Generating Functions for Enumerating the Number of Spanning Trees (and Sum o\ f the Leaves) in Friendship graphs where n people live in a one-sided ST\ RAIGHT street, and every one is friends with all the neighbors distance \ at most r as well as their asymptotics and the BZ constant for r from 2 to, 5 By Shalosh B. Ekhad Theorem number, 1 Part I: Let a(n) be the number of spanning trees of the graph, H[n, 2], whose vertice\ s are 1, ...,n, and where every vertex i is connected to i+1,..., min(n, i + 2), Then infinity ----- \ n 1 ) a[n + 2] x = ------------ / 2 ----- x - 3 x + 1 n = 0 and in Maple notation 1/(x^2-3*x+1) The asympotics, in decimals, is 1.17082039324993690892275210061938287063218550788345771728126917362315627769134\ 1469824324322513634682490852452845232135051275980317334122159075908036460035812\ 6600334519575637497433829935967919569741607284051252601355010270177242193326259\ 2976172917577778816866288985522073220448353277267284954771141720*2.618033988749\ 8948482045868343656381177203091798057628621354486227052604628189024497072072041\ 8939113748475408807538689175212663386222353693179318006076672635443338908659593\ 9582905638322661319928290267880675208766892501711696207032221043216269548626296\ 31361443814975870122034080588795445474924618569536^n Part II: Let b(n) be the sum of the number of leaves in all spanning trees of the abo\ ve-mentioned graph, H[n, 2], Then infinity ----- 2 3 2 \ n 2 (x - x + 1) (x - x - 2 x + 1) ) b[n + 2] x = ---------------------------------- / 2 2 ----- (x - 3 x + 1) n = 0 and in Maple notation 2*(x^2-x+1)*(x^3-x^2-2*x+1)/(x^2-3*x+1)^2 The asympotics in decimals, is: .305572809000084121436330532507489505823752656155389710291641101835791629744878\ 0402342342366484870900121967295396904865982986929102211704545654559513866189164\ 5328873072324833367548934187094405736778569545983298648599863064303437422316542\ 6984361098962949108449480193039023727355289636436200603051443693*2.618033988749\ 8948482045868343656381177203091798057628621354486227052604628189024497072072041\ 8939113748475408807538689175212663386222353693179318006076672635443338908659593\ 9582905638322661319928290267880675208766892501711696207032221043216269548626296\ 31361443814975870122034080588795445474924618569536^n*(1.+n) The BZ constant, in decimals, is .260990336999411149945686272447573459233731406912272027958512287149458591785853\ 7183603603434605903699146228932221665938119091496284518068180418083402936675848\ 2697888493726166427157460690339159842550013178116909459800958549875938043784201\ 1109472307259356240853638648726833908512972544946595778639894035 Theorem number, 2 Part I: Let a(n) be the number of spanning trees of the graph, H[n, 3], whose vertice\ s are 1, ...,n, and where every vertex i is connected to i+1,..., min(n, i + 3), Then infinity ----- 2 \ n 4 x + x + 3 ) a[n + 3] x = - ---------------------------------- / 4 3 2 ----- (x - 1) (x - 4 x - x - 4 x + 1) n = 