Sequences Enumerating Labeled trees where all vertices are restricted to have degrees in a given set S for all possible subsets of three or more of, {1, 2, 3, 4, 5, 6, 7}, that must include 1 of course By Shalosh B. Ekhad In the list below we give the first, 30, terms , followed by the linear recurrence, followed by the asympotics. ------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3], then the first, 30, terms are [0, 1, 3, 16, 120, 1170, 14070, 201600, 3356640, 63730800, 1359666000, 32212857600, 839350512000, 23860289653200, 734964075846000, 24388126963200000, 867393811956672000, 32919980214689568000, 1328053572854936928000, 56752039046079336960000, 2561025679541636186880000, 121704153816369677192640000, 6075255163076158964224320000, 317833937499587100371619840000, 17390368207022203566036864000000, 993258771718938181397164224000000, 59115387099369331836549347904000000, 3660342897283890088556413320192000000, 235438414892005020971515517781504000000, 15709607781751483290617894230142880000000] The enumerating sequence satisfies the recurrence -(n-1)*(n+2)*(n+1)/(n+4)*a(n)-(n+2)*(2*n+3)/(n+4)*a(n+1)+a(n+2) = 0 The asymptotics is CONST*n!*2.414213562^n/n^(5/2) where the CONST. is roughly, 1.042 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 4], then the first, 30, terms are [0, 1, 3, 12, 65, 480, 4620, 54320, 745920, 11692800, 206514000, 4064860800, 88259371200, 2095070577600, 53968306392000, 1499410104192000, 44696761261152000, 1423059292182528000, 48195832584227328000, 1730144058671508480000, 65623332286465873920000, 2622397193856944824320000, 110124935079948075006720000, 4848496036747958859264000000, 223326483425789188693248000000, 10740928221971019655494912000000, 538440526872229039428674304000000, 28087628922692777708337779712000000, 1522351971008326954013726976768000000, 85610533117981348799552645775360000000] The enumerating sequence satisfies the recurrence -17/4*n*(n-1)*(n+3)*(n+2)*(n+1)/(n+4)/(2*n+5)*a(n)+3*n*(2*n+3)*(n+3)*(n+2)/(n+4 )/(2*n+5)*a(n+1)-2*(n+3)*(3*n^2+12*n+10)/(n+4)/(2*n+5)*a(n+2)+a(n+3) = 0 The asymptotics is CONST*n!*2.040041912^n/n^(5/2) where the CONST. is roughly, 0.8218 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 5], then the first, 30, terms are [0, 1, 3, 12, 60, 366, 2730, 25200, 287280, 3934350, 62182890, 1096464600, 21129708600, 440671631400, 9908773875000, 239808711542400, 6234049982160000, 173516376645612000, 5149423328301252000, 162213134225622480000, 5401769292906818832000, 189500705036717084010000, 6984018122815983158910000, 269796476408121256754160000, 10903240664178662894094000000, 460135281914408048214766500000, 20243490484357908839863291500000, 926924668738497361069299162000000, 44105004435931095257796573378000000, 2177633700080818413877438885890000000] The enumerating sequence satisfies the recurrence 49/9*n*(n-1)*(n+4)*(n+3)*(n+1)^2/(3*n+10)/(3*n+14)*a(n)-18*n*(2*n+3)*(n+4)*(n+3 )*(n+1)/(3*n+10)/(3*n+14)*a(n+1)+(n+4)*(n+3)*(n+1)*(54*n^2+216*n+203)/(3*n+10)/ (3*n+14)/(n+2)*a(n+2)-(n+4)*(36*n^3+270*n^2+640*n+483)/(3*n+10)/(3*n+14)/(n+2)* a(n+3)+a(n+4) = 0 The asymptotics is CONST*n!*1.792804743^n/n^(5/2) where the CONST. is roughly, 0.9798 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 6], then the first, 30, terms are [0, 1, 3, 12, 60, 360, 2527, 20496, 192024, 2096640, 26943840, 407167992, 7111560456, 139685049504, 3004291049760, 69365586850560, 1698445034526240, 43822617781662720, 1188462775488468480, 33867347566045593600, 1014828431711007513600, 32001315528532764173760, 1062210039083417598454080, 37087141810547334559173120, 1359848204052189163760448000, 52236308308019568910265088000, 2096362641460689409922732217600, 87652907350655176960444427212800, 3808762556857481353886800255411200, 171637346576171263747252184027136000] The enumerating sequence satisfies the recurrence -6769/192*n*(n-1)*(n+5)*(n+4)*(n+2)^2*(n+1)^2/(2*n+11)/(4*n+7)/(4*n+17)*a(n)+80 *n*(2*n+3)*(n+5)*(n+4)*(n+1)*(n+2)^2/(2*n+11)/(4*n+7)/(4*n+17)*a(n+1)-10*(n+5)* (n+4)*(n+2)*(n+1)*(32*n^2+128*n+123)/(2*n+11)/(4*n+7)/(4*n+17)*a(n+2)+5*(n+5)*( n+4)*(n+2)*(64*n^3+480*n^2+1154*n+891)/(2*n+11)/(4*n+7)/(4*n+17)/(n+3)*a(n+3)-( n+5)*(160*n^4+1920*n^3+8330*n^2+15480*n+10377)/(2*n+11)/(4*n+7)/(4*n+17)/(n+3)* a(n+4)+a(n+5) = 0 The asymptotics is CONST*n!*1.633119605^n/n^(5/2) where the CONST. is roughly, 1.249 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 7], then the first, 30, terms are [0, 1, 3, 12, 60, 360, 2520, 20168, 181944, 1834560, 20623680, 259459200, 3684320640, 59571916404, 1099085567580, 22929726980160, 531217184097600, 13385086031116800, 360254874581521920, 10221201422818626240, 303068047048830423360, 9342667702939213843200, 298595308543770849811200, 9882452209118070495513600, 338664387598579807180416000, 12024059309341204958603971200, 442724815358903467797366960000, 16923749462975329390184408640000, 672297997311714622300143531840000, 27771298003710083814180760771200000] The enumerating sequence satisfies the recurrence 15301/25*n*(n-1)*(n+6)*(n+5)*(n+3)^2*(n+2)^2*(n+1)^2/(5*n+8)/(5*n+14)/(5*n+26)/ (5*n+32)*a(n)-1875*n*(2*n+3)*(n+6)*(n+5)*(n+1)*(n+3)^2*(n+2)^2/(5*n+8)/(5*n+14) /(5*n+26)/(5*n+32)*a(n+1)+125*(n+6)*(n+5)*(n+2)*(n+1)*(75*n^2+300*n+289)*(n+3)^ 2/(5*n+8)/(5*n+14)/(5*n+26)/(5*n+32)*a(n+2)-250*(2*n+5)*(n+6)*(n+5)*(n+3)*(n+2) *(25*n^2+125*n+139)/(5*n+8)/(5*n+14)/(5*n+26)/(5*n+32)*a(n+3)+(n+6)*(n+5)*(n+3) *(9375*n^4+112500*n^3+488625*n^2+906750*n+602059)/(5*n+8)/(5*n+14)/(n+4)/(5*n+ 26)/(5*n+32)*a(n+4)-(2*n+7)*(n+6)*(1875*n^4+26250*n^3+130375*n^2+269500*n+ 193309)/(5*n+8)/(5*n+14)/(n+4)/(5*n+26)/(5*n+32)*a(n+5)+a(n+6) = 0 The asymptotics is CONST*n!*1.524148279^n/n^(5/2) where the CONST. is roughly, 1.595 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 3, 4], then the first, 30, terms are [0, 1, 0, 4, 5, 90, 420, 5600, 52920, 730800, 10256400, 164656800, 2939937000, 56142886800, 1190377188000, 26706423744000, 651950675376000, 16838032891680000, 465265914841728000, 13582334058733440000, 419602472077108320000, 13648210938612169920000, 466562313357047107200000, 16722953511562989250560000, 626966656458493006656000000, 24547892138945458291008000000, 1001638073959707644460000000000, 42533250612485662567137600000000, 1876441464260951549655854160000000, 85894888327259311897211815200000000] The enumerating sequence satisfies the recurrence -15/4*n*(n-1)*(n+2)*(n+1)/(2*n+7)*a(n)-3/4*n*(n+2)*(6*n^2+33*n+41)/(2*n+7)/(n+4 )*a(n+1)+3/4*(n+5)*(n+3)*n/(2*n+7)/(n+4)*a(n+2)+a(n+3) = 0 The asymptotics is CONST*n!*1.660324085^n/n^(5/2) where the CONST. is roughly, 0.3994 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 3, 5], then the first, 30, terms are [0, 1, 0, 4, 0, 96, 0, 5880, 0, 683550, 0, 129313800, 0, 36223387200, 0, 14094984103200, 0, 7275575535228000, 0, 4811642732136240000, 0, 3967691129991388800000, 0, 3991375137025731054600000, 0, 4811392816343744493762900000, 0, 6846804324864301430743110000000, 0, 11357131848177021686356777200000000] The enumerating sequence satisfies the recurrence -8/9*n*(n-1)*(n+3)*(n+2)*(n+1)^2/(3*n+14)/(3*n+4)*a(n)-4/3*(n+3)*(n+1)*(15*n^2+ 60*n+56)/(3*n+14)/(3*n+4)*a(n+2)+a(n+4) = 0 The asymptotics is CONST*n!