[1] m[2] - 16173750 m[2] - 221799814495334 m[1] 2 + 3165312988125 m[1] m[2] + 22493879375 m[2] + 503827497476940 m[1] - 2225696119775 m[2] - 296102822232310) F(m[1] - 3, m[2])/(m[1] 11 2 (-1 + m[1]) (-2 + m[1]) %1) - ----- (3068629701940180 m[1] 10086 2 - 236793879948830 m[1] m[2] - 2060335195500 m[2] - 17927515004882167 m[1] + 90824163499295 m[2] + 13018676616865023) F(m[1] - 6, m[2])/(m[1] (-1 + m[1]) (-2 + m[1]) %1) 30909170369 (16391935 m[1] - 13107964 + 129390 m[2]) F(m[1] - 9, m[2]) - ----------- ---------------------------------------------------------- 1681 m[1] (-1 + m[1]) (-2 + m[1]) %1 %1 := 16391935 m[1] - 62283769 + 129390 m[2] 3 2 F(m[1], m[2]) = 2/62001 (163111935000 m[1] - 1331929196500 m[1] m[2] 2 3 2 + 22587636127210 m[1] m[2] + 9442189741010 m[2] + 2215677957500 m[1] 2 - 105245516256650 m[1] m[2] - 143726489429563 m[2] + 131080005286650 m[1] + 518165285593410 m[2] - 564346813808295) F(m[1], m[2] - 3)/(m[2] 44 2 (m[2] - 1) (m[2] - 2) %1) + ----- (5116245991500 m[1] 20667 2 - 89393762179510 m[1] m[2] - 37299266652310 m[2] + 335098371991640 m[1] + 493077768554633 m[2] - 1264169695356867) F(m[1], m[2] - 6)/(m[2] (m[2] - 1) (m[2] - 2) %1) 122147082904 (951090 m[1] + 387790 m[2] - 3176597) F(m[1], m[2] - 9) + ------------ ------------------------------------------------------- 6889 m[2] (m[2] - 1) (m[2] - 2) %1 %1 := 951090 m[1] + 387790 m[2] - 4339967 and in Maple notation F(m[1],m[2]) = -3/6724*(28377324465560*m[1]^3-708704238860*m[1]^2*m[2]-\ 9411282875*m[1]*m[2]^2-16173750*m[2]^3-221799814495334*m[1]^2+3165312988125*m[1 ]*m[2]+22493879375*m[2]^2+503827497476940*m[1]-2225696119775*m[2]-\ 296102822232310)/m[1]/(-1+m[1])/(-2+m[1])/(16391935*m[1]-62283769+129390*m[2])* F(m[1]-3,m[2])-11/10086*(3068629701940180*m[1]^2-236793879948830*m[1]*m[2]-\ 2060335195500*m[2]^2-17927515004882167*m[1]+90824163499295*m[2]+ 13018676616865023)/m[1]/(-1+m[1])/(-2+m[1])/(16391935*m[1]-62283769+129390*m[2] )*F(m[1]-6,m[2])-30909170369/1681*(16391935*m[1]-13107964+129390*m[2])/m[1]/(-1 +m[1])/(-2+m[1])/(16391935*m[1]-62283769+129390*m[2])*F(m[1]-9,m[2]) F(m[1],m[2]) = 2/62001*(163111935000*m[1]^3-1331929196500*m[1]^2*m[2]+ 22587636127210*m[1]*m[2]^2+9442189741010*m[2]^3+2215677957500*m[1]^2-\ 105245516256650*m[1]*m[2]-143726489429563*m[2]^2+131080005286650*m[1]+ 518165285593410*m[2]-564346813808295)/m[2]/(m[2]-1)/(m[2]-2)/(951090*m[1]+ 387790*m[2]-4339967)*F(m[1],m[2]-3)+44/20667*(5116245991500*m[1]^2-\ 89393762179510*m[1]*m[2]-37299266652310*m[2]^2+335098371991640*m[1]+ 493077768554633*m[2]-1264169695356867)/m[2]/(m[2]-1)/(m[2]-2)/(951090*m[1]+ 387790*m[2]-4339967)*F(m[1],m[2]-6)+122147082904/6889*(951090*m[1]+387790*m[2]-\ 3176597)/m[2]/(m[2]-1)/(m[2]-2)/(951090*m[1]+387790*m[2]-4339967)*F(m[1],m[2]-9 ) Subject to the initial conditions F(0, 0) = 1, F(0, 1) = 0, F(0, 2) = 0, F(0, 3) = 82, F(0, 4) = 0, F(0, 5) = 0, F(0, 6) = 3362, F(0, 7) = 0, F(0, 8) = 0, F(1, 0) = 0, F(1, 1) = 0, F(1, 2) = 15, F(1, 3) = 0, F(1, 4) = 0, F(1, 5) = 1230, F(1, 6) = 0, F(1, 7) = 0, F(1, 8) = 50430, F(2, 0) = 0, F(2, 1) = 70, F(2, 2) = 0, F(2, 3) = 0, F(2, 4) = 11705/2, F(2, 5) = 0, F(2, 6) = 0, F(2, 7) = 244565, F(2, 8) = 0, F(3, 0) = -83, F(3, 1) = 0, F(3, 2) = 0, F(3, 3) = -5756, F(3, 4) = 0, F(3, 5) = 0, F(3, 6) = -384767/2, F(3, 7) = 0, F(3, 8) = 0, F(4, 0) = 0, F(4, 1) = 0, F(4, 2) = 1205, F(4, 3) = 0, F(4, 4) = 0, F(4, 5) = 106685, F(4, 6) = 0, F(4, 7) = 0, F(4, 8) = 37592555/8, F(5, 0) = 0, F(5, 1) = -5810, F(5, 2) = 0, F(5, 3) = 0, F(5, 4) = -898015/2, F(5, 5) = 0, F(5, 6) = 0, F(5, 7) = -17246020, F(5, 8) = 0, F(6, 0) = 6889/2, F(6, 1) = 0, F(6, 2) = 0, F(6, 3) = 757397/3, F(6, 4) = 0, F(6, 5) = 0, F(6, 6) = 56104279/6, F(6, 7) = 0, F(6, 8) = 0, F(7, 0) = 0, F(7, 1) = 0, F(7, 2) = -303365/2, F(7, 3) = 0, F(7, 4) = 0, F(7, 5) = -12234090, F(7, 6) = 0, F(7, 7) = 0, F(7, 8) = -3936286145/8, F(8, 0) = 0, F(8, 1) = 241115, F(8, 2) = 0, F(8, 3) = 0, 217309235 F(8, 4) = ---------, F(8, 5) = 0, F(8, 6) = 0, F(8, 7) = 4064885605/6, 12 F(8, 8) = 0 -------------------------------------------- By the way, it took, 0.984, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000, 2000) = 115866900076342719783872241575516438244428368214076017135219\ 472727650921991082534650816806669212095705624509783743492377636610730192\ 138708006480988541582405199525729455249488264393095756159442715463626906\ 190815649561812486830423236113336523727332184748229031311417764784923714\ 506963985578029307243951901097272915551744602041350004200991050616401956\ 960543727831086052330371719168338653126151112607797472481256432345476564\ 