0 and in Maple notation -(4*x^2+x+3)/(x-1)/(x^4-4*x^3-x^2-4*x+1) The asympotics, in decimals, is 3.90352444042068690574392885294900364177011656020448352936235976515190331308368\ 7835498454541471464014382696565761430084432159251461306785330989222416391545656\ 1571139933622355761432002282808281740729876446902049218252531090574786751156453\ 7355293635077438795012721210885085810274144935941613347601588282*4.419480365787\ 5667074848413155156881333186325196333651926684832652318194029734233993516067056\ 2235617294200632046839265454646838409371545219466019855889276199478293020635623\ 0468040676060014431004669121124061629898241290300356640072093270969917949814978\ 61668038627041193408024058104919310419706116006054^n Part II: Let b(n) be the sum of the number of leaves in all spanning trees of the abo\ ve-mentioned graph, H[n, 3], Then infinity ----- \ n 12 11 10 9 8 7 ) b[n + 3] x = (32 x - 272 x + 452 x + 376 x + 816 x + 12 x / ----- n = 0 6 5 4 3 2 / 2 - 8 x - 232 x + 16 x - 40 x + 18 x - 24 x + 6) / ((x - 1) / 4 3 2 2 (x - 4 x - x - 4 x + 1) ) and in Maple notation (32*x^12-272*x^11+452*x^10+376*x^9+816*x^8+12*x^7-8*x^6-232*x^5+16*x^4-40*x^3+ 18*x^2-24*x+6)/(x-1)^2/(x^4-4*x^3-x^2-4*x+1)^2 The asympotics in decimals, is: 1.21364320047905924642315881530296461130170174296989655686087614232214822164217\ 4453235848888265495556704347669723193585176665388823845515623483985760433451827\ 2980982727890613956829240541410931745617202554157006824176859653665437201560304\ 0149750237124080672548659148010878189269846341986031375390533247*4.419480365787\ 5667074848413155156881333186325196333651926684832652318194029734233993516067056\ 2235617294200632046839265454646838409371545219466019855889276199478293020635623\ 0468040676060014431004669121124061629898241290300356640072093270969917949814978\ 61668038627041193408024058104919310419706116006054^n*(1.+n) The BZ constant, in decimals, is .310909594394204344008503510728871359457534156622455497862965574296392568706818\ 2389717542663389754515614209307854901398957824822098178113415350516967063041587\ 8171787527146830199727656701613986631861076178492467710995347917371238944954205\ 2891443925255860750442024536049386427251672816518078123146621507 Theorem number, 3 Part I: Let a(n) be the number of spanning trees of the graph, H[n, 4], whose vertice\ s are 1, ...,n, and where every vertex i is connected to i+1,..., min(n, i + 4), Then infinity ----- \ n 10 9 8 7 6 5 ) a[n + 4] x = (36 x + 20 x - 76 x + 103 x - 49 x - 608 x / ----- n = 0 4 3 2 / - 112 x + 112 x + 5 x + 13 x + 16) / ( / 6 5 4 3 2 (x - 3 x + 6 x - 10 x + 6 x - 3 x + 1) 8 7 6 5 4 3 2 (x - 4 x - 17 x + 8 x + 49 x + 8 x - 17 x - 4 x + 1)) and in Maple notation (36*x^10+20*x^9-76*x^8+103*x^7-49*x^6-608*x^5-112*x^4+112*x^3+5*x^2+13*x+16)/(x ^6-3*x^5+6*x^4-10*x^3+6*x^2-3*x+1)/(x^8-4*x^7-17*x^6+8*x^5+49*x^4+8*x^3-17*x^2-\ 4*x+1) The asympotics, in decimals, is 22.8362649504052577281914842828321378967389271701504248140234316887942902081234\ 