The first, 6, weighted-generating functions for the number of tilings of an m\
     by n rectangle by rectangles, according to the number of tiles is

[(x-1)/(t*x+x-1), (2*t*x^2-2*t*x+x^2-2*x+1)/(3*t^2*x^2-t^2*x+3*t*x^2-3*t*x+x^2-\
2*x+1), (4*t^3*x^3-4*t^3*x^2+9*t^2*x^3-12*t^2*x^2+5*t*x^3+3*t^2*x-10*t*x^2+x^3+
5*t*x-3*x^2+3*x-1)/(8*t^4*x^3-4*t^4*x^2+20*t^3*x^3-15*t^3*x^2+16*t^2*x^3+t^3*x-\
21*t^2*x^2+6*t*x^3+5*t^2*x-12*t*x^2+x^3+6*t*x-3*x^2+3*x-1), (90*t^8*x^6-100*t^8
*x^5+460*t^7*x^6+10*t^8*x^4-638*t^7*x^5+971*t^6*x^6+186*t^7*x^4-1684*t^6*x^5+
1112*t^5*x^6-8*t^7*x^3+803*t^6*x^4-2422*t^5*x^5+765*t^4*x^6-90*t^6*x^3+1664*t^5
*x^4-2088*t^4*x^5+330*t^3*x^6-366*t^5*x^3+1950*t^4*x^4-1116*t^3*x^5+89*t^2*x^6+
12*t^5*x^2-696*t^4*x^3+1372*t^3*x^4-368*t^2*x^5+14*t*x^6+69*t^4*x^2-720*t^3*x^3
+582*t^2*x^4-70*t*x^5+x^6+138*t^3*x^2-428*t^2*x^3+140*t*x^4-6*x^5-4*t^3*x+137*t
^2*x^2-140*t*x^3+15*x^4-12*t^2*x+70*t*x^2-20*x^3-14*t*x+15*x^2-6*x+1)/(225*t^9*
x^6-160*t^9*x^5+1175*t^8*x^6+15*t^9*x^4-1100*t^8*x^5+2606*t^7*x^6+222*t^8*x^4-\
3250*t^7*x^5+3246*t^6*x^6-7*t^8*x^3+1071*t^7*x^4-5361*t^6*x^5+2536*t^5*x^6-79*t
^7*x^3+2624*t^6*x^4-5451*t^5*x^5+1312*t^4*x^6-371*t^6*x^3+3761*t^5*x^4-3584*t^4
*x^5+458*t^3*x^6+6*t^6*x^2-893*t^5*x^3+3365*t^4*x^4-1554*t^3*x^5+106*t^2*x^6+47
*t^5*x^2-1227*t^4*x^3+1921*t^3*x^4-439*t^2*x^5+15*t*x^6+135*t^4*x^2-1019*t^3*x^
3+696*t^2*x^4-75*t*x^5+x^6-t^4*x+201*t^3*x^2-514*t^2*x^3+150*t*x^4-6*x^5-7*t^3*
x+166*t^2*x^2-150*t*x^3+15*x^4-15*t^2*x+75*t*x^2-20*x^3-15*t*x+15*x^2-6*x+1), (
15552*t^17*x^10-18144*t^17*x^9+164160*t^16*x^10+2592*t^17*x^8-218556*t^16*x^9+
796737*t^15*x^10+57240*t^16*x^8-1216011*t^15*x^9+2365500*t^14*x^10-2844*t^16*x^
7+466905*t^15*x^8-4150453*t^14*x^9+4820959*t^13*x^10-48237*t^15*x^7+2129133*t^
14*x^8-9735938*t^13*x^9+7171839*t^12*x^10+606*t^15*x^6-358361*t^14*x^7+6348495*
t^13*x^8-16667512*t^12*x^9+8081532*t^11*x^10+14211*t^14*x^6-1552839*t^13*x^7+
13398157*t^12*x^8-21586080*t^11*x^9+7061959*t^10*x^10-30*t^14*x^5+120842*t^13*x
^6-4444300*t^12*x^7+20957056*t^11*x^8-21632590*t^10*x^9+4856933*t^9*x^10-1519*t
^13*x^5+560281*t^12*x^6-9007663*t^11*x^7+24999386*t^10*x^8-17014060*t^9*x^9+
2651458*t^8*x^10-18508*t^12*x^5+1658039*t^11*x^6-13492458*t^10*x^7+23150583*t^9
*x^8-10585612*t^8*x^9+1152491*t^7*x^10+43*t^12*x^4-103997*t^11*x^5+3399196*t^10
*x^6-15341027*t^9*x^7+16813432*t^8*x^8-5224137*t^7*x^9+397958*t^6*x^10+1113*t^