042379891913491953586162053025017036645109473240141478998480563499308697\ 267137468446059654832299848016570653349365046474045665370160134261359162\ 670115135439576530984892626421423720329487760317648361033232315568449836\ 351666290593170237621837008605005937603442799587662212196944235332163001\ 147337384194835421341527537284909862288884178295674645635388205323154871\ 527844423490755879019014761468486625559722029474015536400275224503357540\ 032294818599645185808839414711776170107061870971187889420701001795594822\ 947333349568985103188019819241696229297307341242437929560926646504332846\ 632544185186139012096603677363259807024896571565351877020605161589814613\ 772123330840585337321696338350435351735169338895214120981490244006117025\ 731030995341677879353456434424657964820820224336557310303284909937630605\ 195715773041943326929223157627732910009089276340690521848208581470928643\ 130594775848614791919434467947235756213005722983950066342523665994051147\ 946943620468502804254472735113875916370734288308492676023241058591304268\ 879586397565980021159272254721808664146112121693259431580983276043947881\ 322243113865550602794618625732924105394040482443870807189815376405319286\ 333731590732652200886569432046378860603559901597027216563345827033705837\ 123164683042058847233240233196461597652584252868981096295826141510030151\ 017888826590770069392867818093495959854927880428752681372054140807008847\ 569034270769329036203027010042229492023541255145795908969205184145050202\ 820146683456990157651797068327938322155575410342379088888700040843525660\ 183903305921138139274543568804844510936692062449563977664744698633102668\ 435610396705051902835876939731404996754340575088063349296879566313683057\ 129284318667593752947050723190086439936587874409821258824235067730046086\ 984172511612742596577587962515140485257628066460774152294677544344078751\ / 8038199068710531687646626761527840786619313488861 / 255335299290728521\ / 871325744546858413394721454730479909838308365935868192283975321661353735\ 880087663099130851113951557862126253846966532837584899271614680619618354\ 265047826426022596345238275578338276734486487373339847255018947044896047\ 933432193465607626902465962115393282943907833416001601064550940173775641\ 668219532984138445119540059859618198980974015243749721855105958861366961\ 233534517542950825850191637581426163847510368892571845812128083491854326\ 251846205674845871558790992154999387632774558721793988698549156326383684\ 253741698378526705782724001581547845288151475521010769430425949655878575\ 164963933223768147991967428510434032652224190408037005500120965060473377\ 877458262485522331357791124295146482840479983848409864680526672437099733\ 456858048599182324413980494888829228181408086254985367386167348555182825\ 450750339257294643408557995135006003832737594137234535499151828389523069\ 455906552077547902947756196941271488891173490797500812774741916923769633\ 747332157706833502069289202843991934494102114049323693134178484587773995\ 427754578767224923728983889965130565195860847818019098909442681984191308\ 359115432436173785763145186402944170040054049841117069429273182945064344\ 578966094061464304025282908100657827406504835188044225221283429740262581\ 179655453564767278664048207500442816246762505040616496024397565309978295\ 735664532303590614507565957640373268064750926212245077797437877566399936\ 088887589188190179979559736480532219169138615881390235840549453415714805\ 697574324652768411244003268805819544418841611073548861644682649353325964\ 142656223097393424015786102572876539727909153292314874158373511061219766\ 284420536164917708399412796919270593709185958632692869330615418904903753\ 097261727494966694607619126162111539929765402225537785132142061994294322\ 968903847811107142754878977856581232685736584805826199849411980537343511\ 882676370568723405562742236150247156761829299416325038030984813989890239\ 935471175644901691884119884754519472064971349188015454608011477748454084\ 161550751219878528542234341988924661466888619329384584976552153175862997\ 520784382460686085018080337374890812508271768252855667485974255765539486\ 365111016100800877159170169006947529475300282007277994527518039828754901\ 006210096110625218122500200304275310735237975901460124506064816107629013\ 212113442804217695565706166698391211784735175919586146412134400000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000 This took, 1.957, seconds. --------------------------- A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 2, variables , x[1], x[2] 3 2 2 3 exp(-86 x[1] + 53 x[1] x[2] - x[1] x[2] + 17 x[2] ) By Shalosh B. Ekhad