*1.505261323^n/n^(5/2) where the CONST. is roughly, 0.9900 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 3, 6], then the first, 30, terms are [0, 1, 0, 4, 0, 90, 7, 5040, 1512, 529200, 374220, 89829432, 118918800, 22493815344, 48697248600, 7808087327040, 25287667685280, 3591887262163200, 16337096550238464, 2116434676644288000, 12896061183069704640, 1555262915756579789760, 12239201969882093606400, 1395071334433926943073280, 13769644072443316817760000, 1500782440156971752017036800, 18138461814750789518143737600, 1907889375967783614385499520000, 27675021470394503278833093734400, 2830440330434431198094064854400000] The enumerating sequence satisfies the recurrence -10345/192*n*(n-1)*(n+5)*(n+3)*(339416*n^4+64089014*n^3+610903684*n^2+ 1934691319*n+1996422108)*(n+2)^2*(n+1)^2/(4*n+17)/(2*n+11)/(4*n+19)/(339416*n^4 +62731350*n^3+420673138*n^2+903793329*n+608884875)*a(n)+75/32*n*(n+5)*(n+3)*(n+ 1)*(10182480*n^5+1937944140*n^4+20207899430*n^3+75998239570*n^2+117690171773*n+ 60971559951)*(n+2)^2/(4*n+17)/(2*n+11)/(4*n+19)/(339416*n^4+62731350*n^3+ 420673138*n^2+903793329*n+608884875)*a(n+1)+135/16*(n+5)*(n+3)*(n+2)*(n+1)*( 1018248*n^6+196340034*n^5+3171513220*n^4+20426626665*n^3+63781156902*n^2+ 95975042071*n+55599346800)/(4*n+17)/(2*n+11)/(4*n+19)/(339416*n^4+62731350*n^3+ 420673138*n^2+903793329*n+608884875)*a(n+2)-5/2*(n+5)*(n+2)*(13576640*n^7+ 2665385360*n^6+44851534880*n^5+319556016680*n^4+1198473033884*n^3+2473539564884 *n^2+2646846615909*n+1138911464889)/(4*n+17)/(2*n+11)/(4*n+19)/(339416*n^4+ 62731350*n^3+420673138*n^2+903793329*n+608884875)*a(n+3)-675*(2*n+9)*(4*n+15)*( n+5)*(n+2)*(484880*n^3+4727790*n^2+14995239*n+15444263)/(4*n+17)/(2*n+11)/(4*n+ 19)/(n+4)/(339416*n^4+62731350*n^3+420673138*n^2+903793329*n+608884875)*a(n+4)+ a(n+5) = 0 The asymptotics is CONST*n!*1.442871802^n/n^(5/2) where the CONST. is roughly, 0.6166 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 3, 7], then the first, 30, terms are [0, 1, 0, 4, 0, 90, 0, 5048, 0, 531720, 0, 90644400, 0, 22808709924, 0, 7956976547520, 0, 3677659803417600, 0, 2175912673823528640, 0, 1604214273229037870400, 0, 1442207249064746131968000, 0, 1553135575923884622561091200, 0, 1973998285733550786293263680000, 0, 2923877577810685824922548754800000] The enumerating sequence satisfies the recurrence -66564/25*n*(n-1)*(n+6)*(n+5)*(39875*n^3+477050*n^2+1826200*n+2209216)*(n+3)^2* (n+2)^2*(n+1)^2/(5*n+24)/(5*n+26)/(5*n+28)/(5*n+32)/(39875*n^3+237800*n^2+ 396500*n+146016)*a(n)+144*(n+6)*(n+5)*(n+2)*(n+1)*(917125*n^5+14640650*n^4+ 88480900*n^3+249807568*n^2+322610688*n+146779136)*(n+3)^2/(5*n+24)/(5*n+26)/(5* n+28)/(5*n+32)/(39875*n^3+237800*n^2+396500*n+146016)*a(n+2)-6*(n+6)*(n+5)*(n+3 )*(14953125*n^7+358331250*n^6+3582287500*n^5+19272855000*n^4+59840473600*n^3+ 106070044800*n^2+97598166016*n+34742550528)/(5*n+24)/(5*n+26)/(5*n+28)/(n+4)/(5 *n+32)/(39875*n^3+237800*n^2+396500*n+146016)*a(n+4)+a(n+6) = 0 The asymptotics is CONST*n!*1.421588928^n/n^(5/2) where the CONST. is roughly, 1.437 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 4, 5], then the first, 30, terms are [0, 1, 0, 0, 5, 6, 0, 560, 2520, 3150, 277200, 2772000, 9909900, 382582200, 6558552000, 45909864000, 1190713524000, 29010241260000, 332427775680000, 7313411064960000, 218439330236127000, 3655609865197290000, 79288023410275680000, 2621933205677145360000, 58840167700503189000000, 1375163261065433139300000, 47731998302599911300000000, 1340980838219350528200000000, 35272433824747740756135000000, 1264547753437112917923150000000] The enumerating sequence satisfies the recurrence 4/9*n*(n-1)*(n+3)*(n+2)*(100*n^3+1080*n^2+3825*n+4424)*(n+1)^2/(3*n+13)/(3*n+14 )/(100*n^3+780*n^2+1965*n+1579)*a(n)-8/9*n*(n+3)*(n+2)*(n+1)*(1200*n^4+14760*n^ 3+65300*n^2+121653*n+79117)/(3*n+13)/(3*n+14)/(100*n^3+780*n^2+1965*n+1579)*a(n +1)+4/9*n*(n+3)*(n+1)*(300*n^4+4440*n^3+23885*n^2+55577*n+47237)/(3*n+13)/(3*n+ 14)/(100*n^3+780*n^2+1965*n+1579)*a(n+2)-4/9*(n+1)*(800*n^5+13840*n^4+94320*n^3 +315947*n^2+519106*n+333774)/(3*n+13)/(3*n+14)/(100*n^3+780*n^2+1965*n+1579)*a( n+3)+a(n+4) = 0 The asymptotics is CONST*n!*1.141560030^n/n^(5/2) where the CONST. is roughly, 0.4226 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 4, 6], then the first, 30, terms are [0, 1, 0, 0, 5, 0, 7, 560, 0, 5040, 277200, 16632, 7927920, 369969600, 98378280, 24485260800, 1041138178080, 741015475200, 132971110272000, 5330811810804480, 7958320949779200, 1167306230997466560, 45019805491544083200, 121596902292273408000, 15542901818948641737600, 584756963505700457472000, 2583344513927327287737600, 298819293694842910429440000, 11086741941001874039692800000, 74377966674967103888707968000] The enumerating sequence satisfies the recurrence -5425/192*n*(n-1)*(n+4)*(n+3)*(559504*n^4+7701272*n^3+39452069*n^2+89556637*n+ 76398060)*(n+1)^2*(n+2)^2/(4*n+15)/(2*n+11)/(4*n+17)/(559504*n^4+5463256*n^3+ 19705277*n^2+31518299*n+19151724)*a(n)-25/24*n*(n+4)*(n+3)*(n+1)*(33570240*n^5+ 512431680*n^4+3042678980*n^3+8756280500*n^2+12148462949*n+6439165197)*(n+2)^2/( 4*n+15)/(2*n+11)/(4*n+17)/(559504*n^4+5463256*n^3+19705277*n^2+31518299*n+ 19151724)*a(n+1)-25/24*(n+1)*(n+2)*(n+3)*(n+4)*(22380160*n^6+397571520*n^5+ 2868941720*n^4+10775995240*n^3+22269550127*n^2+24106257675*n+10744117164)/(4*n+ 15)/(2*n+11)/(4*n+17)/(559504*n^4+5463256*n^3+19705277*n^2+31518299*n+19151724) *a(n+2)+25/9*(n+4)*(n+2)*(n+1)*(4476032*n^6+90704384*n^5+762137736*n^4+ 3399354724*n^3+8489578294*n^2+11254115691*n+6182886321)/(4*n+15)/(2*n+11)/(4*n+ 17)/(559504*n^4+5463256*n^3+19705277*n^2+31518299*n+19151724)*a(n+3)+5/3*(n+2)* (17904128*n^7+425481984*n^6+4269282320*n^5+23424683880*n^4+75819024227*n^3+ 144571080651*n^2+150117810120*n+65343575880)/(4*n+15)/(2*n+11)/(4*n+17)/(559504 *n^4+5463256*n^3+19705277*n^2+31518299*n+19151724)*a(n+4)+a(n+5) = 0 The asymptotics is CONST*n!*1.072064682^n/n^(5/2) where the CONST. is roughly, 0.4919 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 4, 7], then the first, 30, terms are [0, 1, 0, 0, 5, 0, 0, 568, 0, 0, 286440, 0, 0, 390233844, 0, 0, 1121529208800, 0, 0, 5860721239813440, 0, 0, 50424920320312137600, 0, 0, 665369141234802808003200, 0, 0, 12766007270999032342011720000, 0] The enumerating sequence satisfies the recurrence 20736/25*n*(n+5)*(n-1)*(n+6)*(n+1)^2*(n+2)^2*(n+3)^2/(5*n+32)/(5*n+29)/(5*n+23) /(5*n+26)*a(n)-18*(n+6)*(n+5)*(n+3)*(n+2)*(80*n^3+840*n^2+3114*n+4055)/(5*n+32) /(5*n+29)/(5*n+23)/(5*n+26)*a(n+3)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 5, 6], then the first, 30, terms are [0, 1, 0, 0, 0, 6, 7, 0, 0, 3150, 13860, 16632, 0, 12612600, 122972850, 423783360, 514594080, 192972780000, 3324277756800, 22438874858400, 70220478968640, 8131921578340560, 215903590717815000, 2399938861031712000, 13799648450932344000, 788485627441605693600, 28881732283490423937600, 477028756678820346000000, 4330591495414682097600000, 158066589582635896562400000] The enumerating sequence satisfies the recurrence -5875/576*n*(n-1)*(n+4)*(n+3)*(1795144*n^6+42820752*n^5+419294500*n^4+ 2153481135*n^3+6105351436*n^2+9035281773*n+5434316100)*(n+1)^2*(n+2)^2/(2*n+11) /(4*n+21)/(4*n+19)/(1795144*n^6+32049888*n^5+232117900*n^4+868607775*n^3+ 1759394671*n^2+1811177202*n+729173520)*a(n)-125/192*n*(n+4)*(n+3)*(n+1)*( 35902880*n^7+910269360*n^6+9629355600*n^5+54810220220*n^4+180016434547*n^3+ 337735184998*n^2+330132410991*n+126640950300)*(n+2)^2/(2*n+11)/(4*n+21)/(4*n+19 )/(1795144*n^6+32049888*n^5+232117900*n^4+868607775*n^3+1759394671*n^2+ 1811177202*n+729173520)*a(n+1)+25/192*n*(n+1)*(n+2)*(n+3)*(n+4)*(17951440*n^7+ 500013280*n^6+5878847920*n^5+37822591630*n^4+143784686329*n^3+322857785104*n^2+ 396203980989*n+204717098844)/(2*n+11)/(4*n+21)/(4*n+19)/(1795144*n^6+32049888*n ^5+232117900*n^4+868607775*n^3+1759394671*n^2+1811177202*n+729173520)*a(n+2)-25 /64*(n+4)*(n+2)*(n+1)*(21541728*n^8+653870256*n^7+8592136448*n^6+63827632156*n^ 5+293123023807*n^4+851991625730*n^3+1530226670091*n^2+1552179758712*n+ 680417120880)/(2*n+11)/(4*n+21)/(4*n+19)/(1795144*n^6+32049888*n^5+232117900*n^ 4+868607775*n^3+1759394671*n^2+1811177202*n+729173520)*a(n+3)+5/64*(n+2)*( 403907400*n^9+13673743200*n^8+203777619140*n^7+1754095634655*n^6+9608303981939* n^5+34722512147043*n^4+82763094581993*n^3+125438128893318*n^2+109672950657744*n +42136197062400)/(2*n+11)/(4*n+21)/(4*n+19)/(1795144*n^6+32049888*n^5+232117900 *n^4+868607775*n^3+1759394671*n^2+1811177202*n+729173520)*a(n+4)+a(n+5) = 0 The asymptotics is CONST*n!