11*x^4-344645*t^10*x^5+5065588*t^9*x^6-13454614*t^8*x^7+9616246*t^7*x^8-2040057
*t^6*x^9+108235*t^5*x^10+9167*t^10*x^4-756081*t^9*x^5+5648022*t^8*x^6-9175126*t
^7*x^7+4324592*t^6*x^8-625020*t^5*x^9+22792*t^4*x^10-15*t^10*x^3+38340*t^9*x^4-\
1168516*t^8*x^5+4788486*t^7*x^6-4870068*t^6*x^7+1517289*t^5*x^8-147713*t^4*x^9+
3604*t^3*x^10-276*t^9*x^3+97728*t^8*x^4-1317437*t^7*x^5+3107131*t^6*x^6-1999929
*t^5*x^7+408601*t^4*x^8-26127*t^3*x^9+405*t^2*x^10-1898*t^8*x^3+166094*t^7*x^4-\
1103337*t^6*x^5+1539560*t^5*x^6-625990*t^4*x^7+81998*t^3*x^8-3275*t^2*x^9+29*t*
x^10-6636*t^7*x^3+197398*t^6*x^4-690190*t^5*x^5+575455*t^4*x^6-145166*t^3*x^7+
11620*t^2*x^8-261*t*x^9+x^10+19*t^7*x^2-13786*t^6*x^3+168005*t^5*x^4-320342*t^4
*x^5+158046*t^3*x^6-23660*t^2*x^7+1044*t*x^8-10*x^9+169*t^6*x^2-18509*t^5*x^3+
102983*t^4*x^4-107884*t^3*x^5+30310*t^2*x^6-2436*t*x^7+45*x^8+559*t^5*x^2-16766
*t^4*x^3+44842*t^3*x^4-25130*t^2*x^5+3654*t*x^6-120*x^7+985*t^4*x^2-10338*t^3*x
^3+13300*t^2*x^4-3654*t*x^5+210*x^6-5*t^4*x+1046*t^3*x^2-4220*t^2*x^3+2436*t*x^
4-252*x^5-21*t^3*x+685*t^2*x^2-1044*t*x^3+210*x^4-35*t^2*x+261*t*x^2-120*x^3-29
*t*x+45*x^2-10*x+1)/(46656*t^18*x^10-38880*t^18*x^9+493776*t^17*x^10+5184*t^18*
x^8-479844*t^17*x^9+2414601*t^16*x^10+103068*t^17*x^8-2762808*t^16*x^9+7263723*
t^15*x^10-4392*t^17*x^7+843561*t^16*x^8-9826758*t^15*x^9+15095057*t^14*x^10-\
71736*t^16*x^7+3996593*t^15*x^8-24147516*t^14*x^9+23063173*t^13*x^10+870*t^16*x
^6-540886*t^15*x^7+12578203*t^14*x^8-43510740*t^13*x^9+26910802*t^12*x^10+17946
*t^15*x^6-2455182*t^14*x^7+28278671*t^13*x^8-59614619*t^12*x^9+24580825*t^11*x^
10-42*t^15*x^5+152794*t^14*x^6-7501379*t^13*x^7+47444931*t^12*x^8-63596799*t^11
*x^9+17866504*t^10*x^10-1664*t^14*x^5+744876*t^13*x^6-16430461*t^12*x^7+
61096824*t^11*x^8-53666408*t^10*x^9+10442367*t^9*x^10-19389*t^13*x^5+2378588*t^
12*x^6-26842419*t^11*x^7+61514696*t^10*x^8-36189055*t^9*x^9+4935752*t^8*x^10+44
*t^13*x^4-114088*t^12*x^5+5348602*t^11*x^6-33564630*t^10*x^7+49003365*t^9*x^8-\
19612022*t^8*x^9+1889151*t^7*x^10+955*t^12*x^4-412285*t^11*x^5+8848301*t^10*x^6
-32663314*t^9*x^7+31095614*t^8*x^8-8554179*t^7*x^9+583493*t^6*x^10+7999*t^11*x^
4-1009042*t^10*x^5+11069254*t^9*x^6-24983298*t^8*x^7+15753964*t^7*x^8-2993660*t
^6*x^9+144072*t^5*x^10-11*t^11*x^3+36681*t^10*x^4-1767738*t^9*x^5+10649048*t^8*