m[1] m[2] Theorem: Let , F(m[1], m[2]), be the coefficient of, x[1] x[2] , in \ the Taylor expansion (around the origing) of the multivariable function \ (that may also be viewed as a formal power series) of, x[1], x[2] 3 2 2 3 exp(-86 x[1] + 53 x[1] x[2] - x[1] x[2] + 17 x[2] ) The following PURE recurrence relations hold in each of the, 2, discrete variables, m[1], m[2] 3 2 F(m[1], m[2]) = -1/2601 (686915922331520 m[1] + 5701497825128 m[1] m[2] 2 3 2 + 1352345686 m[1] m[2] + 62191 m[2] - 5508191811299996 m[1] 2 - 28760994548260 m[1] m[2] - 3400429262 m[2] + 12884830905690666 m[1] + 29348221307949 m[2] - 8099678588356278) F(m[1] - 3, m[2])/(m[1] 2 (-1 + m[1]) (-2 + m[1]) %1) - 1/867 (62076491389906196 m[1] 2 - 6471869061209247 m[1] m[2] - 1172129324750 m[2] - 377115061528171907 m[1] + 6573831769724464 m[2] + 319195717665105630) F(m[1] - 6, m[2])/(m[1] (-1 + m[1]) (-2 + m[1]) %1) 336095138 (343980812 m[1] - 351819269 + 62191 m[2]) F(m[1] - 9, m[2]) - --------- ----------------------------------------------------------- 17 m[1] (-1 + m[1]) (-2 + m[1]) %1 %1 := 343980812 m[1] - 1383761705 + 62191 m[2] 3 2 F(m[1], m[2]) = 1/66564 (5779043517767 m[1] - 20041562991774 m[1] m[2] 2 3 2 + 415640758603236 m[1] m[2] - 7914092911240 m[2] + 41475122062746 m[1] 2 - 2057029912253580 m[1] m[2] - 267562677378612 m[2] + 2704889115132273 m[1] + 1494003664392426 m[2] - 2065360288009626) 17 F(m[1], m[2] - 3)/(m[2] (m[2] - 1) (m[2] - 2) %1) + ---- ( 7396 2 2 6640056060118 m[1] - 145188780264095 m[1] m[2] + 2764613724750 m[2] + 573353471575910 m[1] + 98999418281015 m[2] - 444476491543092) F(m[1], m[2] - 6)/(m[2] (m[2] - 1) (m[2] - 2) %1) 1129435987 (38817571 m[1] - 739790 m[2] - 30207387) F(m[1], m[2] - 9) + ---------- ---------------------------------------------------------- 7396 m[2] (m[2] - 1) (m[2] - 2) %1 %1 := 38817571 m[1] - 739790 m[2] - 27988017 and in Maple notation F(m[1],m[2]) = -1/2601*(686915922331520*m[1]^3+5701497825128*m[1]^2*m[2]+ 1352345686*m[1]*m[2]^2+62191*m[2]^3-5508191811299996*m[1]^2-28760994548260*m[1] *m[2]-3400429262*m[2]^2+12884830905690666*m[1]+29348221307949*m[2]-\ 8099678588356278)/m[1]/(-1+m[1])/(-2+m[1])/(343980812*m[1]-1383761705+62191*m[2 ])*F(m[1]-3,m[2])-1/867*(62076491389906196*m[1]^2-6471869061209247*m[1]*m[2]-\ 1172129324750*m[2]^2-377115061528171907*m[1]+6573831769724464*m[2]+ 319195717665105630)/m[1]/(-1+m[1])/(-2+m[1])/(343980812*m[1]-1383761705+62191*m [2])*F(m[1]-6,m[2])-336095138/17*(343980812*m[1]-351819269+62191*m[2])/m[1]/(-1 +m[1])/(-2+m[1])/(343980812*m[1]-1383761705+62191*m[2])*F(m[1]-9,m[2]) F(m[1],m[2]) = 1/66564*(5779043517767*m[1]^3-20041562991774*m[1]^2*m[2]+ 415640758603236*m[1]*m[2]^2-7914092911240*m[2]^3+41475122062746*m[1]^2-\ 2057029912253580*m[1]*m[2]-267562677378612*m[2]^2+2704889115132273*m[1]+ 1494003664392426*m[2]-2065360288009626)/m[2]/(m[2]-1)/(m[2]-2)/(38817571*m[1]-\ 739790*m[2]-27988017)*F(m[1],m[2]-3)+17/7396*(6640056060118*m[1]^2-\ 145188780264095*m[1]*m[2]+2764613724750*m[2]^2+573353471575910*m[1]+ 98999418281015*m[2]-444476491543092)/m[2]/(m[2]-1)/(m[2]-2)/(38817571*m[1]-\ 739790*m[2]-27988017)*F(m[1],m[2]-6)+1129435987/7396*(38817571*m[1]-739790*m[2] -30207387)/m[2]/(m[2]-1)/(m[2]-2)/(38817571*m[1]-739790*m[2]-27988017)*F(m[1],m [2]-9) Subject to the initial conditions F(0, 0) = 1, F(0, 1) = 0, F(0, 2) = 0, F(0, 3) = 17, F(0, 4) = 0, F(0, 5) = 0, F(0, 6) = 289/2, F(0, 7) = 0, F(0, 8) = 0, F(1, 0) = 0, F(1, 1) = 0, F(1, 2) = -1, F(1, 3) = 0, F(1, 4) = 0, F(1, 5) = -17, F(1, 6) = 0, F(1, 7) = 0, F(1, 8) = -289/2, F(2, 0) = 0, F(2, 1) = 53, F(2, 2) = 0, F(2, 3) = 0, F(2, 4) = 1803/2, F(2, 5) = 0, F(2, 6) = 0, F(2, 7) = 7667, F(2, 8) = 0, F(3, 0) = -86, F(3, 1) = 0, F(3, 2) = 0, F(3, 3) = -1515, F(3, 4) = 0, F(3, 5) = 0, F(3, 6) = -79969/6, F(3, 7) = 0, F(3, 8) = 0, F(4, 0) = 0, F(4, 1) = 0, F(4, 2) = 2981/2, F(4, 3) = 0, F(4, 4) = 0, 5179867 F(4, 5) = 25365, F(4, 6) = 0, F(4, 7) = 0, F(4, 8) = -------, F(5, 0) = 0, 24 F(5, 1) = -4558, F(5, 2) = 0, F(5, 3) = 0, F(5, 4) = -157867/2, F(5, 5) = 0, F(5, 6) = 0, F(5, 7) = -2049742/3, F(5, 8) = 0, F(6, 0) = 3698, F(6, 1) = 0, F(6, 2) = 0, F(6, 3) = 553421/6, F(6, 4) = 0, F(6, 5) = 0, F(6, 6) = 4137527/4, F(6, 7) = 0, F(6, 8) = 0, F(7, 0) = 0, F(7, 1) = 0, F(7, 2) = -124485, F(7, 3) = 0, F(7, 4) = 0, F(7, 5) = -12860021/6, F(7, 6) = 0, F(7, 7) = 0, F(7, 8) = -55346644/3, F(8, 0) = 0, 90799297 F(8, 1) = 195994, F(8, 2) = 0, F(8, 3) = 0, F(8, 4) = --------, F(8, 5) = 0, 24 864196843 F(8, 6) = 0, F(8, 7) = ---------, F(8, 8) = 0 24 -------------------------------------------- By the