*.8483939058^n/n^(5/2) where the CONST. is roughly, 0.5246 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 5, 7], then the first, 30, terms are [0, 1, 0, 0, 0, 6, 0, 8, 0, 3150, 0, 27720, 0, 12696684, 0, 353152800, 0, 197089532640, 0, 12485596163040, 0, 8403700839534000, 0, 1005307722898839360, 0, 804466403589015367200, 0, 160853174957399324400000, 0, 151153281437039180447148000] The enumerating sequence satisfies the recurrence -13924/25*n*(n-1)*(n+5)*(n+4)*(115625*n^4+2220000*n^3+15804820*n^2+49437264*n+ 57343744)*(n+1)^2*(n+2)^2*(n+3)^2/(5*n+32)/(5*n+28)/(5*n+26)/(115625*n^4+ 1295000*n^3+5259820*n^2+9157984*n+5778496)/(5*n+24)*a(n)-2/5*(n+5)*(n+4)*(n+2)* (n+1)*(95390625*n^6+2213062500*n^5+20523151500*n^4+96542979200*n^3+240104691040 *n^2+294101858432*n+134615842816)*(n+3)^2/(5*n+32)/(5*n+28)/(5*n+26)/(115625*n^ 4+1295000*n^3+5259820*n^2+9157984*n+5778496)/(5*n+24)*a(n+2)+1/2*(n+5)*(n+3)*(n +2)*(320859375*n^7+9369093750*n^6+115527375500*n^5+778840326600*n^4+ 3095304479200*n^3+7235646771840*n^2+9182626935296*n+4857717874688)/(5*n+32)/(5* n+28)/(5*n+26)/(115625*n^4+1295000*n^3+5259820*n^2+9157984*n+5778496)/(5*n+24)* a(n+4)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 6, 7], then the first, 30, terms are [0, 1, 0, 0, 0, 0, 7, 8, 0, 0, 0, 16632, 72072, 84084, 0, 0, 514594080, 4940103168, 16621388784, 19554575040, 0, 85825029850560, 1454508400625280, 9599755444126848, 29205605187158400, 34515715221187200, 50323912792490937600, 1327384366410630528000, 14435304984715606992000, 80837707914407399155200] The enumerating sequence satisfies the recurrence 4329/100*n*(n-1)*(n+5)*(n+4)*(5228934375*n^10+248765250000*n^9+5358731512500*n^ 8+68897483986500*n^7+585751932961425*n^6+3440097717358500*n^5+14121688734913960 *n^4+39951533095882864*n^3+74407103520600548*n^2+82201001271384560*n+ 40812264916432128)*(n+1)^2*(n+2)^2*(n+3)^2/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/ (5228934375*n^10+196475906250*n^9+3355146309375*n^8+34355708761500*n^7+ 233715822624675*n^6+1102371048643950*n^5+3640932063609085*n^4+8282308454286024* n^3+12362415534952466*n^2+10882214700911916*n+4270394731492512)*a(n)-27/25*n*(n +5)*(n+4)*(n+1)*(313736062500*n^11+15396519093750*n^10+343870983890625*n^9+ 4622183872693125*n^8+41611914426289500*n^7+263713623877494650*n^6+ 1200642219018789675*n^5+3922898585870900535*n^4+8993407935884599636*n^3+ 13727632750074569404*n^2+12495042545813371736*n+5106097055699033184)*(n+3)^2*(n +2)^2/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/(5228934375*n^10+196475906250*n^9+ 3355146309375*n^8+34355708761500*n^7+233715822624675*n^6+1102371048643950*n^5+ 3640932063609085*n^4+8282308454286024*n^3+12362415534952466*n^2+ 10882214700911916*n+4270394731492512)*a(n+1)+9/25*n*(n+5)*(n+4)*(n+2)*(n+1)*( 78434015625*n^11+4045214812500*n^10+94968049265625*n^9+1341357157635000*n^8+ 12676040650735875*n^7+84184740969455700*n^6+400814633681556675*n^5+ 1366814955382051320*n^4+3266601592859657668*n^3+5200422163279201456*n^2+ 4951548639401648828*n+2130434209997862608)*(n+3)^2/(5*n+28)/(5*n+29)/(5*n+31)/( 5*n+32)/(5228934375*n^10+196475906250*n^9+3355146309375*n^8+34355708761500*n^7+ 233715822624675*n^6+1102371048643950*n^5+3640932063609085*n^4+8282308454286024* n^3+12362415534952466*n^2+10882214700911916*n+4270394731492512)*a(n+2)-3/50*(n+ 1)*(n+2)*(n+3)*(n+4)*(n+5)*(2143863093750*n^12+115928862609375*n^11+ 2877452965734375*n^10+43381225577677500*n^9+442629523926270000*n^8+ 3220061054334380325*n^7+17118963933647002925*n^6+66955489514837448000*n^5+ 190971362737549048478*n^4+386773010582086280040*n^3+527033697709778963648*n^2+ 432992167463734957824*n+161862871819734005760)/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+ 32)/(5228934375*n^10+196475906250*n^9+3355146309375*n^8+34355708761500*n^7+ 233715822624675*n^6+1102371048643950*n^5+3640932063609085*n^4+8282308454286024* n^3+12362415534952466*n^2+10882214700911916*n+4270394731492512)*a(n+3)+18/25*(n +5)*(n+3)*(n+2)*(836629500000*n^13+48168735000000*n^12+1281365208140625*n^11+ 20862503141227500*n^10+231996832032539625*n^9+1860548418748085600*n^8+ 11066048975131695075*n^7+49386328495142850020*n^6+165207062562268214999*n^5+ 408598866836738239936*n^4+725141003360976706876*n^3+872996577991124620384*n^2+ 637866359780427424320*n+213210930555533352960)/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+ 32)/(5228934375*n^10+196475906250*n^9+3355146309375*n^8+34355708761500*n^7+ 233715822624675*n^6+1102371048643950*n^5+3640932063609085*n^4+8282308454286024* n^3+12362415534952466*n^2+10882214700911916*n+4270394731492512)*a(n+4)-6/25*(n+ 3)*(12047464800000*n^14+747843375600000*n^13+21558802843875000*n^12+ 382672466067667875*n^11+4672944393538087575*n^10+41522877443118649200*n^9+ 276771248182954287990*n^8+1404938795598479072301*n^7+5452919962724890075801*n^6 +16085540603139468978736*n^5+35453637635892244479050*n^4+ 56530324507598090150864*n^3+61537046462274347239224*n^2+40858559750131786242144 *n+12457905664269800851200)/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/(5228934375*n^ 10+196475906250*n^9+3355146309375*n^8+34355708761500*n^7+233715822624675*n^6+ 1102371048643950*n^5+3640932063609085*n^4+8282308454286024*n^3+ 12362415534952466*n^2+10882214700911916*n+4270394731492512)*a(n+5)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3, 4], then the first, 30, terms are [0, 1, 3, 16, 125, 1290, 16590, 255920, 4609080, 94978800, 2204848800, 56949631200, 1620257238600, 50353072770000, 1697293852254000, 61682913163872000, 2404329063248112000, 100063862555713056000, 4428789208130038752000, 207722713185360172800000, 10292152416527198337120000, 537184746331461797849280000, 29459678797959895162117440000, 1693611870329614431532976640000, 101851015232620436039345088000000, 6395057909890043467680839232000000, 418486823618120042177079271584000000, 28494974731169644390364306386752000000, 2015805191037831850078647493251984000000, 147950021363423827353837395054090400000000] The enumerating sequence satisfies the recurrence -1/2*n*(n-1)*(n+1)*(n+3)^2/(n+4)/(2*n+7)*a(n)-1/2*n*(n+3)/(n+4)/(2*n+7)*a(n+1)-\ 1/4*(n+3)*(21*n^3+147*n^2+326*n+224)/(n+4)/(n+2)/(2*n+7)*a(n+2)+a(n+3) = 0 The asymptotics is CONST*n!*2.660324085^n/n^(5/2) where the CONST. is roughly, 0.5055 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3, 5], then the first, 30, terms are [0, 1, 3, 16, 120, 1176, 14280, 207480, 3515400, 68118750, 1486713690, 36103082400, 965761196400, 28221827634000, 894612348630000, 30577827966285600, 1121095370254860000, 43890717759941052000, 1827526484494066356000, 80645297058051485760000, 3759652073792707092000000, 184645454826917409605280000, 9528860462088151731587040000, 515521793821135826058614280000, 29176874556418241422484715000000, 1724155281905078594439967799700000, 106191252387899334923345399821500000, 6805583776568075515719101190792000000, 453157948647777327631264058034564000000, 31306363004174687460742110258238380000000] The enumerating sequence satisfies the recurrence -107/9*n*(n-1)*(n+3)*(n+2)*(n+1)^2/(3*n+14)/(3*n+4)*a(n)+2*n*(2*n+3)*(n+3)*(n+2 )*(n+1)/(3*n+14)/(3*n+4)*a(n+1)+1/3*(n+3)*(n+1)*(102*n^2+408*n+349)/(3*n+14)/(3 *n+4)*a(n+2)-(2*n+5)*(18*n^2+90*n+83)/(3*n+14)/(3*n+4)*a(n+3)+a(n+4) = 0 The asymptotics is CONST*n!*2.505261323^n/n^(5/2) where the CONST. is roughly, 0.6386 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3, 6], then the first, 30, terms are [0, 1, 3, 16, 120, 1170, 14077, 201936, 3368736, 64134000, 1373013180, 32664931992, 855240442056, 24443875664208, 757430035001640, 25295685484829760, 905866770919350240, 34630349600257931520, 1407713794166896280064, 60634330096389429219840, 2758752175800352562550720, 132213878894536147573824960, 6657476152971635678544137280, 351406584382039758693831790080, 19402853135869179898450008480000, 1118514454710794546402660267404800, 67200208745246054153581153305657600, 4200928874327782840186740170466892800, 272842253967142489364381930012355225600, 18384969590852840870438418557616565632000] The enumerating sequence satisfies the recurrence -5929/192*n*(n-1)*(n+5)*(n+3)*(4408*n^4+265582*n^3+2390492*n^2+7541291*n+ 7897386)*(n+2)^2*(n+1)^2/(4*n+19)/(2*n+11)/(4*n+17)/(4408*n^4+247950*n^3+ 1620194*n^2+3539421*n+2485413)*a(n)-77/32*n*(n+5)*(n+3)*(n+1)*(220400*n^5+ 13609700*n^4+146716450*n^3+629515750*n^2+1199204691*n+844937793)*(n+2)^2/(4*n+ 19)/(2*n+11)/(4*n+17)/(4408*n^4+247950*n^3+1620194*n^2+3539421*n+2485413)*a(n+1 )+1/16*(n+5)*(n+3)*(n+2)*(n+1)*(374680*n^6+24073190*n^5-40945260*n^4-2824744325 *n^3-17000708536*n^2-39190967529*n-31929391368)/(4*n+19)/(2*n+11)/(4*n+17)/( 4408*n^4+247950*n^3+1620194*n^2+3539421*n+2485413)*a(n+2)+1/2*(n+5)*(n+2)*( 1939520*n^7+131402480*n^6+2080555520*n^5+15163714400*n^4+60076025648*n^3+ 133867841768*n^2+157930983363*n+76806159633)/(4*n+19)/(2*n+11)/(4*n+17)/(4408*n ^4+247950*n^3+1620194*n^2+3539421*n+2485413)*a(n+3)-(n+5)*(705280*n^8+50956480* n^7+964026160*n^6+8745773020*n^5+45037255276*n^4+138996753724*n^3+254726987649* n^2+255325996764*n+107595273618)/(4*n+19)/(2*n+11)/(4*n+17)/(n+4)/(4408*n^4+ 247950*n^3+1620194*n^2+3539421*n+2485413)*a(n+4)+a(n+5) = 0 The asymptotics is CONST*n!