----------------------------------- Theorem Number , 1 Let , A(1, n), be the number of spanning trees of the, 1, by n rectangle, and let infinity ----- \ n f[1](t) = ) A(1, n) t / ----- n = 1 Then, f[1](t), equals the following rational function t ----- 1 - t and in Maple format t/(1-t) For the sake of the OEIS, here are the first 30 terms [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] ----------------------------------- Theorem Number , 2 Let , A(2, n), be the number of spanning trees of the, 2, by n rectangle, and let infinity ----- \ n f[2](t) = ) A(2, n) t / ----- n = 1 Then, f[2](t), equals the following rational function t ------------ 2 t - 4 t + 1 and in Maple format t/(t^2-4*t+1) For the sake of the OEIS, here are the first 30 terms [1, 4, 15, 56, 209, 780, 2911, 10864, 40545, 151316, 564719, 2107560, 7865521, 29354524, 109552575, 408855776, 1525870529, 5694626340, 21252634831, 79315912984, 296011017105, 1104728155436, 4122901604639, 15386878263120, 57424611447841, 214311567528244, 799821658665135, 2984975067132296, 11140078609864049, 41575339372323900] ----------------------------------- Theorem Number , 3 Let , A(3, n), be the number of spanning trees of the, 3, by n rectangle, and let infinity ----- \ n f[3](t) = ) A(3, n) t / ----- n = 1 Then, f[3](t), equals the following rational function 3 -t + t ----------------------------- 4 3 2 t - 15 t + 32 t - 15 t + 1 and in Maple format (-t^3+t)/(t^4-15*t^3+32*t^2-15*t+1) For the sake of the OEIS, here are the first 30 terms [1, 15, 192, 2415, 30305, 380160, 4768673, 59817135, 750331584, 9411975375, 118061508289, 1480934568960, 18576479568193, 233018797965135, 2922930580320960, 36664523428884015, 459910778352898337, 5769007865476035840, 72365017995700730081, 907729015392142395375, 11386329862223500207296, 142827325702873981372815, 1791590900164844149159297, 22473276298898655194173440, 281899259233885701359889025, 3536075083120298342440801935, 44355657504903447676859022528, 556386475525911555059983837935, 6979175319719333589675520988129, 87545061366451265500023776544000] ----------------------------------- Theorem Number , 4 Let , A(4, n), be the number of spanning trees of the, 4, by n rectangle, and let infinity ----- \ n f[4](t) = ) A(4, n) t / ----- n = 1 Then, f[4](t), equals the following rational function 7 5 4 3 t - 49 t + 112 t - 49 t + t --------------------------------------------------------------------- 8 7 6 5 4 3 2 t - 56 t + 672 t - 2632 t + 4094 t - 2632 t + 672 t - 56 t + 1 and in Maple format (t^7-49*t^5+112*t^4-49*t^3+t)/(t^8-56*t^7+672*t^6-2632*t^5+4094*t^4-2632*t^3+ 672*t^2-56*t+1) For the sake of the OEIS, here are the first 30 terms [1, 56, 2415, 100352, 4140081, 170537640, 7022359583, 289143013376, 11905151192865, 490179860527896, 20182531537581071, 830989874753525760, 34214941811800329425, 1408756312731277540744, 58003732850974438010175, 2388229243489107516194816, 98332273432993735566334273, 4048705133573796221764525560, 166700236720421701398973625263, 6863668260790646684044452390912, 282602730032000912315954740344945, 11635804643672459612539385026592872, 479089319803727354015166006771993439, 19725887755842805930245125686879764480, 812188107043419732226106284169171254881, 33440802735348892291948093536540743747800, 1376882126057448000490859816205636406470735, 56691354093976384771073887779803938974152704, 2334193732481147942455005866608760491340320529, 96107430627295371442302076835629206411100969800] ----------------------------------- Theorem Number , 5 Let , A(5, n), be the number of spanning trees of the, 5, by n rectangle, and let infinity ----- \ n f[5](t) = ) A(5, n) t / ----- n = 1 Then, f[5](t), equals the following rational function 15 13 12 11 10 9 7 (-t + 1440 t - 26752 t + 185889 t - 574750 t + 708928 t - 708928 t 6 5 4 3 / 16 15 + 574750 t - 185889 t + 26752 t - 1440 t + t) / (t - 209 t / 14 13 12 11 10 + 11936 t - 274208 t + 3112032 t - 19456019 t + 70651107 t 9 8 7 6 5 - 152325888 t + 196664896 t - 152325888 t + 70651107 t - 19456019 t 4 3 2 + 3112032 t - 274208 t + 11936 t - 209 t + 1) and in Maple format (-t^15+1440*t^13-26752*t^12+185889*t^11-574750*t^10+708928*t^9-708928*t^7+ 574750*t^6-185889*t^5+26752*t^4-1440*t^3+t)/(t^16-209*t^15+11936*t^14-274208*t^ 13+3112032*t^12-19456019*t^11+70651107*t^10-152325888*t^9+196664896*t^8-\ 152325888*t^7+70651107*t^6-19456019*t^5+3112032*t^4-274208*t^3+11936*t^2-209*t+ 1) For the sake of the OEIS, here are the first 30 terms [1, 209, 30305, 4140081, 557568000, 74795194705, 10021992194369, 1342421467113969, 179796299139278305, 24080189412483072000, 3225041354570508955681, 431926215138756947267505, 57847355494807961811035009, 7747424602888405489208931601, 1037602902862756514154816000000, 138964858389586339640223412108401, 18611389483734199394023624777573409, 2492600085599977923424220468405177105, 333830807688353225138019865387722924481, 44709541971379003103897461691112357888000, 5987892960038182781131697625354150226327105, 801951004627869433685025226859351146717402769, 107404293649401297954327034703922488508540561569, 14384522530358739351890623742584897464468359377905, 1926501086648879747745673025840512108858205299200000, 258013877695694120804712221064093162848578908856571281, 34555475491252459965101163680103428812170683289730088705, 4627971553665255661896001375366896994510897198018850677809, 619818433896435032361580416286522616772627078698025714103201, 83011506562431376380390013640740802987778069424647853056000000] ----------------------------------- Theorem Number , 6 Let , A(6, n), be the number of spanning trees of the, 6, by n rectangle, and let infinity ----- \ n f[6](t) = ) A(6, n) t / ----- n = 1 Then, f[6](t), equals the following rational function 31 29 28 27 26 (t - 33359 t + 3642600 t - 173371343 t + 4540320720 t 25 24 23 - 70164186331 t + 634164906960 t - 2844883304348 t 22 21 20 - 1842793012320 t + 104844096982372 t - 678752492380560 t 19 18 17 + 2471590551535210 t - 5926092273213840 t + 9869538714631398 t 16 15 14 - 11674018886109840 t + 9869538714631398 t - 5926092273213840 t 13 12 11 + 2471590551535210 t - 678752492380560 t + 104844096982372 t 10 9 8 7 - 1842793012320 t - 2844883304348 t + 634164906960 t - 70164186331 t 6 5 4 3 / 32 + 4540320720 t - 173371343 t + 3642600 t - 33359 t + t) / (t / 31 30 29 28 27 - 780 t + 194881 t - 22377420 t + 1419219792 t - 55284715980 t 26 25 24 + 1410775106597 t - 24574215822780 t + 300429297446885 t 23 22 21 - 2629946465331120 t + 16741727755133760 t - 78475174345180080 t 20 19 + 273689714665707178 t - 716370537293731320 t 18 17 + 1417056251105102122 t - 2129255507292156360 t 16 15 + 2437932520099475424 t - 2129255507292156360 t 14 13 + 1417056251105102122 t - 716370537293731320 t 12 11 10 + 273689714665707178 t - 78475174345180080 t + 16741727755133760 t 9 8 7 - 2629946465331120 t + 300429297446885 t - 24574215822780 t 6 5 4 3 + 1410775106597 t - 55284715980 t + 1419219792 t - 22377420 t 2 + 194881 t - 780 t + 1) and in Maple format (t^31-33359*t^29+3642600*t^28-173371343*t^27+4540320720*t^26-70164186331*t^25+ 634164906960*t^24-2844883304348*t^23-1842793012320*t^22+104844096982372*t^21-\ 678752492380560*t^20+2471590551535210*t^19-5926092273213840*t^18+ 9869538714631398*t^17-11674018886109840*t^16+9869538714631398*t^15-\ 5926092273213840*t^14+2471590551535210*t^13-678752492380560*t^12+ 104844096982372*t^11-1842793012320*t^10-2844883304348*t^9+634164906960*t^8-\ 70164186331*t^7+4540320720*t^6-173371343*t^5+3642600*t^4-33359*t^3+t)/(t^32-780 *t^31+194881*t^30-22377420*t^29+1419219792*t^28-55284715980*t^27+1410775106597* t^26-24574215822780*t^25+300429297446885*t^24-2629946465331120*t^23+ 16741727755133760*t^22-78475174345180080*t^21+273689714665707178*t^20-\ 716370537293731320*t^19+1417056251105102122*t^18-2129255507292156360*t^17+ 2437932520099475424*t^16-2129255507292156360*t^15+1417056251105102122*t^14-\ 716370537293731320*t^13+273689714665707178*t^12-78475174345180080*t^11+ 16741727755133760*t^10-2629946465331120*t^9+300429297446885*t^8-24574215822780* t^7+1410775106597*t^6-55284715980*t^5+1419219792*t^4-22377420*t^3+194881*t^2-\ 780*t+1) For the sake of the OEIS, here are the first 30 terms [1, 780, 380160, 170537640, 74795194705, 32565539635200, 14143261515284447, 6136973985625588560, 