way, it took, 1.032, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000, 2000) = 514108999241420773392363629103561307921542407119783032697436\ 429146430450515406019648243602123373070692682113947086275233815935592072\ 127590178271960794163098013140767257678989729151028443045416981088698478\ 684545272421850287095652374278458908743985541849821776422127463834565870\ 772729064039151374607998468951939540141282049307409224180427353228580531\ 633766426662003537634462742028445758895974823640741395563502081281445774\ 639154939658767165651853309572450337593819171326708066491155928566135457\ 437239580928050922194346935110715380498144370683952490861875769656574889\ 112837907293620353956306171287713102310822060251045714093713554219695653\ 028081286375020993353057737387462015104082330954019866855409705718233964\ 550535546452286091271213089497898059425632544258668607341118773508143940\ 245990066634472701673336561673714473063770879989335307042744996446689454\ 787763820209257405910262468374967661213467260662607642543907135193502292\ 118617418829497056754134640009407360855881874148854029380257069600190888\ 978917336590555964505156923582473783516579957946093714494387630198172636\ 910018173571693725184727418289530643848798068388320593662217656471464899\ 653227063001693745558934664268639625255485732865677040660764060570970591\ 111670588776136406215430337041348155704279748354158105394173386513381644\ 396722206487558591652530280980209806079583396291891152526519425059200603\ 127063236457446730617005375777683202336622525961520162244430504256909247\ 185809555618477996978090345808038508547307385350711812734023648916819460\ 129978756219365398551370192174794748615539211220482376987022924039083687\ 720967259062568699072108847039944695045028268168962298340955507676908599\ 861354984723637199376777317502736265054379859275999078137025263176078690\ 193207342649286268829774329983324943605284251542075243461388145273189399\ 137983775318224530500327838826020972807342887356938384901791927285895624\ / 4756749915363731401134868977627 / 365806600070085248676093121748185441\ / 563068058373376938676181736754489556726026382356403825897244913454981807\ 827451996416937878975449825962637567117881767809997658640079157963562566\ 934297444302290708641773632069200795432238633676001289343722405905719142\ 176261307169533942835619132568894225318160920433302627123596123120466532\ 112689511886870371922056743158499406866753315069646970724906062563901097\ 617403503381443863461933516216704145210678305532163213203837805993819934\ 628071179371301266222184801119598737306342546688458378335225099705337589\ 374568616899382515608302225842032846738644345643636601530137297583479979\ 731622396243791058513658321537634801149827466447964768087599935186468149\ 977654463038383242575065528019393805560737776014396860894587917250548091\ 773216014978043389474445163314768131964332139461999001259493170895418139\ 381001566686869638308591272487527954670761468461025238115317303956932000\ 014444122861210556092821789250837737188898199448343740176859324748968674\ 055511720170872930762701341898642069875424678795179602646576438581098327\ 412528081721785925693857883287965617426281618101730685482976929354534550\ 177676740256374715783405447857233256133298457232336635476658321074197464\ 059847306169914652054689803200683196622023218422882257549296494522079030\ 056082634516802354659507570152933143696842518144229711157097341619447143\ 798990592916513043949841812060039425145615454193483509807623772981332686\ 482869592760085443800556489598727546759722297476551222392543145995165975\ 548900076385847671332415677285820747168411337213809655390021033175703179\ 605012689745963322026136158192193951784286748041693070352307482258059717\ 251329024894377181946152100406036599908627474747059913036274441835060478\ 622783454274869267179107145491647172481743919110155663496274219928244440\ 351928959678295418907929916835724381247626038740763324892991930113658351\ 499411949063761298665387863523033835400631164940107745598194657586207286\ 410363358445594075525131426453702865848434807518058729808967564671894096\ 560995456906718057776615633402565680688715241593040200188703300517726204\ 625426298329776226239451736327114800869201319086247001131378601145799117\ 553746262806310935009672892867635870142550856247953634631140674703256728\ 239706731450674350146208391032143710763724542156840209600778094273294792\ 039469917276503354641758167123434490373609116139520000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000 This took, 1.981, seconds. --------------------------- A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 