7290691076113386367136933759580535126599419403679340008886093103421903102328275\ 7451729674322624596670223528379154112580879327429630552553977310234416167540654\ 5557313903967868834321861804099112984858812614027410530821227387*6.298096056225\ 7735736763280347455186344983566539488734555762957580607021719381702047684955966\ 0868174082297905849451741105870880197908448107777793195646538809910654441251005\ 5905938348446142400780573792080060295441158235077834737768247256185377889131512\ 94989027077865744774017135367553226025262757348097^n Part II: Let b(n) be the sum of the number of leaves in all spanning trees of the abo\ ve-mentioned graph, H[n, 4], Then infinity ----- \ n 31 30 29 28 ) b[n + 4] x = (2016 x - 26424 x + 76784 x + 177840 x / ----- n = 0 27 26 25 24 23 - 805992 x + 336572 x + 1406432 x - 6628092 x + 9345048 x 22 21 20 19 18 + 17151224 x - 35171192 x + 20561596 x + 9552176 x - 140184504 x 17 16 15 14 13 + 76273480 x + 138809200 x - 6890160 x - 26763408 x - 13830680 x 12 11 10 9 8 - 8491192 x + 1139888 x + 2000920 x + 599424 x + 149240 x 7 6 5 4 3 2 - 31696 x - 9420 x - 3120 x - 2644 x + 976 x - 144 x - 184 x + 36) / 6 5 4 3 2 2 / ((x - 3 x + 6 x - 10 x + 6 x - 3 x + 1) / 8 7 6 5 4 3 2 2 (x - 4 x - 17 x + 8 x + 49 x + 8 x - 17 x - 4 x + 1) ) and in Maple notation (2016*x^31-26424*x^30+76784*x^29+177840*x^28-805992*x^27+336572*x^26+1406432*x^ 25-6628092*x^24+9345048*x^23+17151224*x^22-35171192*x^21+20561596*x^20+9552176* x^19-140184504*x^18+76273480*x^17+138809200*x^16-6890160*x^15-26763408*x^14-\ 13830680*x^13-8491192*x^12+1139888*x^11+2000920*x^10+599424*x^9+149240*x^8-\ 31696*x^7-9420*x^6-3120*x^5-2644*x^4+976*x^3-144*x^2-184*x+36)/(x^6-3*x^5+6*x^4 -10*x^3+6*x^2-3*x+1)^2/(x^8-4*x^7-17*x^6+8*x^5+49*x^4+8*x^3-17*x^2-4*x+1)^2 The asympotics in decimals, is: 7.54183968047861397010472399332927094262603687549895250260964949220026463503088\ 0547814087092747742345526115709475747835705252667297289111783060846777410436687\ 2432494677657563400524577388747820060925324189242161237625802070162023757395888\ 0962412464229917475488818345618034146223205385634740799712054508*6.298096056225\ 7735736763280347455186344983566539488734555762957580607021719381702047684955966\ 0868174082297905849451741105870880197908448107777793195646538809910654441251005\ 5905938348446142400780573792080060295441158235077834737768247256185377889131512\ 94989027077865744774017135367553226025262757348097^n*(1.+n) The BZ constant, in decimals, is .330257145678491294152201301215982579594511047966878645657116548401719850150894\ 8011993845958733621898749861243663556491188833571288442784842579638132249880330\ 3136653393885225076822939108298232248635475703397612277447225929476675876480347\ 1615233791622060126409183346294148549392148036565874334392134124 Theorem number, 4 Part I: Let a(n) be the number of spanning trees of the graph, H[n, 5], whose vertice\ s are 1, ...,n, and where