*2.442871802^n/n^(5/2) where the CONST. is roughly, 0.8022 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3, 7], then the first, 30, terms are [0, 1, 3, 16, 120, 1170, 14070, 201608, 3357144, 63753480, 1360580760, 32248616400, 840753753840, 23916562113444, 737292284508780, 24488026181498880, 871851366533347200, 33127076934900748800, 1338075580171626616320, 57257126271435550299840, 2587522017819781933450560, 123149926533195124956993600, 6157241526411781052189342400, 322661336339652585088932518400, 17685219341107609617598848576000, 1011922337376717340967467189315200, 60338525812146975845747844909360000, 3743259334957578893675591379726720000, 241247320412940874930671755826324480000, 16129805627513092660598894217499522800000] The enumerating sequence satisfies the recurrence -24389/25*n*(n-1)*(n+6)*(n+5)*(68370671875*n^6+1430811696875*n^5+12163735041875 *n^4+53675218766125*n^3+129421745905050*n^2+161412576897400*n+81293111241408)*( n+1)^2*(n+2)^2*(n+3)^2/(5*n+24)/(5*n+26)/(5*n+28)/(5*n+32)/(68370671875*n^6+ 1020587665625*n^5+6035236635625*n^4+17960982129875*n^3+28095942967300*n^2+ 21683635671300*n+6428355499808)*a(n)-n*(n+6)*(n+5)*(n+1)*(93941303156250*n^7+ 2106847226240625*n^6+19661874854795625*n^5+98818939739180125*n^4+ 288313237389650325*n^3+487028747135253250*n^2+438937087166227992*n+ 160972642651029440)*(n+3)^2*(n+2)^2/(5*n+24)/(5*n+26)/(5*n+28)/(5*n+32)/( 68370671875*n^6+1020587665625*n^5+6035236635625*n^4+17960982129875*n^3+ 28095942967300*n^2+21683635671300*n+6428355499808)*a(n+1)-(n+6)*(n+5)*(n+2)*(n+ 1)*(55585356234375*n^8+1385591334496875*n^7+14869061050878750*n^6+ 90035835399439000*n^5+337673183445835025*n^4+805287058138357925*n^3+ 1192674993692208554*n^2+999184220801703864*n+359630553385225408)*(n+3)^2/(5*n+ 24)/(5*n+26)/(5*n+28)/(5*n+32)/(68370671875*n^6+1020587665625*n^5+6035236635625 *n^4+17960982129875*n^3+28095942967300*n^2+21683635671300*n+6428355499808)*a(n+ 2)-2*(n+6)*(n+5)*(n+3)*(n+2)*(119648675781250*n^9+3401285537890625*n^8+ 42173155029531250*n^7+299166305720218750*n^6+1336891888799393750*n^5+ 3898321854836095625*n^4+7405715705185697750*n^3+8818277640995247000*n^2+ 5951553041995564800*n+1724816050683506176)/(5*n+24)/(5*n+26)/(5*n+28)/(5*n+32)/ (68370671875*n^6+1020587665625*n^5+6035236635625*n^4+17960982129875*n^3+ 28095942967300*n^2+21683635671300*n+6428355499808)*a(n+3)+(n+6)*(n+5)*(n+3)*( 487141037109375*n^10+16040225785546875*n^9+233641025460468750*n^8+ 1979843822318031250*n^7+10791602604746512500*n^6+39463188786156885000*n^5+ 97841577038389226625*n^4+162007144413734410675*n^3+170992824299097473550*n^2+ 103583589878382150952*n+27271259694805482048)/(5*n+24)/(5*n+26)/(5*n+28)/(n+4)/ (5*n+32)/(68370671875*n^6+1020587665625*n^5+6035236635625*n^4+17960982129875*n^ 3+28095942967300*n^2+21683635671300*n+6428355499808)*a(n+4)-(n+6)*( 256390019531250*n^11+9852369205078125*n^10+169205649912890625*n^9+ 1711731724955000000*n^8+11313117914446656250*n^7+51182754759780012500*n^6+ 161340055804606365000*n^5+353301804133146214625*n^4+524865830351676548275*n^3+ 501817637250932263950*n^2+276753622270045549864*n+66476506702224489024)/(5*n+24 )/(5*n+26)/(5*n+28)/(n+4)/(5*n+32)/(68370671875*n^6+1020587665625*n^5+ 6035236635625*n^4+17960982129875*n^3+28095942967300*n^2+21683635671300*n+ 6428355499808)*a(n+5)+a(n+6) = 0 The asymptotics is CONST*n!*2.421588928^n/n^(5/2) where the CONST. is roughly, 0.9378 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 4, 5], then the first, 30, terms are [0, 1, 3, 12, 65, 486, 4830, 59360, 854280, 14014350, 258717690, 5323764600, 120952131300, 3006595792200, 81139567509000, 2362254517022400, 73800885922764000, 2462955318771948000, 87450030435611844000, 3291627712621968720000, 130922747971074538479000, 5486879912937516663570000, 241667735891134807103850000, 11160348279786344995836480000, 539233408679063159124063000000, 27206285337271941408178971300000, 1430797751630925598118850151500000, 78304811981431283628590212002000000, 4452860038058722414936568309553000000, 262734885740016052535329349602410000000] The enumerating sequence satisfies the recurrence 25*n*(n-1)*(n+3)*(n+1)*(40*n^2+320*n+657)*(n+2)^2/(3*n+13)/(3*n+14)/(40*n^2+240 *n+377)*a(n)-10*n*(n+3)*(n+2)*(240*n^4+2760*n^3+11302*n^2+18962*n+10557)/(3*n+ 13)/(3*n+14)/(40*n^2+240*n+377)*a(n+1)+2*(n+3)*(n+1)*(1320*n^4+17160*n^3+82461* n^2+172810*n+132734)/(3*n+13)/(3*n+14)/(40*n^2+240*n+377)*a(n+2)-2/9*(n+2)*( 7120*n^4+103240*n^3+560186*n^2+1348190*n+1214829)/(3*n+13)/(3*n+14)/(40*n^2+240 *n+377)*a(n+3)+a(n+4) = 0 The asymptotics is CONST*n!*2.141560030^n/n^(5/2) where the CONST. is roughly, 0.5788 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 4, 6], then the first, 30, terms are [0, 1, 3, 12, 65, 480, 4627, 54656, 756504, 11980080, 213998400, 4262465592, 93692374776, 2252433447264, 58786185898440, 1655322450862080, 50023502804315040, 1614897034078648320, 55466339190746964480, 2019599093916309185280, 77707036716937537029120, 3150419809034816706548160, 134235014424576305351791680, 5997005023505322660028715520, 280315744907164936591519497600, 13682264395696963253937740390400, 696127275929443346177948536377600, 36857328817910585080780451780812800, 2027694324389399375986851291568051200, 115747592675156731210491514387707264000] The enumerating sequence satisfies the recurrence -4787/576*n*(n-1)*(n+4)*(n+3)*(559504*n^4+8348632*n^3+45871189*n^2+109807653*n+ 96334452)*(n+1)^2*(n+2)^2/(4*n+17)/(2*n+11)/(4*n+15)/(559504*n^4+6110616*n^3+ 24182317*n^2+40873155*n+24608860)*a(n)+1/72*n*(n+4)*(n+3)*(n+1)*(1376379840*n^5 +22602204480*n^4+144036941060*n^3+443371856420*n^2+654905171657*n+367803290517) *(n+2)^2/(4*n+17)/(2*n+11)/(4*n+15)/(559504*n^4+6110616*n^3+24182317*n^2+ 40873155*n+24608860)*a(n+1)-1/72*(n+1)*(n+2)*(n+3)*(n+4)*(4364131200*n^6+ 82575854400*n^5+635964640840*n^4+2548424489720*n^3+5593212541517*n^2+ 6357807293017*n+2912179367100)/(4*n+17)/(2*n+11)/(4*n+15)/(559504*n^4+6110616*n ^3+24182317*n^2+40873155*n+24608860)*a(n+2)+1/9*(n+4)*(n+2)*(649024640*n^7+ 14552097920*n^6+137833617160*n^5+714407241260*n^4+2185802199970*n^3+ 3940813285679*n^2+3866672541098*n+1586882378949)/(4*n+17)/(2*n+11)/(4*n+15)/( 559504*n^4+6110616*n^3+24182317*n^2+40873155*n+24608860)*a(n+3)-1/3*(179041280* n^8+4820057600*n^7+56082423280*n^6+368003819720*n^5+1487777048023*n^4+ 3789021649625*n^3+5924393888225*n^2+5185075680759*n+1936942571916)/(4*n+17)/(2* n+11)/(4*n+15)/(559504*n^4+6110616*n^3+24182317*n^2+40873155*n+24608860)*a(n+4) +a(n+5) = 0 The asymptotics is CONST*n!