2, variables , x[1], x[2] 3 2 2 3 exp(-96 x[1] + 85 x[1] x[2] - 98 x[1] x[2] - 15 x[2] ) By Shalosh B. Ekhad m[1] m[2] Theorem: Let , F(m[1], m[2]), be the coefficient of, x[1] x[2] , in \ the Taylor expansion (around the origing) of the multivariable function \ (that may also be viewed as a formal power series) of, x[1], x[2] 3 2 2 3 exp(-96 x[1] + 85 x[1] x[2] - 98 x[1] x[2] - 15 x[2] ) The following PURE recurrence relations hold in each of the, 2, discrete variables, m[1], m[2] 3 2 F(m[1], m[2]) = -2/2025 (814392711356978 m[1] - 823429298352486 m[1] m[2] 2 3 + 247904015054640 m[1] m[2] - 23346014379352 m[2] 2 - 7299002056464333 m[1] + 5006894175420606 m[1] m[2] 2 - 773406834413976 m[2] + 19652891596717995 m[1] - 6800549713710858 m[2] - 15448993673263794) F(m[1] - 3, m[2])/(m[1] (-1 + m[1]) (-2 + m[1]) %1) 16 2 - --- (15949982084513347 m[1] - 8555758035603028 m[1] m[2] 675 2 + 1131277205011996 m[2] - 107266291457768355 m[1] + 29470764247524150 m[2] + 124877459330159382) F(m[1] - 6, m[2])/(m[1] (-1 + m[1]) (-2 + m[1]) %1) - 16866628096 (209801267 m[1] - 330476997 - 49609462 m[2]) F(m[1] - 9, m[2]) ----------- -------------------------------------------------------------- 75 m[1] (-1 + m[1]) (-2 + m[1]) %1 %1 := 209801267 m[1] - 959880798 - 49609462 m[2] 3 2 F(m[1], m[2]) = 5/82944 (1485119839594375 m[1] + 36554402604976950 m[1] m[2] 2 3 + 93452820134969172 m[1] m[2] - 289535047716018104 m[2] 2 - 84562939397009250 m[1] - 356388245566283160 m[1] m[2] 2 + 2299354939192918968 m[2] + 102662277968710485 m[1] - 5323120508473993962 m[2] + 3156472279431165870) F(m[1], m[2] - 3)/(m[2] 2 (m[2] - 1) (m[2] - 2) %1) + 1/13824 (4613597517204872650 m[1] 2 + 18929989082708680142 m[1] m[2] - 25596858781251553796 m[2] + 9668208556778715033 m[1] + 157418892445412862672 m[2] - 102788063116439886351) F(m[1], m[2] - 6)/(m[2] (m[2] - 1) (m[2] - 2) %1) 164713165 - --------- 192 (12091348175 m[1] + 62574626114 m[2] - 42772445139) F(m[1], m[2] - 9) --------------------------------------------------------------------- m[2] (m[2] - 1) (m[2] - 2) %1 %1 := 12091348175 m[1] + 62574626114 m[2] - 230496323481 and in Maple notation F(m[1],m[2]) = -2/2025*(814392711356978*m[1]^3-823429298352486*m[1]^2*m[2]+ 247904015054640*m[1]*m[2]^2-23346014379352*m[2]^3-7299002056464333*m[1]^2+ 5006894175420606*m[1]*m[2]-773406834413976*m[2]^2+19652891596717995*m[1]-\ 6800549713710858*m[2]-15448993673263794)/m[1]/(-1+m[1])/(-2+m[1])/(209801267*m[ 1]-959880798-49609462*m[2])*F(m[1]-3,m[2])-16/675*(15949982084513347*m[1]^2-\ 8555758035603028*m[1]*m[2]+1131277205011996*m[2]^2-107266291457768355*m[1]+ 29470764247524150*m[2]+124877459330159382)/m[1]/(-1+m[1])/(-2+m[1])/(209801267* m[1]-959880798-49609462*m[2])*F(m[1]-6,m[2])-16866628096/75*(209801267*m[1]-\ 330476997-49609462*m[2])/m[1]/(-1+m[1])/(-2+m[1])/(209801267*m[1]-959880798-\ 49609462*m[2])*F(m[1]-9,m[2]) F(m[1],m[2]) = 5/82944*(1485119839594375*m[1]^3+36554402604976950*m[1]^2*m[2]+ 93452820134969172*m[1]*m[2]^2-289535047716018104*m[2]^3-84562939397009250*m[1]^ 2-356388245566283160*m[1]*m[2]+2299354939192918968*m[2]^2+102662277968710485*m[ 1]-5323120508473993962*m[2]+3156472279431165870)/m[2]/(m[2]-1)/(m[2]-2)/( 12091348175*m[1]+62574626114*m[2]-230496323481)*F(m[1],m[2]-3)+1/13824*( 4613597517204872650*m[1]^2+18929989082708680142*m[1]*m[2]-25596858781251553796* m[2]^2+9668208556778715033*m[1]+157418892445412862672*m[2]-\ 102788063116439886351)/m[2]/(m[2]-1)/(m[2]-2)/(12091348175*m[1]+62574626114*m[2 ]-230496323481)*F(m[1],m[2]-6)-164713165/192*(12091348175*m[1]+62574626114*m[2] -42772445139)/m[2]/(m[2]-1)/(m[2]-2)/(12091348175*m[1]+62574626114*m[2]-\ 230496323481)*F(m[1],m[2]-9) Subject to the initial conditions F(0, 0) = 1, F(0, 1) = 0, F(0, 2) = 0, F(0, 3) = -15, F(0, 4) = 0, F(0, 5) = 0, F(0, 6) = 225/2, F(0, 7) = 0, F(0, 8) = 0, F(1, 0) = 0, F(1, 1) = 0, F(1, 2) = -98, F(1, 3) = 0, F(1, 4) = 0, F(1, 5) = 1470, F(1, 6) = 0, F(1, 7) = 0, F(1, 8) = -11025, F(2, 0) = 0, F(2, 1) = 85, F(2, 2) = 0, F(2, 3) = 0, F(2, 4) = 3527, F(2, 5) = 0, F(2, 6) = 0, F(2, 7) = -124935/2, F(2, 8) = 0, F(3, 0) = -96, F(3, 1) = 0, F(3, 2) = 0, F(3, 3) = -6890, F(3, 4) = 0, F(3, 5) = 0, F(3, 6) = -128146/3, F(3, 7) = 0, F(3, 8) = 0, F(4, 0) = 0, F(4, 1) = 0, F(4, 2) = 26041/2, F(4, 3) = 0, F(4, 4) = 0, -9774517 F(4, 5) = 425725/2, F(4, 6) = 0, F(4, 7) = 0, F(4, 8) = --------, 12 F(5, 0) = 0, F(5, 1) = -8160, F(5, 2) = 0, F(5, 3) = 0, F(5, 4) = -692617, F(5, 5) = 0, F(5, 6) = 0, F(5, 7) = -6078895/3, F(5, 8) = 0, F(6, 0) = 4608, F(6, 1) = 0, F(6, 2) = 0, F(6, 