every vertex i is connected to i+1,..., min(n, i + 5), Then infinity ----- \ n 36 35 34 33 32 ) a[n + 5] x = - (576 x + 1072 x + 880 x + 9397 x + 37352 x / ----- n = 0 31 30 29 28 27 - 38820 x - 162696 x - 409945 x - 940188 x - 6830079 x 26 25 24 23 22 - 10208568 x - 12657852 x - 11172468 x - 10598248 x + 2897948 x 21 20 19 18 17 + 13501001 x + 22499688 x + 26110019 x + 28239293 x + 18386448 x 16 15 14 13 12 + 10158616 x + 2700085 x - 4176485 x - 10893240 x - 8747319 x 11 10 9 8 7 - 7446180 x - 6025584 x - 4322484 x - 802676 x - 189560 x 6 5 4 3 2 / - 61343 x + 7796 x + 19551 x + 5928 x + 1012 x + 296 x + 125) / ( / 8 7 6 5 4 3 2 16 15 (x - 1) (x + 3 x + 6 x - x + 15 x - x + 6 x + 3 x + 1) (x - 5 x 14 13 12 11 10 9 8 7 + 10 x - 10 x - 28 x + 10 x + 110 x + 110 x + 88 x + 110 x 6 5 4 3 2 16 15 14 + 110 x + 10 x - 28 x - 10 x + 10 x - 5 x + 1) (x - 5 x - 23 x 13 12 11 10 9 8 7 6 - 10 x - 94 x - 485 x + 242 x + 110 x + 649 x + 110 x + 242 x 5 4 3 2 - 485 x - 94 x - 10 x - 23 x - 5 x + 1)) and in Maple notation -(576*x^36+1072*x^35+880*x^34+9397*x^33+37352*x^32-38820*x^31-162696*x^30-\ 409945*x^29-940188*x^28-6830079*x^27-10208568*x^26-12657852*x^25-11172468*x^24-\ 10598248*x^23+2897948*x^22+13501001*x^21+22499688*x^20+26110019*x^19+28239293*x ^18+18386448*x^17+10158616*x^16+2700085*x^15-4176485*x^14-10893240*x^13-8747319 *x^12-7446180*x^11-6025584*x^10-4322484*x^9-802676*x^8-189560*x^7-61343*x^6+ 7796*x^5+19551*x^4+5928*x^3+1012*x^2+296*x+125)/(x-1)/(x^8+3*x^7+6*x^6-x^5+15*x ^4-x^3+6*x^2+3*x+1)/(x^16-5*x^15+10*x^14-10*x^13-28*x^12+10*x^11+110*x^10+110*x ^9+88*x^8+110*x^7+110*x^6+10*x^5-28*x^4-10*x^3+10*x^2-5*x+1)/(x^16-5*x^15-23*x^ 14-10*x^13-94*x^12-485*x^11+242*x^10+110*x^9+649*x^8+110*x^7+242*x^6-485*x^5-94 *x^4-10*x^3-23*x^2-5*x+1) The asympotics, in decimals, is 194.678094852967184875601993571069359907375209042413772215025409112573627014652\ 3972593080764502034494900121097687671652165572051004739595064980606739366842186\ 1062706435920216383563776226326567439358647016945324278356900653340105125107179\ 5810664178845416144727987284388548896566189456823153531782198280*8.216271114701\ 8080784197270517005864370282025075930083403333260117064480028303210164396856662\ 1743991959463897966104229258622840833148397780121598105156859723905359918565442\ 5456249668868065723070529646912028862036409706909970198249278968640386785790148\ 21722529481650260900175511656134298623976970864339^n Part II: Let b(n) be the sum of the number of leaves in all spanning trees of the abo\ ve-mentioned graph, H[n, 5], Then infinity ----- \ n 86 85 84 83 ) b[n + 5] x = (286720 x - 4407200 x + 12552832 x + 51804128 x / ----- n = 0 82 81 80 79 - 140078160 x - 223042112 x + 3392556960 x - 6772574688 x 78 77 76 75 - 32998308048 x + 37874334944 x + 159383937616 x - 1048771754582 x 74 73 72 + 1215139671232 x + 8239060471446 x - 1765281505312 x 71 70 69 - 31306266572154 x + 140453959471792 x - 109339756667154 x 68 67 