x^6-15084878*t^7*x^7+6357278*t^6*x^8-833202*t^5*x^9+27936*t^4*x^10-182*t^10*x^3
+106450*t^9*x^4-2290473*t^8*x^5+7948853*t^7*x^6-7183835*t^6*x^7+2026913*t^5*x^8
-181296*t^4*x^9+4123*t^3*x^10-1329*t^9*x^3+210799*t^8*x^4-2236499*t^7*x^5+
4613876*t^6*x^6-2679943*t^5*x^7+502380*t^4*x^8-29915*t^3*x^9+438*t^2*x^10-5427*
t^8*x^3+297230*t^7*x^4-1660222*t^6*x^5+2073051*t^5*x^6-771489*t^4*x^7+93988*t^3
*x^8-3543*t^2*x^9+30*t*x^10+7*t^8*x^2-13724*t^7*x^3+305646*t^6*x^4-936997*t^5*x
^5+711594*t^4*x^6-166628*t^3*x^7+12576*t^2*x^8-270*t*x^9+x^10+82*t^7*x^2-22947*
t^6*x^3+231647*t^5*x^4-398151*t^4*x^5+181762*t^3*x^6-25620*t^2*x^7+1080*t*x^8-\
10*x^9+371*t^6*x^2-26433*t^5*x^3+129096*t^4*x^4-124418*t^3*x^5+32844*t^2*x^6-\
2520*t*x^7+45*x^8+893*t^5*x^2-21375*t^4*x^3+51940*t^3*x^4-27258*t^2*x^5+3780*t*
x^6-120*x^7-t^5*x+1314*t^4*x^2-12068*t^3*x^3+14448*t^2*x^4-3780*t*x^5+210*x^6-9
*t^4*x+1243*t^3*x^2-4596*t^2*x^3+2520*t*x^4-252*x^5-27*t^3*x+750*t^2*x^2-1080*t
*x^3+210*x^4-39*t^2*x+270*t*x^2-120*x^3-30*t*x+45*x^2-10*x+1), (36578304000*t^
39*x^20-54393292800*t^39*x^19+820087027200*t^38*x^20+20253542400*t^39*x^18-\
1318973564160*t^38*x^19+8879593165920*t^37*x^20-2506291200*t^39*x^17+
593098060800*t^38*x^18-15455863475400*t^37*x^19+61886025307530*t^36*x^20+
67737600*t^39*x^16-99957634560*t^38*x^17+8184825564632*t^37*x^18-\
116620959240194*t^36*x^19+312199359345855*t^35*x^20+5842368000*t^38*x^16-\
1757451953608*t^37*x^17+71442890081958*t^36*x^18-637032560411570*t^35*x^19+
1215483255656829*t^34*x^20-96257280*t^38*x^15+153700740616*t^37*x^16-\
18812402170318*t^36*x^17+445504613968692*t^35*x^18-2685341480945184*t^34*x^19+
3802480219700859*t^33*x^20-4843656112*t^37*x^15+2210076107056*t^36*x^16-\
140272848674192*t^35*x^17+2122177098889667*t^34*x^18-9093646514718545*t^33*x^19
+9826492731491328*t^32*x^20+39613952*t^37*x^14-107433225472*t^36*x^15+
20990759867853*t^35*x^16-784492430443613*t^34*x^17+8056218579414982*t^33*x^18-\
25428701639027419*t^32*x^19+21402163968074467*t^31*x^20+1809426672*t^36*x^14-\
1428021529582*t^35*x^15+144659708737806*t^34*x^16-3447690879447774*t^33*x^17+
25091248496613040*t^32*x^18-59898408190407552*t^31*x^19+39886089479715092*t^30*
x^20-6287232*t^36*x^13+39022315544*t^35*x^14-12998036697675*t^34*x^15+
765309120930830*t^33*x^16-12293018316765610*t^32*x^17+65480218811696885*t^31*x^