3) = 4997485/6, F(6, 4) = 0, F(6, 5) = 0, F(6, 6) = 38788369/2, F(6, 7) = 0, F(6, 8) = 0, F(7, 0) = 0, F(7, 1) = 0, F(7, 2) = -798384, F(7, 3) = 0, F(7, 4) = 0, F(7, 5) = -111717805/3, F(7, 6) = 0, F(7, 7) = 0, F(7, 8) = -861648167/3, F(8, 0) = 0, 1257932209 F(8, 1) = 391680, F(8, 2) = 0, F(8, 3) = 0, F(8, 4) = ----------, 24 22590100745 F(8, 5) = 0, F(8, 6) = 0, F(8, 7) = -----------, F(8, 8) = 0 24 -------------------------------------------- By the way, it took, 0.976, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000, 2000) = -51137460997961948159257881937679347869235548479492823148611\ 149061351907479180097755342159189618725767676382447489868074497292475782\ 004009492113195301962292105086650041301945412798103212127787623938618877\ 521413377200766380474301198531613180715687133616996798780347193738317926\ 375913564278269461040536972957806121598997808258114524933029930431468324\ 253349499208872885134403604604939093079585665044820953217236366717084278\ 388371648323178950526119279575078999142536962010615820309203278024068043\ 699054890924053518379043570440096070120601261916023416101309692453935106\ 326879325199135686500562968272240864719314508636903395774785337976386245\ 611143386592665603482388565842461518554613774772876092164149599320490044\ 420290086971284207538190865706308104846016476657239793196643148548230323\ 855590548035122868745408180276482693329321748326072133288988629155448045\ 972319502348727493261190452331854792213866823931672580997039049697962548\ 533498134090034558421181162885791206418779848955915409526267847949694353\ 497586483915667899991383256121259570284506179558823209233466520946743066\ 110695018414053389402529826045848542334384124154210735291079291869485023\ 452816175853331132623958880683026883607277212633808688176421887000025288\ 606875023587616518411448579837989192480797423506663537437903802326405810\ 958841212049319474627795221708468467482768213140627734628722293394507595\ 209704191933184760689445756921417520916120059911701125875903234126274184\ 630116170911646873400287360820642185074291340882443353175510788836610048\ 065492334880631389900710176991662965392928144785990698743664654658126467\ 617179730089706143203151167740220098778454653407356462043464748556150398\ 865842641456335984753534601215753931754702593417579373885529143429739743\ 808338403895137260517517524189890450528060516865400725726663245423193800\ 482234072396959655148022814418216702345616301999431762379774182493341056\ 093215481614409729755060822686400142006539922955172557714136292820638011\ 015003029556246796515735322369707721404066041747360198357054304102917996\ / 127210371801429609921833302444843092681195751223 / 3698412316885053065\ / 659029723557021927567783678407303607939337412039876033075634380441582798\ 593468794122058718843724228480070467582672872784019520493289931902549857\ 574329722058077423048963131732718561635579000699640394995059715474277741\ 526605206767381033032053657044920377933649318251687939797582835263169943\ 345181391842952070992265285620932069064911925315710915013130942799739593\ 726072323715911832528895714334450825442342535279177938710901986078856015\ 847624876775753015541160820849553243055177217201825469089860306592869633\ 903930235991464598456116531151845286480996768623493854894382171029414169\ 881902824832977736257211726729505187178530824369469496919211517396702618\ 532719932951424310435844754983654106917022801666665313573635603086732968\ 603370486908114898442441327903012209760015437925045451845269792354769314\ 204434632214705453300567310253277593193477938752580424061007493631688988\ 298244801035158088381328448045327313438455589852219769373051648834357667\ 229276261248764167840651582609928527931723125813476809402271568701264364\ 992960316536839707295780237995997410507754334712887676184832535954997739\ 258038807812389753634599395974376722074871255909527405759451313929874071\ 914744054978099360899191514585534165994106466183377907024131803907816942\ 134431470351901958287600165151494394535462565884434357229106341531734212\ 066241505343675167085382832795628201698768320459957423347215069970881073\ 422666822421562087602541838567040332542390979398899829372192345810720079\ 336373718701126591027482389930445206485820266676496524680629328448443207\ 068227357474940234977150355887496821955420717016666791936722636450940968\ 635461677241412582504245656101239890999699544546325918193843954244171914\ 888804099435354103722994482558420899788767095963344574723513888062235568\ 436595973539236069734575879100413242635320850361184001584454435798158615\ 