*2.072064682^n/n^(5/2) where the CONST. is roughly, 0.6839 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 4, 7], then the first, 30, terms are [0, 1, 3, 12, 65, 480, 4620, 54328, 746424, 11712960, 207188520, 4085928000, 88909460640, 2115513163764, 54633922122780, 1522015334199360, 45499765025114400, 1452918222583027200, 49357808609605816320, 1777437052439232824640, 67634552532762223968960, 2711665586925202638931200, 114255269208968812433308800, 5047464189689718303617510400, 233293319545066668843411456000, 11259468500372351168823580867200, 566427216882597388268946755760000, 29652850599628797310485324248640000, 1612964273721374655430817877569160000, 91034761474776616938754519470264000000] The enumerating sequence satisfies the recurrence 72361/25*n*(n-1)*(n+6)*(n+5)*(n+1)^2*(n+2)^2*(n+3)^2/(5*n+23)/(5*n+29)/(5*n+26) /(5*n+32)*a(n)-4035*n*(2*n+5)*(n+6)*(n+5)*(n+1)*(n+3)^2*(n+2)^2/(5*n+23)/(5*n+ 29)/(5*n+26)/(5*n+32)*a(n+1)+(n+6)*(n+5)*(n+2)*(n+1)*(13695*n^2+82170*n+128372) *(n+3)^2/(5*n+23)/(5*n+29)/(5*n+26)/(5*n+32)*a(n+2)-2*(n+6)*(n+5)*(n+3)*(n+2)*( 6970*n^3+73185*n^2+259401*n+310495)/(5*n+23)/(5*n+29)/(5*n+26)/(5*n+32)*a(n+3)+ (n+6)*(n+5)*(n+3)*(9375*n^4+150000*n^3+900375*n^2+2403750*n+2408944)/(5*n+23)/( 5*n+29)/(n+4)/(5*n+26)/(5*n+32)*a(n+4)-(n+6)*(3750*n^5+84375*n^4+756500*n^3+ 3379125*n^2+7521138*n+6674608)/(5*n+23)/(5*n+29)/(n+4)/(5*n+26)/(5*n+32)*a(n+5) +a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 5, 6], then the first, 30, terms are [0, 1, 3, 12, 60, 366, 2737, 25536, 297864, 4216590, 69182190, 1265794992, 25246677456, 543426988104, 12583999037610, 313285118065920, 8376401880887040, 239922182049469920, 7332771062201064480, 238048415853940692000, 8172467754903994782240, 295604441270931385953360, 11231879728610910450561480, 447265388338422408870635520, 18629959767288090893944272000, 810295390458399469301851773600, 36740718806695054821043371597600, 1733931564820623791310094940332800, 85040647692360649433746498427731200, 4328090567718006333774704673635136000] The enumerating sequence satisfies the recurrence -101/18*n*(n-1)*(n+4)*(n+3)*(5609825*n^6+142024350*n^5+1491832700*n^4+ 8323859310*n^3+26030083979*n^2+43279580844*n+29907694080)*(n+2)^2*(n+1)^2/(2*n+ 11)/(4*n+19)/(4*n+21)/(5609825*n^6+108365400*n^5+865858325*n^4+3664575510*n^3+ 8673406124*n^2+10900122816*n+5689756080)*a(n)+1/3*n*(n+4)*(n+3)*(n+1)*( 1009768500*n^7+27079035750*n^6+307014653750*n^5+1904386702825*n^4+6964341543790 *n^3+14968838223626*n^2+17428045476771*n+8417905878960)*(n+2)^2/(2*n+11)/(4*n+ 19)/(4*n+21)/(5609825*n^6+108365400*n^5+865858325*n^4+3664575510*n^3+8673406124 *n^2+10900122816*n+5689756080)*a(n+1)-1/12*(n+4)*(n+3)*(n+2)*(n+1)*(13407481750 *n^8+393068123500*n^7+4978051875250*n^6+35548838369050*n^5+156469528967095*n^4+ 434456220621241*n^3+742821132895845*n^2+714816546158349*n+296375698800720)/(2*n +11)/(4*n+19)/(4*n+21)/(5609825*n^6+108365400*n^5+865858325*n^4+3664575510*n^3+ 8673406124*n^2+10900122816*n+5689756080)*a(n+2)+1/16*(n+4)*(n+2)*(21990514000*n ^9+721664307000*n^8+10422962029750*n^7+86937538975750*n^6+461442753933955*n^5+ 1616172161328907*n^4+3735306174491655*n^3+5494069861352299*n^2+4667699496013116 *n+1745928635512320)/(2*n+11)/(4*n+19)/(4*n+21)/(5609825*n^6+108365400*n^5+ 865858325*n^4+3664575510*n^3+8673406124*n^2+10900122816*n+5689756080)*a(n+3)-1/ 64*(51133554875*n^10+1908154608750*n^9+31793913767500*n^8+311445852131450*n^7+ 1986126072012865*n^6+8615544546800432*n^5+25747130592316952*n^4+ 52349290929217976*n^3+69318986040098928*n^2+53997457621864032*n+ 18797236916129280)/(2*n+11)/(4*n+19)/(4*n+21)/(5609825*n^6+108365400*n^5+ 865858325*n^4+3664575510*n^3+8673406124*n^2+10900122816*n+5689756080)*a(n+4)+a( n+5) = 0 The asymptotics is CONST*n!*1.848393906^n/n^(5/2) where the CONST. is roughly, 0.7744 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 5, 7], then the first, 30, terms are [0, 1, 3, 12, 60, 366, 2730, 25208, 287784, 3954510, 62848170, 1116450720, 21704482800, 456987374844, 10375665480180, 253489417140960, 6649326547063200, 186680273674134240, 5587160234183367840, 177504611638722207840, 5962485735497331200160, 211039018819322454961200, 7848627443457359096538000, 305988750950188399124391360, 12480186824032777377215688000, 531551579986531566833226055200, 23601138713804986318000876860000, 1090628071907584717309057295040000, 52373121559757092524557467660320000, 2609774318180274348651670543969548000] The enumerating sequence satisfies the recurrence 56277/50*n*(n-1)*(n+5)*(n+4)*(788779296875*n^8+27134007812500*n^7+ 403750823031250*n^6+3393443641462500*n^5+17620132346075975*n^4+ 57893865256200440*n^3+117620158867809244*n^2+135245319111910704*n+ 67504156437496320)*(n+1)^2*(n+2)^2*(n+3)^2/(5*n+28)/(5*n+24)/(5*n+26)/(5*n+32)/ (788779296875*n^8+20823773437500*n^7+235898588656250*n^6+1496581226712500*n^5+ 5814700761575975*n^4+14178274458709040*n^3+21233454762757524*n^2+ 17934408850027136*n+6589225236323520)*a(n)-72/5*n*(n+5)*(n+4)*(n+1)*( 473267578125000*n^9+16990306054687500*n^8+266671100850000000*n^7+ 2399225801896484375*n^6+13620675297239819375*n^5+50537616837913143875*n^4+ 122373047982111134225*n^3+186125163487206886170*n^2+160920778756778639160*n+ 59996525317993870464)*(n+3)^2*(n+2)^2/(5*n+28)/(5*n+24)/(5*n+26)/(5*n+32)/( 788779296875*n^8+20823773437500*n^7+235898588656250*n^6+1496581226712500*n^5+ 5814700761575975*n^4+14178274458709040*n^3+21233454762757524*n^2+ 17934408850027136*n+6589225236323520)*a(n+1)+3/10*(n+5)*(n+4)*(n+2)*(n+1)*( 45670321289062500*n^10+1753740337500000000*n^9+29833915683890234375*n^8+ 295854052415356562500*n^7+1892506365460846858750*n^6+8152796614197357862500*n^5 +23933229326911451909675*n^4+47230004353432598030600*n^3+ 59900103227015094903244*n^2+44034205385648518129392*n+14227310561449307896320)* (n+3)^2/(5*n+28)/(5*n+24)/(5*n+26)/(5*n+32)/(788779296875*n^8+20823773437500*n^ 7+235898588656250*n^6+1496581226712500*n^5+5814700761575975*n^4+ 14178274458709040*n^3+21233454762757524*n^2+17934408850027136*n+ 6589225236323520)*a(n+2)-1/2*(n+5)*(n+4)*(n+3)*(n+2)*(28474932617187500*n^11+ 1193099676660156250*n^10+22426582545217187500*n^9+249495666249019296875*n^8+ 1824290711669014260000*n^7+9200209559095735981500*n^6+32635799414520416850400*n ^5+81382360926874691112275*n^4+139730594635368855184280*n^3+ 157247531946044976057676*n^2+104356181230865370491376*n+30940065468261299870208 )/(5*n+28)/(5*n+24)/(5*n+26)/(5*n+32)/(788779296875*n^8+20823773437500*n^7+ 235898588656250*n^6+1496581226712500*n^5+5814700761575975*n^4+14178274458709040 *n^3+21233454762757524*n^2+17934408850027136*n+6589225236323520)*a(n+3)+1/2*(n+ 5)*(n+3)*(16978474365234375*n^12+787801210546875000*n^11+16573252501001562500*n ^10+208927946558325625000*n^9+1756908512991553330625*n^8+ 10376907798408267508500*n^7+44116974381784639229850*n^6+ 135959256137607459797100*n^5+301288544493591082634770*n^4+ 468022084514600496583136*n^3+483657693118300415444392*n^2+ 298584446355889191333408*n+83332039060955965271040)/(5*n+28)/(5*n+24)/(5*n+26)/ (5*n+32)/(788779296875*n^8+20823773437500*n^7+235898588656250*n^6+ 1496581226712500*n^5+5814700761575975*n^4+14178274458709040*n^3+ 21233454762757524*n^2+17934408850027136*n+6589225236323520)*a(n+4)-( 2957922363281250*n^13+153516170654296875*n^12+3642192129335937500*n^11+ 52277631848736328125*n^10+506222471660351312500*n^9+3490393793420305556250*n^8+ 17620399440457872465000*n^7+65907010471322786795625*n^6+ 182562286034940531648550*n^5+369733164148003867456545*n^4+ 532028854681790555398352*n^3+515138363003223801795924*n^2+ 300992686313396670177552*n+80259800535088296998400)/(5*n+28)/(5*n+24)/(5*n+26)/ (5*n+32)/(788779296875*n^8+20823773437500*n^7+235898588656250*n^6+ 1496581226712500*n^5+5814700761575975*n^4+14178274458709040*n^3+ 21233454762757524*n^2+17934408850027136*n+6589225236323520)*a(n+5)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 6, 7], then the first, 30, terms are [0, 1, 3, 12, 60, 360, 2527, 20504, 192528, 2116800, 27609120, 427126392, 7682442768, 155679928404, 3450719972700, 81922065825600, 2057437871929440, 54330197459586048, 1505321246430329328, 43768510092419112000, 1337084331816644611200, 42971396544076929888960, 1453843662338854263597120, 51763112274896081474583168, 1936641046607231862145248000, 75958021110724366033339920000, 3114185055780754568991916137600, 133069259143964925933703692940800, 5910175089460117644670647876931200, 272226962227797329429670233291251200] The enumerating sequence satisfies the recurrence 5725/4*n*(n-1)*(n+5)*(n+4)*(15059331*n^10+713442060*n^9+16264795680*n^8+ 229906998432*n^7+2167688377917*n^6+13917335398956*n^5+60844194533732*n^4+ 178607491067312*n^3+340670173042420*n^2+391079858110960*n+212929916313600)*(n+1 )^2*(n+2)^2*(n+3)^2/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/(15059331*n^10+ 562848750*n^9+10521487035*n^8+123665427432*n^7+956986994403*n^6+4914522288594*n ^5+16777902720137*n^4+38244337763964*n^3+58857199789774*n^2+60251254829220*n+ 