18-120653513908714142*t^30*x^19+64357370715904991*t^29*x^20-325221288*t^35*x^13
+519518771874*t^34*x^14-87352492644422*t^33*x^15+3227207851333028*t^32*x^16-\
36393650235454944*t^31*x^17+145478449791245433*t^30*x^18-210264284102841420*t^
29*x^19+90748465456083459*t^28*x^20+338688*t^35*x^12-7660385328*t^34*x^13+
4790983707073*t^33*x^14-454907404886478*t^32*x^15+11142266982269630*t^31*x^16-\
91033200950546284*t^30*x^17+278575234731612010*t^29*x^18-319969171673052554*t^
28*x^19+112669040612606377*t^27*x^20+26415624*t^34*x^12-109802001451*t^33*x^13+
32745076167515*t^32*x^14-1898447962285356*t^31*x^15+32135129967448086*t^30*x^16
-195006814447060790*t^29*x^17+464280931014304243*t^28*x^18-428349932460864197*t
^27*x^19+123920426359175270*t^26*x^20+785773120*t^33*x^12-1079998168771*t^32*x^
13+173520006721694*t^31*x^14-6506861111988690*t^30*x^15+78623805741308212*t^29*
x^16-361592960768241885*t^28*x^17+678764851419408188*t^27*x^18-\
507529653549790046*t^26*x^19+121343381488192784*t^25*x^20-714672*t^33*x^11+
13240033885*t^32*x^12-7810051546207*t^31*x^13+736214912281324*t^30*x^14-\
18661393736464967*t^29*x^15+165189886359765276*t^28*x^16-585383355546241303*t^
27*x^17+876033032992724724*t^26*x^18-534857309787281224*t^25*x^19+
106215475772332698*t^24*x^20-36790518*t^32*x^11+147512514804*t^31*x^12-\
43477306309766*t^30*x^13+2561592593807032*t^29*x^14-45439670183905733*t^28*x^15
+300957014188875996*t^27*x^16-833174567844654969*t^26*x^17+1003312149475984034*
t^25*x^18-503356071789867319*t^24*x^19+83384834576608916*t^23*x^20-842118609*t^
31*x^11+1180706398912*t^30*x^12-192558793785514*t^29*x^13+7444652482156048*t^28
*x^14-95031505235574231*t^27*x^15+479234304373706874*t^26*x^16-\
1048510815673289351*t^25*x^17+1024003758513044313*t^24*x^18-424413647108385109*
t^23*x^19+58864631749634711*t^22*x^20+535188*t^31*x^10-11602952694*t^30*x^11+
7156108839357*t^29*x^12-695453034827988*t^28*x^13+18334414570713094*t^27*x^14-\
172300896919131763*t^26*x^15+671289228175245497*t^25*x^16-1172094025761300480*t
^24*x^17+934553280277676919*t^23*x^18-321447633352692052*t^22*x^19+
37443355960765817*t^21*x^20+23422729*t^30*x^10-109279400825*t^29*x^11+
34069887521854*t^28*x^12-2087476570605405*t^27*x^13+38705203575229466*t^26*x^14
-272886510274531634*t^25*x^15+831519162315851894*t^24*x^16-1168253087480000559*
t^23*x^17+764771806691323222*t^22*x^18-219136780229786490*t^21*x^19+