646058116957942891997731102666704306359619943707143556440204792593001104\ 393512163095845726575364837225675395934233171866481180452925443656843775\ 642135304587164579201240832696598010609165411239175080492525420517869671\ 025492928028481848235008530907664052053280422424873905846704512820511959\ 967982180702013137032361112931452467928678144801465598867701671624531232\ 483959027330720598011740851063803172434092188744100016912656216659230060\ 301984409013090728340228943052883181120716763196488413703768637440000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000 This took, 2.001, seconds. --------------------------- A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 2, variables , x[1], x[2] 3 2 2 3 exp(-75 x[1] + 54 x[1] x[2] - 18 x[1] x[2] + 82 x[2] ) By Shalosh B. Ekhad m[1] m[2] Theorem: Let , F(m[1], m[2]), be the coefficient of, x[1] x[2] , in \ the Taylor expansion (around the origing) of the multivariable function \ (that may also be viewed as a formal power series) of, x[1], x[2] 3 2 2 3 exp(-75 x[1] + 54 x[1] x[2] - 18 x[1] x[2] + 82 x[2] ) The following PURE recurrence relations hold in each of the, 2, discrete variables, m[1], m[2] 3 2 F(m[1], m[2]) = -9/1681 (28219332960 m[1] + 1237874727 m[1] m[2] 2 3 2 + 12331908 m[1] m[2] + 33714 m[2] - 235002694473 m[1] 2 - 6880569819 m[1] m[2] - 34418628 m[2] + 572496664113 m[1] + 8293718906 m[2] - 390608558352) F(m[1] - 3, m[2])/(m[1] (-1 + m[1]) 243 2 (-2 + m[1]) %1) - ---- (230776549920 m[1] + 2217732709 m[1] m[2] 1681 2 + 2854452 m[2] - 1464242094891 m[1] - 10814789452 m[2] + 1469924683776) F(m[1] - 6, m[2])/(m[1] (-1 + m[1]) (-2 + m[1]) %1) 18113627025 (231840 m[1] - 311967 + 1873 m[2]) F(m[1] - 9, m[2]) - ----------- ---------------------------------------------------- 1681 m[1] (-1 + m[1]) (-2 + m[1]) %1 %1 := 231840 m[1] - 1007487 + 1873 m[2] 3 2 F(m[1], m[2]) = 6/625 (2854764 m[1] + 12266316 m[1] m[2] 2 3 2 + 664522893 m[1] m[2] - 24301662 m[2] - 16289748 m[1] 2 - 3275609733 m[1] m[2] + 1400666658 m[2] + 4325867136 m[1] - 6465119272 m[2] + 8281287768) F(m[1], m[2] - 3)/(m[2] (m[2] - 1) 108 2 (m[2] - 2) (8811 m[1] - 322 m[2] + 17274)) - --- (164201796 m[1] 625 2 + 8915356983 m[1] m[2] - 326033050 m[2] - 35052146064 m[1] + 18139422554 m[2] - 66109550956) F(m[1], m[2] - 6)/(m[2] (m[2] - 1) (m[2] - 2) (8811 m[1] - 322 m[2] + 17274)) 8801880984 (8811 m[1] - 322 m[2] + 16308) F(m[1], m[2] - 9) + ---------- --------------------------------------------------------- 625 m[2] (m[2] - 1) (m[2] - 2) (8811 m[1] - 322 m[2] + 17274) and in Maple notation F(m[1],m[2]) = -9/1681*(28219332960*m[1]^3+1237874727*m[1]^2*m[2]+12331908*m[1] *m[2]^2+33714*m[2]^3-235002694473*m[1]^2-6880569819*m[1]*m[2]-34418628*m[2]^2+ 572496664113*m[1]+8293718906*m[2]-390608558352)/m[1]/(-1+m[1])/(-2+m[1])/( 231840*m[1]-1007487+1873*m[2])*F(m[1]-3,m[2])-243/1681*(230776549920*m[1]^2+ 2217732709*m[1]*m[2]+2854452*m[2]^2-1464242094891*m[1]-10814789452*m[2]+ 1469924683776)/m[1]/(-1+m[1])/(-2+m[1])/(231840*m[1]-1007487+1873*m[2])*F(m[1]-\ 6,m[2])-18113627025/1681*(231840*m[1]-311967+1873*m[2])/m[1]/(-1+m[1])/(-2+m[1] )/(231840*m[1]-1007487+1873*m[2])*F(m[1]-9,m[2]) F(m[1],m[2]) = 6/625*(2854764*m[1]^3+12266316*m[1]^2*m[2]+664522893*m[1]*m[2]^2 -24301662*m[2]^3-16289748*m[1]^2-3275609733*m[1]*m[2]+1400666658*m[2]^2+ 4325867136*m[1]-6465119272*m[2]+8281287768)/m[2]/(m[2]-1)/(m[2]-2)/(8811*m[1]-\ 322*m[2]+17274)*F(m[1],m[2]-3)-108/625*(164201796*m[1]^2+8915356983*m[1]*m[2]-\ 326033050*m[2]^2-35052146064*m[1]+18139422554*m[2]-66109550956)/m[2]/(m[2]-1)/( m[2]-2)/(8811*m[1]-322*m[2]+17274)*F(m[1],m[2]-6)+8801880984/625*(8811*m[1]-322 *m[2]+16308)/m[2]/(m[2]-1)/(m[2]-2)/(8811*m[1]-322*m[2]+17274)*F(m[1],m[2]-9) Subject to the initial conditions F(0, 0) = 1, F(0, 1) = 0, F(0, 2) = 0, F(0, 3) = 82, F(0, 4) = 0, F(0, 5) = 0, F(0, 6) = 3362, F(0, 7) = 0, F(0, 8) = 0, F(1, 0) = 0, F(1, 1) = 0, F(1, 2) = -18, F(1, 3) = 0, F(1, 4) = 0, F(1, 5) = -1476, F(1, 6) = 0, F(1, 7) = 0, F(1, 8) = -60516, F(2, 0) = 0, F(2, 1) = 54, F(2, 2) = 0, F(2, 3) = 0, F(2, 4) = 4590, F(2, 5) = 0, F(2, 6) = 0, F(2, 7) = 194832, F(2, 8) = 0, F(3, 0) = -75, F(3, 1) = 0, F(3, 2) = 0, F(3, 3) = -7122, F(3, 4) = 0, F(3, 5) = 0, F(3, 6) = -332826, F(3, 7) = 0, F(3, 8) = 0, F(4, 0) = 0, F(4, 1) = 0, F(4, 2) = 2808, F(4, 3) = 0, F(4, 4) = 0, F(4, 5) = 239004, F(4, 6) = 0, F(4, 7) = 0, F(4, 8) = 10162206, F(5, 0) = 0, F(5, 1) = -4050, F(5, 2) = 0, F(5, 3) = 0, F(5, 4) = -370494, F(5, 5) = 0, F(5, 6) = 0, F(5, 7) = -16816896, F(5, 8) = 0, F(6, 0) = 5625/2, F(6, 1) = 0, F(6, 2) = 0, F(6, 3) = 329769, F(6, 4) = 0, F(6, 5) = 0, F(6, 