2662079368040434932480, 1154617875754582889149500, 500769437567956298239402223, 217185579535490113365186969600, 94193702839904633186530210863025, 40851869157273984726590135085017940, 17717469746416280095776019395706656000, 7684070867169415429692559499446691755680, 3332583081296808509759455619848802299528513, 1445341907485491645328460310146924377335398400, 626845049313054375044367343971643549398400207439, 271862811296852944176805652529210910158678393501000, 117906950273496738441417982275113800569185953508401920, 51136265564260810934348514677358578182539093981985579140, 22177807578786871771975685968088359261519721558786693144351, 9618519137859358557484551263002772100493327681623268157030400, 4171553480859669166764609074130946419690927672433431476417301025, 1809203495268378561427549408448430750759030036616841097919332985500, 784651881439272378819594430425368142664597935508249012776181518286080, 340303662167183966505537475986213779843792970000024727053163782358717080, 147589759514663135524738411416132378729266135389120483181714254960506345457 , 64009705258157188975303879517278636748220260668937846227901842986863001\ 600000] ----------------------------------- Theorem Number , 7 Let , A(7, n), be the number of spanning trees of the, 7, by n rectangle, and let infinity ----- \ n f[7](t) = ) A(7, n) t / ----- n = 1 Then, f[7](t), equals the following rational function 47 46 45 44 43 (-t - 142 t + 661245 t - 279917500 t + 53184503243 t 42 41 40 - 5570891154842 t + 341638600598298 t - 11886702497030032 t 39 38 + 164458937576610742 t + 4371158470492451828 t 37 36 - 288737344956855301342 t + 7736513993329973661368 t 35 34 - 131582338768322853956994 t + 1573202877300834187134466 t 33 32 - 13805721749199518460916737 t + 90975567796174070740787232 t 31 30 - 455915282590547643587452175 t + 1747901867578637315747826286 t 29 28 - 5126323837327170557921412877 t + 11416779122947828869806142972 t 27 26 - 18924703166237080216745900796 t + 22194247945745188489023284104 t 25 23 - 15563815847174688069871470516 t + 15563815847174688069871470516 t 22 21 - 22194247945745188489023284104 t + 18924703166237080216745900796 t 20 19 - 11416779122947828869806142972 t + 5126323837327170557921412877 t 18 17 - 1747901867578637315747826286 t + 455915282590547643587452175 t 16 15 - 90975567796174070740787232 t + 13805721749199518460916737 t 14 13 - 1573202877300834187134466 t + 131582338768322853956994 t 12 11 - 7736513993329973661368 t + 288737344956855301342 t 10 9 8 - 4371158470492451828 t - 164458937576610742 t + 11886702497030032 t 7 6 5 4 - 341638600598298 t + 5570891154842 t - 53184503243 t + 279917500 t 3 2 / 48 47 46 - 661245 t + 142 t + t) / (t - 2769 t + 2630641 t / 45 44 43 - 1195782497 t + 305993127089 t - 48551559344145 t 42 41 40 + 5083730101530753 t - 366971376492201338 t + 18871718211768417242 t 39 38 - 709234610141846974874 t + 19874722637854592209338 t 37 36 - 422023241997789381263002 t + 6880098547452856483997402 t 35 34 - 87057778313447181201990522 t + 862879164715733847737203343 t 33 32 - 6750900711491569851736413311 t + 41958615314622858303912597215 t 31 30 - 208258356862493902206466194607 t + 828959040281722890327985220255 t 29 - 2654944041424536277948746010303 t 28 + 6859440538554030239641036025103 t 27 - 14324708604336971207868317957868 t 26 + 24214587194571650834572683444012 t 25 - 33166490975387358866518005011884 t 24 + 36830850383375837481096026357868 t 23 - 33166490975387358866518005011884 t 22 + 24214587194571650834572683444012 t 21 - 14324708604336971207868317957868 t 20 + 6859440538554030239641036025103 t 19 18 - 2654944041424536277948746010303 t + 828959040281722890327985220255 t 17 16 - 208258356862493902206466194607 t + 41958615314622858303912597215 t 15 14 - 6750900711491569851736413311 t + 862879164715733847737203343 t 13 12 - 87057778313447181201990522 t + 6880098547452856483997402 t 11 10 - 422023241997789381263002 t + 19874722637854592209338 t 9 8 - 709234610141846974874 t + 18871718211768417242 t 7 6 5 - 366971376492201338 t + 5083730101530753 t - 48551559344145 t 4 3 2 + 305993127089 t - 1195782497 t + 2630641 t - 2769 t + 1) and in Maple format (-t^47-142*t^46+661245*t^45-279917500*t^44+53184503243*t^43-5570891154842*t^42+ 341638600598298*t^41-11886702497030032*t^40+164458937576610742*t^39+ 4371158470492451828*t^38-288737344956855301342*t^37+7736513993329973661368*t^36 -131582338768322853956994*t^35+1573202877300834187134466*t^34-\ 13805721749199518460916737*t^33+90975567796174070740787232*t^32-\ 455915282590547643587452175*t^31+1747901867578637315747826286*t^30-\ 5126323837327170557921412877*t^29+11416779122947828869806142972*t^28-\ 18924703166237080216745900796*t^27+22194247945745188489023284104*t^26-\ 15563815847174688069871470516*t^25+15563815847174688069871470516*t^23-\ 22194247945745188489023284104*t^22+18924703166237080216745900796*t^21-\ 11416779122947828869806142972*t^20+5126323837327170557921412877*t^19-\ 1747901867578637315747826286*t^18+455915282590547643587452175*t^17-\ 90975567796174070740787232*t^16+13805721749199518460916737*t^15-\ 1573202877300834187134466*t^14+131582338768322853956994*t^13-\ 7736513993329973661368*t^12+288737344956855301342*t^11-4371158470492451828*t^10 -164458937576610742*t^9+11886702497030032*t^8-341638600598298*t^7+5570891154842 *t^6-53184503243*t^5+279917500*t^4-661245*t^3+142*t^2+t)/(t^48-2769*t^47+ 2630641*t^46-1195782497*t^45+305993127089*t^44-48551559344145*t^43+ 5083730101530753*t^42-366971376492201338*t^41+18871718211768417242*t^40-\ 709234610141846974874*t^39+19874722637854592209338*t^38-\ 422023241997789381263002*t^37+6880098547452856483997402*t^36-\ 87057778313447181201990522*t^35+862879164715733847737203343*t^34-\ 6750900711491569851736413311*t^33+41958615314622858303912597215*t^32-\ 208258356862493902206466194607*t^31+828959040281722890327985220255*t^30-\ 2654944041424536277948746010303*t^29+6859440538554030239641036025103*t^28-\ 14324708604336971207868317957868*t^27+24214587194571650834572683444012*t^26-\ 33166490975387358866518005011884*t^25+36830850383375837481096026357868*t^24-\ 33166490975387358866518005011884*t^23+24214587194571650834572683444012*t^22-\ 14324708604336971207868317957868*t^21+6859440538554030239641036025103*t^20-\ 2654944041424536277948746010303*t^19+828959040281722890327985220255*t^18-\ 208258356862493902206466194607*t^17+41958615314622858303912597215*t^16-\ 6750900711491569851736413311*t^15+862879164715733847737203343*t^14-\ 87057778313447181201990522*t^13+6880098547452856483997402*t^12-\ 422023241997789381263002*t^11+19874722637854592209338*t^10-\ 709234610141846974874*t^9+18871718211768417242*t^8-366971376492201338*t^7+ 5083730101530753*t^6-48551559344145*t^5+305993127089*t^4-1195782497*t^3+2630641 *t^2-2769*t+1) For the sake of the OEIS, here are the first 30 terms [1, 2911, 4768673, 7022359583, 10021992194369, 14143261515284447, 19872369301840986112, 27873182693625548898079, 39067130344394503972142977, 54740416599810921320592441119, 76692291658239649098972455530913, 107441842254735898225957962027174559, 150517199699838971875005120330439121217, 210860422397100784567572728149075575177216, 295394542455170446994290978914210208936856257, 413817852397080403148772215737537261617562185247, 579716568874689646501757437321208369959290374622113, 812123531133033168142883595132576030926156951442592543, 1137701760634799958175744040984009442816388920028037144833, 1593803394109789292797576569479470546332292105269153758133407, 2232754928999722159130672264241999812738025912226830847970115584, 3127860409030804832077623110864882699311021626916094040318841846367, 4381811268884140282919155969069868192446292013819070322586801748900801, 6138467661942008213109992731398215374098731636135137023549118053901775903, 859936289009737997841230251314733191189585697229929623343089598925676857\ 4881, 1204682441484683464364257766284002745370263948307629072619488692250\ 8852622617567, 1687636401951607174354363475118209245934741672124114915459\ 1948423807265185066276481, 2364205310054134336585173809675912909548670026\ 0436370416654508151840816551118573142016, 3312008879127708028824865881856\ 4781485527366292764043666969724060788126481246419278707073, 4639784357433\ 985514633579783342721488093846906680396507737077533745532742038715512880\ 9566943]