21493149350164220*t^20*x^20+468258014*t^29*x^10-755257578342*t^28*x^11+
130910934643152*t^27*x^12-5285698605084391*t^26*x^13+70693139305825482*t^25*x^
14-379867405401059464*t^24*x^15+914729306612753214*t^23*x^16-\
1041369726940734170*t^22*x^17+562354122822564801*t^21*x^18-134663958412131215*t
^20*x^19+11144581864520443*t^19*x^20-176100*t^29*x^9+5725343952*t^28*x^10-\
4010095438591*t^27*x^11+414445141904857*t^26*x^12-11425486909184546*t^25*x^13+
112553777506728919*t^24*x^14-467112690382455576*t^23*x^15+896740060203015417*t^
22*x^16-832130738572568064*t^21*x^17+372155610104942887*t^20*x^18-\
74670547186987075*t^19*x^19+5222616000029237*t^18*x^20-7568330*t^28*x^9+
48327750325*t^27*x^10-16895158495300*t^26*x^11+1098645980582529*t^25*x^12-\
21286199453339961*t^24*x^13+157173133259943342*t^23*x^14-509469895148784056*t^
22*x^15+785554463974648425*t^21*x^16-597127418887978047*t^20*x^17+
221891205133145550*t^19*x^18-37379281231257396*t^18*x^19+2211995891338540*t^17*
x^20-145300545*t^27*x^9+301317173899*t^26*x^10-57858096284653*t^25*x^11+
2470111990845655*t^24*x^12-34446224449256593*t^23*x^13+193459587958206257*t^22*
x^14-494460984740235529*t^21*x^15+616203600234245825*t^20*x^16-\
385270652529536996*t^19*x^17+119263331134287144*t^18*x^18-16893102424326681*t^
17*x^19+846343431734077*t^16*x^20+27140*t^27*x^8-1684057007*t^26*x^9+
1449657965483*t^25*x^10-164056082807714*t^24*x^11+4758940526789686*t^23*x^12-\
48723223685500724*t^22*x^13+210731507302466562*t^21*x^14-428110706619515229*t^
20*x^15+433467782734725161*t^19*x^16-223665278540961900*t^18*x^17+
57791453900434953*t^17*x^18-6889425971802044*t^16*x^19+292239245875903*t^15*x^
20+1298386*t^26*x^8-13369257100*t^25*x^9+5550078098278*t^24*x^10-\
390798944501483*t^23*x^11+7922094923894999*t^22*x^12-60545749469762391*t^21*x^
13+203779854023856986*t^20*x^14-331283131470573543*t^19*x^15+273702886532771873
*t^18*x^16-116856983651905340*t^17*x^17+25236632820930579*t^16*x^18-\
2532920872626896*t^15*x^19+90922402228414*t^14*x^20+26034603*t^25*x^8-\
77992014165*t^24*x^9+17300507248695*t^23*x^10-791156541339871*t^22*x^11+
11471214286239816*t^21*x^12-66364850591328518*t^20*x^13+175357010627893586*t^19
*x^14-229406142205275409*t^18*x^15+155189695770499434*t^17*x^16-\
54928788084409630*t^16*x^17+9922054498869455*t^15*x^18-838193889766688*t^14*x^
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