66 - 756837798553936 x - 727369523814740 x + 2864334083478544 x 65 64 63 - 10477687896754602 x + 11279194115203312 x + 15663965414461944 x 62 61 60 + 50243800946136208 x + 7585150625375352 x + 83669848441900164 x 59 58 57 - 54080107834236800 x - 101534367824757412 x - 318108985457311982 x 56 55 54 - 224696925497309220 x - 429754497077154194 x - 24249322681081988 x 53 52 51 + 197334828968756032 x + 845234595728857624 x + 859162871503526066 x 50 49 + 1326621361070739180 x + 653400209972170158 x 48 47 + 184838822003090368 x - 988422155012723648 x 46 45 - 1328611424340516992 x - 2177738311926879038 x 44 43 - 1596665204326268112 x - 1105428881624370722 x 42 41 + 185590541186208052 x + 776623498697357200 x 40 39 + 1788809753262611180 x + 1610816195760422306 x 38 37 + 1401845678707693904 x + 577974837834111886 x 36 35 34 + 119759331037872964 x - 611599022007774384 x - 690592798199163780 x 33 32 31 - 708917213285279822 x - 440606003342926672 x - 276442108433622098 x 30 29 28 + 5593730558339024 x + 88017606988549760 x + 120986611230999088 x 27 26 25 + 93058401778691618 x + 71699199535821380 x + 28745246203186526 x 24 23 22 + 7558736132415480 x + 1959786699784544 x + 15320630140164 x 21 20 19 - 592380205488542 x - 286638698161740 x - 88033161671554 x 18 17 16 - 28497194371004 x - 6453784987872 x + 617443648412 x 15 14 13 12 + 684261616584 x + 260166304016 x + 87564069752 x + 22059801984 x 11 10 9 8 7 + 2826671114 x + 237136688 x - 65794300 x - 34020480 x - 6653454 x 6 5 4 3 2 - 1109280 x - 610662 x - 66432 x + 3114 x - 3840 x - 1370 x + 320) / 2 8 7 6 5 4 3 2 2 16 / ((x - 1) (x + 3 x + 6 x - x + 15 x - x + 6 x + 3 x + 1) (x / 15 14 13 12 11 10 9 8 - 5 x + 10 x - 10 x - 28 x + 10 x + 110 x + 110 x + 88 x 7 6 5 4 3 2 2 16 15 + 110 x + 110 x + 10 x - 28 x - 10 x + 10 x - 5 x + 1) (x - 5 x 14 13 12 11 10 9 8 7 - 23 x - 10 x - 94 x - 485 x + 242 x + 110 x + 649 x + 110 x 6 5 4 3 2 2 + 242 x - 485 x - 94 x - 10 x - 23 x - 5 x + 1) ) and in Maple notation (286720*x^86-4407200*x^85+12552832*x^84+51804128*x^83-140078160*x^82-223042112* x^81+3392556960*x^80-6772574688*x^79-32998308048*x^78+37874334944*x^77+ 159383937616*x^76-1048771754582*x^75+1215139671232*x^74+8239060471446*x^73-\ 1765281505312*x^72-31306266572154*x^71+140453959471792*x^70-109339756667154*x^ 69-756837798553936*x^68-727369523814740*x^67+2864334083478544*x^66-\ 10477687896754602*x^65+11279194115203312*x^64+15663965414461944*x^63+ 50243800946136208*x^62+7585150625375352*x^61+83669848441900164*x^60-\ 54080107834236800*x^59-101534367824757412*x^58-318108985457311982*x^57-\ 224696925497309220*x^56-429754497077154194*x^55-24249322681081988*x^54+ 197334828968756032*x^53+845234595728857624*x^52+859162871503526066*x^51+ 1326621361070739180*x^50+653400209972170158*x^49+184838822003090368*x^48-\ 988422155012723648*x^47-1328611424340516992*x^46-2177738311926879038*x^45-\ 1596665204326268112*x^44-1105428881624370722*x^43+185590541186208052*x^42+ 