32792947104960)*a(n)-75/2*n*(n+5)*(n+4)*(n+1)*(2861272890*n^11+139845900735*n^ 10+3293527595310*n^9+48364690919130*n^8+479003500203588*n^7+3284303209568085*n^ 6+15687210597586772*n^5+51944043849195622*n^4+117371225050609632*n^3+ 174890527342632620*n^2+158351807097010336*n+67556312888944320)*(n+3)^2*(n+2)^2/ (5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/(15059331*n^10+562848750*n^9+10521487035*n^ 8+123665427432*n^7+956986994403*n^6+4914522288594*n^5+16777902720137*n^4+ 38244337763964*n^3+58857199789774*n^2+60251254829220*n+32792947104960)*a(n+1)+ 15/2*(n+5)*(n+4)*(n+2)*(n+1)*(31474001790*n^12+1616989912560*n^11+ 40075509778515*n^10+623199733248240*n^9+6620550085609704*n^8+49623592405302786* n^7+265596091351550437*n^6+1016876823049245334*n^5+2767956274024770794*n^4+ 5272047755410220120*n^3+6788003953008610248*n^2+5434645184869643232*n+ 2093447130952803840)*(n+3)^2/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/(15059331*n^10 +562848750*n^9+10521487035*n^8+123665427432*n^7+956986994403*n^6+4914522288594* n^5+16777902720137*n^4+38244337763964*n^3+58857199789774*n^2+60251254829220*n+ 32792947104960)*a(n+2)-1/2*(n+5)*(n+4)*(n+3)*(n+2)*(557647026930*n^13+ 30601112183775*n^12+810385191467010*n^11+13542587240603445*n^10+ 156515645413531512*n^9+1299262222656225099*n^8+7875046217107710038*n^7+ 35058489448352800643*n^6+114511763677915317158*n^5+272335818589655728526*n^4+ 463789302658845218264*n^3+546066290648625533840*n^2+408205589406551885760*n+ 149318190215030016000)/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/(15059331*n^10+ 562848750*n^9+10521487035*n^8+123665427432*n^7+956986994403*n^6+4914522288594*n ^5+16777902720137*n^4+38244337763964*n^3+58857199789774*n^2+60251254829220*n+ 32792947104960)*a(n+3)+3/5*(n+5)*(n+3)*(307586835675*n^14+18263096103600*n^13+ 523009997673795*n^12+9492610688366580*n^11+120380741045670195*n^10+ 1113254137920363636*n^9+7659079454536971653*n^8+39545652257440598344*n^7+ 153525433975539413254*n^6+446595392801562521344*n^5+964692932736474018708*n^4+ 1520806478102108199936*n^3+1688040152674868382000*n^2+1208694764132677718080*n+ 428228985417004108800)/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/(15059331*n^10+ 562848750*n^9+10521487035*n^8+123665427432*n^7+956986994403*n^6+4914522288594*n ^5+16777902720137*n^4+38244337763964*n^3+58857199789774*n^2+60251254829220*n+ 32792947104960)*a(n+4)-3/25*(539997490998*n^15+35032561479945*n^14+ 1094686947629100*n^13+21740418796473621*n^12+304053361729596396*n^11+ 3138907008431284527*n^10+24478494683724573736*n^9+145798439783358498331*n^8+ 665879186238212192638*n^7+2329570083917357117068*n^6+6210590890277909285084*n^5 +12493582862328711975068*n^4+18632245989581486588528*n^3+ 19859093310098484713840*n^2+13823916057487866333120*n+4796265203228297664000)/( 5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/(15059331*n^10+562848750*n^9+10521487035*n^8 +123665427432*n^7+956986994403*n^6+4914522288594*n^5+16777902720137*n^4+ 38244337763964*n^3+58857199789774*n^2+60251254829220*n+32792947104960)*a(n+5)+a (n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 3, 4, 5], then the first, 30, terms are [0, 1, 0, 4, 5, 96, 420, 6440, 55440, 885150, 11503800, 206929800, 3544440900, 73220347200, 1536340806000, 36361268143200, 895996917816000, 24067397878524000, 678318876275040000, 20454364144534320000, 647627848594774047000, 21696796412967836160000, 761682411990993352860000, 28089660942200858920920000, 1082348648359622425782000000, 43586777189058524860338900000, 1828444028146879598885430000000, 79850220942831621547855590000000, 3622255848644666183076291165000000, 170522773104074386765088919600000000] The enumerating sequence satisfies the recurrence -92/9*n*(n-1)*(n+3)*(n+2)*(116*n^3+1280*n^2+4799*n+6162)*(n+1)^2/(3*n+14)/(3*n+ 13)/(116*n^3+932*n^2+2587*n+2527)*a(n)-4/9*n*(n+1)*(n+2)*(n+3)*(7656*n^4+95964* n^3+445294*n^2+905897*n+676968)/(3*n+14)/(3*n+13)/(116*n^3+932*n^2+2587*n+2527) *a(n+1)-4/9*(n+3)*(n+1)*(5220*n^5+78480*n^4+471269*n^3+1425183*n^2+2190976*n+ 1379742)/(3*n+14)/(3*n+13)/(116*n^3+932*n^2+2587*n+2527)*a(n+2)+2/9*(n+1)*(4408 *n^5+77292*n^4+540046*n^3+1882893*n^2+3281635*n+2290470)/(3*n+14)/(3*n+13)/(116 *n^3+932*n^2+2587*n+2527)*a(n+3)+a(n+4) = 0 The asymptotics is CONST*n!*1.707993250^n/n^(5/2) where the CONST. is roughly, 0.3357 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 3, 4, 6], then the first, 30, terms are [0, 1, 0, 4, 5, 90, 427, 5600, 54432, 735840, 10630620, 167999832, 3066783720, 58158884784, 1250978208480, 28069640639040, 691502890958880, 17934982800134400, 498849364394244864, 14649718425397735680, 455203622929898082240, 14901809173447613376960, 512360464041721889193600, 18483146549116859281351680, 697094152100097920786217600, 27467504565796368490005388800, 1127650404987434691318853497600, 48186536759712663873667666560000, 2139131814482845267451139187814400, 98536764045707280805356786509568000] The enumerating sequence satisfies the recurrence -9115/64*n*(n-1)*(n+4)*(n+3)*(441590648*n^5+8203157282*n^4+60147743337*n^3+ 217183042687*n^2+385263333724*n+267936651660)*(n+1)^2*(n+2)^2/(2*n+11)/(4*n+17) /(4*n+19)/(441590648*n^5+5995204042*n^4+31751020689*n^3+81542849888*n^2+ 100735802473*n+47470183920)*a(n)-25/32*n*(n+4)*(n+3)*(n+1)*(135126738288*n^6+ 2712856235724*n^5+22204942062164*n^4+94577763156492*n^3+220193106666289*n^2+ 264089026431624*n+126235879690635)*(n+2)^2/(2*n+11)/(4*n+17)/(4*n+19)/( 441590648*n^5+5995204042*n^4+31751020689*n^3+81542849888*n^2+100735802473*n+ 47470183920)*a(n+1)-5/16*(n+1)*(n+2)*(n+3)*(n+4)*(176194668552*n^7+ 3977838429726*n^6+37930034241955*n^5+198010852039830*n^4+611006693730378*n^3+ 1113453269915334*n^2+1107501931039565*n+462290396392200)/(2*n+11)/(4*n+17)/(4*n +19)/(441590648*n^5+5995204042*n^4+31751020689*n^3+81542849888*n^2+100735802473 *n+47470183920)*a(n+2)+5/36*(n+4)*(n+2)*(38859977024*n^8+1013327668496*n^7+ 10982556429672*n^6+63702803615280*n^5+210238770376080*n^4+377638714662948*n^3+ 280398015341527*n^2-86152078323566*n-180496309927065)/(2*n+11)/(4*n+17)/(4*n+19 )/(441590648*n^5+5995204042*n^4+31751020689*n^3+81542849888*n^2+100735802473*n+ 47470183920)*a(n+3)+5/6*(n+2)*(28261801472*n^8+807620080768*n^7+9947676453136*n ^6+68829261047740*n^5+291719329548910*n^4+772160915759314*n^3+1238316876640601* n^2+1088284369958142*n+393580523340405)/(2*n+11)/(4*n+17)/(4*n+19)/(441590648*n ^5+5995204042*n^4+31751020689*n^3+81542849888*n^2+100735802473*n+47470183920)*a (n+4)+a(n+5) = 0 The asymptotics is CONST*n!*1.671181366^n/n^(5/2) where the CONST. is roughly, 0.3739 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 3, 4, 7], then the first, 30, terms are [0, 1, 0, 4, 5, 90, 420, 5608, 52920, 733320, 10265640, 165488400, 2947864920, 56496123684, 1196279884800, 26908568406720, 656714387128800, 16991237841177600, 469723106458045440, 13730314302631967040, 424542599806683758400, 13824090735056248046400, 473063756620244463273600, 16973770893363807659136000, 637063996788440634276864000, 24969626913162457696477507200, 1019972573924304136091155200000, 43358626290497251635927305280000, 1914980110078542826855030414920000, 87755589905163672291288932242800000] The enumerating sequence satisfies the recurrence -126504/25*n*(n-1)*(n+6)*(n+4)*(4028795000000*n^8+139581595000000*n^7+ 2098786350550000*n^6+17882676486430000*n^5+94406354488526500*n^4+ 316118109760527700*n^3+655494899344086245*n^2+769408810295856069*n+ 391354675863772620)*(n+1)^2*(n+2)^2*(n+3)^2/(5*n+28)/(5*n+29)/(5*n+26)/(5*n+32) /(4028795000000*n^8+107351235000000*n^7+1234521445550000*n^6+7995559358130000*n ^5+31871427139626500*n^4+80003472964721700*n^3+123415320152612145*n^2+ 106913428069010679*n+39809566704121596)*a(n)-432/5*n*(n+6)*(n+4)*(n+1)*( 604319250000000*n^9+21843718125000000*n^8+346643600925500000*n^7+ 3166926788820950000*n^6+18337590425621845000*n^5+69708013189753233500*n^4+ 173729241083365334050*n^3+273291520896276670765*n^2+245755773060072490033*n+ 96016800736307394012)*(n+3)^2*(n+2)^2/(5*n+28)/(5*n+29)/(5*n+26)/(5*n+32)/( 4028795000000*n^8+107351235000000*n^7+1234521445550000*n^6+7995559358130000*n^5 +31871427139626500*n^4+80003472964721700*n^3+123415320152612145*n^2+ 106913428069010679*n+39809566704121596)*a(n+1)-36*(n+6)*(n+4)*(n+2)*(n+1)*( 777557435000000*n^10+30049477575000000*n^9+517814095119150000*n^8+ 5238851426410290000*n^7+34458880607973654500*n^6+153967244680967706100*n^5+ 473286810921473109985*n^4+988420470113142480287*n^3+1342439249585996815858*n^2+ 1071041914828765356048*n+381349173962002264800)*(n+3)^2/(5*n+28)/(5*n+29)/(5*n+ 26)/(5*n+32)/(4028795000000*n^8+107351235000000*n^7+1234521445550000*n^6+ 7995559358130000*n^5+31871427139626500*n^4+80003472964721700*n^3+ 123415320152612145*n^2+106913428069010679*n+39809566704121596)*a(n+2)+6*(n+6)*( n+4)*(n+3)*(n+2)*(2256125200000000*n^11+95086632200000000*n^10+ 1797068782978000000*n^9+20084169435721800000*n^8+147319961714184540000*n^7+ 743721820720346802000*n^6+2632663113047585604200*n^5+6521992359106983630940*n^4 +11054292393259061660750*n^3+12169107571573535384397*n^2+7795958568134520356715 *n+2187470763597627920340)/(5*n+28)/(5*n+29)/(5*n+26)/(5*n+32)/(4028795000000*n ^8+107351235000000*n^7+1234521445550000*n^6+7995559358130000*n^5+ 31871427139626500*n^4+80003472964721700*n^3+123415320152612145*n^2+ 106913428069010679*n+39809566704121596)*a(n+3)+6*(n+6)*(n+3)*(1510798125000000* n^12+70472675625000000*n^11+1492940628231250000*n^10+18983629765656250000*n^9+ 161274184716469687500*n^8+963718028772098587500*n^7+4150581847380926184375*n^6+ 12970951242180311838375*n^5+29165680714075166424275*n^4+45972118952458842492305 *n^3+48165833146665710263986*n^2+30082393655090016108168*n+ 8459401254894744468480)/(5*n+28)/(5*n+29)/(5*n+26)/(5*n+32)/(4028795000000*n^8+ 107351235000000*n^7+1234521445550000*n^6+7995559358130000*n^5+31871427139626500 *n^4+80003472964721700*n^3+123415320152612145*n^2+106913428069010679*n+ 39809566704121596)*a(n+4)-120*(n+6)*(n+3)*(5*n+27)*(10629680000000*n^9+ 392571199250000*n^8+6385436116500000*n^7+60023030621670000*n^6+ 359233137235482000*n^5+1419183579262706175*n^4+3699891276474837820*n^3+ 6136674605071386247*n^2+5874924466117770900*n+2473294868978497536)/(5*n+28)/(5* n+29)/(n+5)/(5*n+26)/(5*n+32)/(4028795000000*n^8+107351235000000*n^7+ 1234521445550000*n^6+7995559358130000*n^5+31871427139626500*n^4+ 80003472964721700*n^3+123415320152612145*n^2+106913428069010679*n+ 39809566704121596)*a(n+5)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 3, 5, 6], then the first, 30, terms are [0, 1, 0, 4, 0, 96, 7, 5880, 1512, 683550, 388080, 129330432, 130810680, 36241549344, 57674266650, 14111935437600, 32582038769280, 7292248383420000, 23072083285515264, 4829975292895552800, 20073542110130789280, 3990655419053532989760, 21086525218747602814200, 4024291354477879838633280, 26343335697751531643184000, 4865297104057347942715130400, 38631896080022969969582937600, 6947248969119164413563535680000, 65750321346578230337900458934400, 11569024529798518302528084476400000] The enumerating sequence satisfies the recurrence -6535/576*n*(n-1)*(n+4)*(n+3)*(237881440136*n^6+5477434676688*n^5+ 51686178627935*n^4+255541770052200*n^3+697384055457959*n^2+994962856309962*n+ 579158328119760)*(n+1)^2*(n+2)^2/(2*n+11)/(4*n+19)/(4*n+21)/(237881440136*n^6+ 4050146035872*n^5+27867226846535*n^4+98813773504620*n^3+189669691904129*n^2+ 186035225781528*n+72484382606940)*a(n)+25/96*n*(n+4)*(n+3)*(n+1)*( 70412906280256*n^7+1726940023720032*n^6+17736094115870600*n^5+98681070061421771 *n^4+320535182219693820*n^3+606308217343728211*n^2+616502659919839230*n+ 258903536358261600)*(n+2)^2/(2*n+11)/(4*n+19)/(4*n+21)/(237881440136*n^6+ 4050146035872*n^5+27867226846535*n^4+98813773504620*n^3+189669691904129*n^2+ 186035225781528*n+72484382606940)*a(n+1)-5/96*(n+1)*(n+2)*(n+3)*(n+4)*( 784770871008664*n^8+21209140482428368*n^7+245915938479882793*n^6+ 1596685821712075075*n^5+6345368697896983531*n^4+15794122330160099617*n^3+ 24024034692419567052*n^2+20389735801320159060*n+7377119155361844000)/(2*n+11)/( 4*n+19)/(4*n+21)/(237881440136*n^6+4050146035872*n^5+27867226846535*n^4+ 98813773504620*n^3+189669691904129*n^2+186035225781528*n+72484382606940)*a(n+2) -5/32*(n+4)*(n+2)*(166517008095200*n^9+5083081834395600*n^8+67663434532979380*n ^7+514716319137495545*n^6+2461354562089789054*n^5+7656190869642207470*n^4+ 15449479901191977800*n^3+19436906850659289065*n^2+13775886196758440646*n+ 4168288464232394880)/(2*n+11)/(4*n+19)/(4*n+21)/(237881440136*n^6+4050146035872 *n^5+27867226846535*n^4+98813773504620*n^3+189669691904129*n^2+186035225781528* n+72484382606940)*a(n+3)+5/64*(n+2)*(236216270055048*n^9+7801255334501664*n^8+ 112895908135929823*n^7+938453270106683004*n^6+4930908527216870842*n^5+ 16952826798058617156*n^4+38053877949927360427*n^3+53626040749740533376*n^2+ 42888614287997685780*n+14754238310723688000)/(2*n+11)/(4*n+19)/(4*n+21)/( 237881440136*n^6+4050146035872*n^5+27867226846535*n^4+98813773504620*n^3+ 189669691904129*n^2+186035225781528*n+72484382606940)*a(n+4)+a(n+5) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 3, 5, 7], then the first, 30, terms are [0, 1, 0, 4, 0, 96, 0, 5888, 0, 686070, 0, 130173120, 0, 36586714164, 0, 14288299945920, 0, 7403775329190240, 0, 4916025904323774240, 0, 4070452649858573378400, 0, 4111965641718108573759360, 0, 4977941773678512302959495200, 0, 7114477978621690908541390080000, 0, 11852760388645812506030236023948000] The enumerating sequence satisfies the recurrence -784/25*n*(n-1)*(n+5)*(n+4)*(7625*n^4+137250*n^3+909560*n^2+2622656*n+2765056)* (n+1)^2*(n+2)^2*(n+3)^2/(5*n+28)/(5*n+24)/(5*n+26)/(5*n+32)/(7625*n^4+76250*n^3 +269060*n^2+387416*n+181984)*a(n)-8/5*(n+5)*(n+4)*(n+2)*(n+1)*(1944375*n^6+ 42776250*n^5+377209300*n^4+1693240780*n^3+4035447560*n^2+4760705312*n+ 2111805696)*(n+3)^2/(5*n+28)/(5*n+24)/(5*n+26)/(5*n+32)/(7625*n^4+76250*n^3+ 269060*n^2+387416*n+181984)*a(n+2)-1/2*(n+5)*(n+3)*(18871875*n^8+566156250*n^7+ 7272071000*n^6+52061043600*n^5+226163553200*n^4+606486418080*n^3+970601861888*n ^2+833546085888*n+285046874112)/(5*n+28)/(5*n+24)/(5*n+26)/(5*n+32)/(7625*n^4+ 76250*n^3+269060*n^2+387416*n+181984)*a(n+4)+a(n+6) = 0 The asymptotics is CONST*n!*1.508789333^n/n^(5/2) where the CONST. is roughly, 0.9586 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 3, 6, 7], then the first, 30, terms are [0, 1, 0, 4, 0, 90, 7, 5048, 1512, 531720, 374220, 90661032, 118990872, 22826872068, 48795626880, 7973504098560, 25395732442080, 3693226068499968, 16456787170872048, 2192086614821463360, 13039381180644698880, 1623169675146862402560, 12429682208265379337280, 1467433112030360342195328, 14053375913518851743433600, 1591260282631485637566595200, 18613266953427414993742617600, 2039248493812895189356179648000, 28566674683606307814588989366400, 3049819572130618900972005722275200] The enumerating sequence satisfies the recurrence -184491/100*n*(n-1)*(n+5)*(n+4)*(9500172633901*n^10+372884394450800*n^9+ 6371324064097860*n^8+61945921116594492*n^7+375376298764784175*n^6+ 1455489980809441956*n^5+3541800876656521940*n^4+4972322504878256968*n^3+ 3012690481032655284*n^2-626164399873351536*n-1140548061619520640)*(n+3)^2*(n+2) ^2*(n+1)^2/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/(9500172633901*n^10+ 277882668111790*n^9+3442872282566205*n^8+23259146087972292*n^7+ 90824671862614821*n^6+191891774276798826*n^5+127892502039436175*n^4-\ 306011809039968592*n^3-713190700371088422*n^2-490677429344377476*n-\ 68266472254220160)*a(n)+27/50*n*(n+5)*(n+4)*(n+1)*(29355533438754090*n^11+ 1196246079011103135*n^10+21560999169785190970*n^9+226584596543775350775*n^8+ 1542241256018801223446*n^7+7151356214645726813485*n^6+23167982725515520513254*n ^5+52921531221443409979865*n^4+84544709872538909913984*n^3+ 90763844826895758387780*n^2+58627993119263500687776*n+16783344497940318815040)* (n+3)^2*(n+2)^2/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/(9500172633901*n^10+ 277882668111790*n^9+3442872282566205*n^8+23259146087972292*n^7+ 90824671862614821*n^6+191891774276798826*n^5+127892502039436175*n^4-\ 306011809039968592*n^3-713190700371088422*n^2-490677429344377476*n-\ 