6) = 17894529, F(6, 7) = 0, F(6, 8) = 0, F(7, 0) = 0, F(7, 1) = 0, F(7, 2) = -159975, F(7, 3) = 0, F(7, 4) = 0, F(7, 5) = -14246442, F(7, 6) = 0, F(7, 7) = 0, F(7, 8) = -632117520, F(8, 0) = 0, F(8, 1) = 151875, F(8, 2) = 0, F(8, 3) = 0, F(8, 4) = 15231969, F(8, 5) = 0, F(8, 6) = 0, F(8, 7) = 746605836, F(8, 8) = 0 -------------------------------------------- By the way, it took, 1.006, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000, 2000) = 243021699142181386637270323630212849861432435243930864568642\ 610084396486361356684043057165375794069757780647043222667036074405725124\ 016006054559167116500586298663593489524556440241872089228763444604458343\ 417140343022310724088759164262522001767433362330713522610583684240312869\ 779537990608697916099911603109325246446326118907608446157728191413294409\ 665268202741885984169005475088360146124011408867847426186976535123617127\ 854813745460563553321900930410799399913658997578504022032728789381042551\ 491846943282997824689732764063589153235234546481497157577982250257211859\ 568259051946345118735309330443366547364815012956208038258996729710050986\ 660199410259064461517532593518999839868916351134246039210052132834126729\ 899561431586256353121381444377215985161283936420729267908082945305499020\ 082821789548101618067701144017938656109750368715406310670482914506685145\ 332846013449659527360976735596691775486381626283043779170272091502021423\ 268375457745987747678385231338446554308233729626153160935799290212981021\ 097351148122914984980762231901232727292661180504243033702715923334783412\ 811686088017312618745073298971506127350366542295553538621753053919696811\ 867178839840500187562642252519264851147949106441935772250081841459059105\ 375904077407599631480507792537079191050545142117787714919683393552193712\ 288522049832140730537372148750744914411229210607141357986368575527329274\ 509510169217625711586305744855120816454787714478866074563529847980271155\ 153891095479896133969637894309969209316717506838274256906253225193455028\ 108668348875288034437218849506792242962546713792567636336388675769152948\ 411008375094645480411567006067220795730130410022603308421362264521487865\ 352480505090237425453742433469215257373429771496578952074529739654560292\ 615488407791803074423234232105860589062620444700879753831464569206335077\ 689630253111586453237132503544738325137886039512247904938345812451525299\ 515318635054070006166035744936604852234230663342920072289915387620143467\ / 85673216220304327821 / 31965350973387412767230906520599051403148616773\ / 509474343235848714361161242510411082345824113440947876036170496572296331\ 096452222358815901805307890817851945353289276964895791775201727344091812\ 108327751945072314547738732782989542852017585927480672279774589024410777\ 330716346797191525488683135193223255752137450225894364096172903265029495\ 160851266939195358851632480968828433755048993361810331258967839925593803\ 665145154040256997970128763542686570321863018537386733216152347605877582\ 962149000727422180436786713283036204801140820374244725940491521543091573\ 698989944620099579050098923740660656954198360933220463800326244719727374\ 386378655291077175393914633239328220894489716826092618125674712970742664\ 855964116136174170149170787717246883420789263446399161818658046371082336\ 992401666772138420235588679055074558640435080572454631904922478204377235\ 565894748634810326475089818160998627198687553852302136735513878305932045\ 395136838986188761990575823844075040976521624086486127192547040304836028\ 220949507526767961759253105487935467698081824766531024512596970836880666\ 115208056435473822485322836805321510654755152923426528446984864425760489\ 710680219950458036633132850148616638299105057030468024143586073039040134\ 461782011783564688673197873172170894967484436788602067504784544646554115\ 783339871391635649322068046746728562585641476916612442667713862405742361\ 256043611854750740076989719390904654202381042544069335449187828095730778\ 490946694746008284908475548505336414036076871607472397357403458121677052\ 316372904660904822811384237744023746119538503358329511463806868746196880\ 160427365326821140007402743793250224328089645685601599301476695602988578\ 161039449841102178095710088295071859800268614651238438386766645921808355\ 999384530759260501667053697070717592927532505217315491689819326004591551\ 834903392504486264515552703774433751130312719537010945443092077509288219\ 594244790040927482912279559486830258797470436261865406977217727529553855\ 231503412163690287935541152317440000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000000 This took, 2.085, seconds.