776623498697357200*x^41+1788809753262611180*x^40+1610816195760422306*x^39+ 1401845678707693904*x^38+577974837834111886*x^37+119759331037872964*x^36-\ 611599022007774384*x^35-690592798199163780*x^34-708917213285279822*x^33-\ 440606003342926672*x^32-276442108433622098*x^31+5593730558339024*x^30+ 88017606988549760*x^29+120986611230999088*x^28+93058401778691618*x^27+ 71699199535821380*x^26+28745246203186526*x^25+7558736132415480*x^24+ 1959786699784544*x^23+15320630140164*x^22-592380205488542*x^21-286638698161740* x^20-88033161671554*x^19-28497194371004*x^18-6453784987872*x^17+617443648412*x^ 16+684261616584*x^15+260166304016*x^14+87564069752*x^13+22059801984*x^12+ 2826671114*x^11+237136688*x^10-65794300*x^9-34020480*x^8-6653454*x^7-1109280*x^ 6-610662*x^5-66432*x^4+3114*x^3-3840*x^2-1370*x+320)/(x-1)^2/(x^8+3*x^7+6*x^6-x ^5+15*x^4-x^3+6*x^2+3*x+1)^2/(x^16-5*x^15+10*x^14-10*x^13-28*x^12+10*x^11+110*x ^10+110*x^9+88*x^8+110*x^7+110*x^6+10*x^5-28*x^4-10*x^3+10*x^2-5*x+1)^2/(x^16-5 *x^15-23*x^14-10*x^13-94*x^12-485*x^11+242*x^10+110*x^9+649*x^8+110*x^7+242*x^6 -485*x^5-94*x^4-10*x^3-23*x^2-5*x+1)^2 The asympotics in decimals, is: 66.2255687894783828512695965406398914633981056592340874687572609744185698894093\ 7005420587009707538176488990876930450857391120957206400398541316589515715574409\ 3101503335463146674786158673089376142928053859192691722091096421772456863383401\ 6384764301556658156405373510993035251149371406374651866689681330*8.216271114701\ 8080784197270517005864370282025075930083403333260117064480028303210164396856662\ 1743991959463897966104229258622840833148397780121598105156859723905359918565442\ 5456249668868065723070529646912028862036409706909970198249278968640386785790148\ 21722529481650260900175511656134298623976970864339^n*(1.+n) The BZ constant, in decimals, is .340179868924112836772060183916870487107478540491420808722455473554087114986963\ 7106305885610306541613274353498978178840660433379254041238701241735752591276975\ 0644465452717178780407367931860440780427263450567398621830464946602601346845038\ 5989127893935952397101240552277896176385243683381867509462708188 -------------------------------- This took, 3462.954, seconds. [[1/(x^2-3*x+1), 2*(x^2-x+1)*(x^3-x^2-2*x+1)/(x^2-3*x+1)^2], [-(4*x^2+x+3)/(x-1 )/(x^4-4*x^3-x^2-4*x+1), (32*x^12-272*x^11+452*x^10+376*x^9+816*x^8+12*x^7-8*x^ 6-232*x^5+16*x^4-40*x^3+18*x^2-24*x+6)/(x-1)^2/(x^4-4*x^3-x^2-4*x+1)^2], [(36*x ^10+20*x^9-76*x^8+103*x^7-49*x^6-608*x^5-112*x^4+112*x^3+5*x^2+13*x+16)/(x^6-3* x^5+6*x^4-10*x^3+6*x^2-3*x+1)/(x^8-4*x^7-17*x^6+8*x^5+49*x^4+8*x^3-17*x^2-4*x+1 ), (2016*x^31-26424*x^30+76784*x^29+177840*x^28-805992*x^27+336572*x^26+1406432 *x^25-6628092*x^24+9345048*x^23+17151224*x^22-35171192*x^21+20561596*x^20+ 9552176*x^19-140184504*x^18+76273480*x^17+138809200*x^16-6890160*x^15-26763408* x^14-13830680*x^13-8491192*x^12+1139888*x^11+2000920*x^10+599424*x^9+149240*x^8 -31696*x^7-9420*x^6-3120*x^5-2644*x^4+976*x^3-144*x^2-184*x+36)/(x^6-3*x^5+6*x^ 4-10*x^3+6*x^2-3*x+1)^2/(x^8-4*x^7-17*x^6+8*x^5+49*x^4+8*x^3-17*x^2-4*x+1)^2], [-(576*x^36+1072*x^35+880*x^34+9397*x^33+37352*x^32-38820*x^31-162696*x^30-\ 409945*x^29-940188*x^28-6830079*x^27-10208568*x^26-12657852*x^25-11172468*x^24-\ 10598248*x^23+2897948*x^22+13501001*x^21+22499688*x^20+26110019*x^19+28239293*x ^18+18386448*x^17+10158616*x^16+2700085*x^15-4176485*x^14-10893240*x^13-8747319 *x^12-7446180*x^11-6025584*x^10-4322484*x^9-802676*x^8-189560*x^7-61343*x^6+ 7796*x^5+19551*x^4+5928*x^3+1012*x^2+296*x+125)/(x-1)/(x^8+3*x^7+6*x^6-x^5+15*x ^4-x^3+6*x^2+3*x+1)/(x^16-5*x^15+10*x^14-10*x^13-28*x^12+10*x^11+110*x^10+110*x ^9+88*x^8+110*x^7+110*x^6+10*x^5-28*x^4-10*x^3+10*x^2-5*x+1)/(x^16-5*x^15-23*x^ 14-10*x^13-94*x^12-485*x^11+242*x^10+110*x^9+649*x^8+110*x^7+242*x^6-485*x^5-94 *x^4-10*x^3-23*x^2-5*x+1), (286720*x^86-4407200*x^85+12552832*x^84+51804128*x^ 83-140078160*x^82-223042112*x^81+3392556960*x^80-6772574688*x^79-32998308048*x^ 78+37874334944*x^77+159383937616*x^76-1048771754582*x^75+1215139671232*x^74+ 8239060471446*x^73-1765281505312*x^72-31306266572154*x^71+140453959471792*x^70-\ 109339756667154*x^69-756837798553936*x^68-727369523814740*x^67+2864334083478544 *x^66-10477687896754602*x^65+11279194115203312*x^64+15663965414461944*x^63+ 50243800946136208*x^62+7585150625375352*x^61+83669848441900164*x^60-\ 54080107834236800*x^59-101534367824757412*x^58-318108985457311982*x^57-\ 224696925497309220*x^56-429754497077154194*x^55-24249322681081988*x^54+ 197334828968756032*x^53+845234595728857624*x^52+859162871503526066*x^51+ 1326621361070739180*x^50+653400209972170158*x^49+184838822003090368*x^48-\ 988422155012723648*x^47-1328611424340516992*x^46-2177738311926879038*x^45-\ 1596665204326268112*x^44-1105428881624370722*x^43+185590541186208052*x^42+ 776623498697357200*x^41+1788809753262611180*x^40+1610816195760422306*x^39+ 1401845678707693904*x^38+577974837834111886*x^37+119759331037872964*x^36-\ 611599022007774384*x^35-690592798199163780*x^34-708917213285279822*x^33-\ 440606003342926672*x^32-276442108433622098*x^31+5593730558339024*x^30+ 88017606988549760*x^29+120986611230999088*x^28+93058401778691618*x^27+ 71699199535821380*x^26+28745246203186526*x^25+7558736132415480*x^24+ 1959786699784544*x^23+15320630140164*x^22-592380205488542*x^21-286638698161740* x^20-88033161671554*x^19-28497194371004*x^18-6453784987872*x^17+617443648412*x^ 16+684261616584*x^15+260166304016*x^14+87564069752*x^13+22059801984*x^12+ 2826671114*x^11+237136688*x^10-65794300*x^9-34020480*x^8-6653454*x^7-1109280*x^ 6-610662*x^5-66432*x^4+3114*x^3-3840*x^2-1370*x+320)/(x-1)^2/(x^8+3*x^7+6*x^6-x ^5+15*x^4-x^3+6*x^2+3*x+1)^2/(x^16-5*x^15+10*x^14-10*x^13-28*x^12+10*x^11+110*x ^10+110*x^9+88*x^8+110*x^7+110*x^6+10*x^5-28*x^4-10*x^3+10*x^2-5*x+1)^2/(x^16-5 *x^15-23*x^14-10*x^13-94*x^12-485*x^11+242*x^10+110*x^9+649*x^8+110*x^7+242*x^6 -485*x^5-94*x^4-10*x^3-23*x^2-5*x+1)^2]]