68266472254220160)*a(n+1)+54/25*(n+5)*(n+4)*(n+2)*(n+1)*(12302723560901795*n^12 +532096185057393180*n^11+10165561035599576385*n^10+112412471936888350695*n^9+ 789806746168224707416*n^8+3618160641547951811346*n^7+10460970590054567396763*n^ 6+15988480458422430683075*n^5-1816226894534404099921*n^4-\ 60264882498728652261196*n^3-118344111146209066422218*n^2-\ 103721599025127445230520*n-35881908550836312662400)*(n+3)^2/(5*n+29)/(5*n+31)/( 5*n+28)/(5*n+32)/(9500172633901*n^10+277882668111790*n^9+3442872282566205*n^8+ 23259146087972292*n^7+90824671862614821*n^6+191891774276798826*n^5+ 127892502039436175*n^4-306011809039968592*n^3-713190700371088422*n^2-\ 490677429344377476*n-68266472254220160)*a(n+2)-3/50*(n+5)*(n+4)*(n+3)*(n+2)*( 440523005033989370*n^13+20594571908438516275*n^12+433152710426603156130*n^11+ 5409278953871733859625*n^10+44556608398318667621236*n^9+ 254046286598863780824963*n^8+1021417294778774775109774*n^7+ 2881344063520553978614967*n^6+5498203498798990662354474*n^5+ 6341973096683470203011242*n^4+2503876499833715984443256*n^3-\ 3839900544593368159615872*n^2-5911143639716554383354240*n-\ 2579559427712708565004800)/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/(9500172633901*n ^10+277882668111790*n^9+3442872282566205*n^8+23259146087972292*n^7+ 90824671862614821*n^6+191891774276798826*n^5+127892502039436175*n^4-\ 306011809039968592*n^3-713190700371088422*n^2-490677429344377476*n-\ 68266472254220160)*a(n+3)-3/25*(n+5)*(n+3)*(176133200632524540*n^14+ 9026875080708126480*n^13+208915161774617858965*n^12+2877954208165193304610*n^11 +26150430250973535300737*n^10+163696366389871529121872*n^9+ 711952507003223409611439*n^8+2087405886201093223698918*n^7+ 3647315450495564243122787*n^6+1555451804772933898801208*n^5-\ 9362561318291366781830084*n^4-25534206047915444010805088*n^3-\ 31136947994918995403565984*n^2-19026774327139284648595200*n-\ 4562607579323613714547200)/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/(9500172633901*n ^10+277882668111790*n^9+3442872282566205*n^8+23259146087972292*n^7+ 90824671862614821*n^6+191891774276798826*n^5+127892502039436175*n^4-\ 306011809039968592*n^3-713190700371088422*n^2-490677429344377476*n-\ 68266472254220160)*a(n+4)+6/25*(n+3)*(37487681213373346*n^14+ 2014973198096770317*n^13+49186286572306666219*n^12+720273150352633974480*n^11+ 7036407290316110826070*n^10+48189670861075045583091*n^9+ 236168291164511406363887*n^8+826722845826497965390140*n^7+ 2005505847981084419715812*n^6+3063984755893345499045172*n^5+ 1890479735987098498752194*n^4-2753027153738240825925120*n^3-\ 7490291066472350397952728*n^2-7037263782869913841018080*n-\ 2570127696106663433760000)/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/(9500172633901*n ^10+277882668111790*n^9+3442872282566205*n^8+23259146087972292*n^7+ 90824671862614821*n^6+191891774276798826*n^5+127892502039436175*n^4-\ 306011809039968592*n^3-713190700371088422*n^2-490677429344377476*n-\ 68266472254220160)*a(n+5)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 4, 5, 6], then the first, 30, terms are [0, 1, 0, 0, 5, 6, 7, 560, 2520, 8190, 291060, 2788632, 17837820, 443122680, 6779903130, 70818908160, 1551443974080, 31603795423200, 476701430325120, 10539123986270880, 256311575226546840, 5106138694701604560, 122270110250378290200, 3357754460469468259200, 82676658238462169481600, 2209273417640734690845600, 66904935193026829235937600, 1940317647982812949490640000, 58262726073288487461230520000, 1929164770627497971974774368000] The enumerating sequence satisfies the recurrence -18275/576*n*(n-1)*(n+4)*(n+3)*(1199192744*n^6+28297674512*n^5+275658334485*n^4 +1420412444440*n^3+4090336213846*n^2+6257445077673*n+3986743944960)*(n+1)^2*(n+ 2)^2/(2*n+11)/(4*n+19)/(4*n+21)/(1199192744*n^6+21102518048*n^5+152157853085*n^ 4+576771996740*n^3+1218060033476*n^2+1369669861457*n+647782489410)*a(n)-25/288* n*(n+4)*(n+3)*(n+1)*(107927346960*n^7+2708681726520*n^6+29512029743930*n^5+ 183221648373670*n^4+708032931040648*n^3+1711944699237059*n^2+2386304158213677*n +1457413430634756)*(n+2)^2/(2*n+11)/(4*n+19)/(4*n+21)/(1199192744*n^6+ 21102518048*n^5+152157853085*n^4+576771996740*n^3+1218060033476*n^2+ 1369669861457*n+647782489410)*a(n+1)-25/288*(n+1)*(n+2)*(n+3)*(n+4)*( 636771347064*n^8+17573150554128*n^7+207955854965275*n^6+1376453261690355*n^5+ 5566666651526461*n^4+14069819149647402*n^3+21685010492270554*n^2+ 18623669133322785*n+6823481630449176)/(2*n+11)/(4*n+19)/(4*n+21)/(1199192744*n^ 6+21102518048*n^5+152157853085*n^4+576771996740*n^3+1218060033476*n^2+ 1369669861457*n+647782489410)*a(n+2)+25/288*(n+4)*(n+2)*(n+1)*(376546521616*n^8 +11333022187272*n^7+147497162923026*n^6+1083842240538354*n^5+4916210440480500*n ^4+14088074538907107*n^3+24891031390366889*n^2+24769477281647112*n+ 10617251611524768)/(2*n+11)/(4*n+19)/(4*n+21)/(1199192744*n^6+21102518048*n^5+ 152157853085*n^4+576771996740*n^3+1218060033476*n^2+1369669861457*n+ 647782489410)*a(n+3)-5/192*(n+2)*(188273260808*n^9+6325467506464*n^8+ 90051312134017*n^7+701176188400036*n^6+3178986245199763*n^5+7953063995616505*n^ 4+7200676454219174*n^3-12996174246641253*n^2-38600684417013642*n-\ 28131256993532112)/(2*n+11)/(4*n+19)/(4*n+21)/(1199192744*n^6+21102518048*n^5+ 152157853085*n^4+576771996740*n^3+1218060033476*n^2+1369669861457*n+ 647782489410)*a(n+4)+a(n+5) = 0 The asymptotics is CONST*n!*1.161768135^n/n^(5/2) where the CONST. is roughly, 0.3871 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 4, 5, 7], then the first, 30, terms are [0, 1, 0, 0, 5, 6, 0, 568, 2520, 3150, 286440, 2799720, 9909900, 402846444, 6722515800, 46263016800, 1271619148800, 30249383804640, 339076331193600, 7867753935481440, 231820232617373400, 3784705014264174000, 85545237743743377600, 2828799935701875495360, 61929755659239708240000, 1487405985380383651783200, 52236570950222207749200000, 1435699391607087427220400000, 38331233134450887233322720000, 1400168830760736660370895148000] The enumerating sequence satisfies the recurrence -26614/25*n*(n-1)*(n+5)*(n+4)*(867854188437500*n^9+34640118545312500*n^8+ 609540104666434375*n^7+6204319614864396875*n^6+40246018180353149125*n^5+ 172481082816901069425*n^4+488195787146447951780*n^3+879656244760671847892*n^2+ 915180327019629296912*n+418657103142063840000)*(n+3)^2*(n+2)^2*(n+1)^2/(5*n+29) /(5*n+26)/(5*n+28)/(5*n+32)/(867854188437500*n^9+26829430849375000*n^8+ 363661907087684375*n^7+2834562449639356250*n^6+13989945678367514750*n^5+ 45297341145204823800*n^4+96112202161703680955*n^3+128698331012658749102*n^2+ 98532511354603021268*n+32800850147761197000)*a(n)-2*n*(n+5)*(n+4)*(n+1)*( 3473152462126875000*n^10+143839483111530937500*n^9+2647532240633706356250*n^8+ 28496251655397110618125*n^7+198427103911315440663375*n^6+ 932927221823052990950525*n^5+2995123691041087758239455*n^4+ 6472439931260185115145350*n^3+8991194059582259984370492*n^2+ 7230883310677738006136568*n+2547683640297447749709088)*(n+3)^2*(n+2)^2/(5*n+29) /(5*n+26)/(5*n+28)/(5*n+32)/(867854188437500*n^9+26829430849375000*n^8+ 363661907087684375*n^7+2834562449639356250*n^6+13989945678367514750*n^5+ 45297341145204823800*n^4+96112202161703680955*n^3+128698331012658749102*n^2+ 98532511354603021268*n+32800850147761197000)*a(n+1)-2/5*(n+5)*(n+4)*(n+2)*(n+1) *(9177558042726562500*n^11+403029485787585937500*n^10+7949649410586699296875*n^ 9+92920789628920663093750*n^8+714768629892890362987500*n^7+ 3797310491795279913852500*n^6+14210682900265178177765375*n^5+ 37446381396790219189558650*n^4+68072388848208155703856150*n^3+ 81298975451795556376097736*n^2+57426462962966173127970888*n+ 18188686773321332530276000)*(n+3)^2/(5*n+29)/(5*n+26)/(5*n+28)/(5*n+32)/( 867854188437500*n^9+26829430849375000*n^8+363661907087684375*n^7+ 2834562449639356250*n^6+13989945678367514750*n^5+45297341145204823800*n^4+ 96112202161703680955*n^3+128698331012658749102*n^2+98532511354603021268*n+ 32800850147761197000)*a(n+2)-(n+5)*(n+4)*(n+3)*(n+2)*(1466673578459375000*n^12+ 69541852180023437500*n^11+1494119120118223781250*n^10+19221848186092872453125*n ^9+164788505541788541661875*n^8+990871327780137741755750*n^7+ 4280360663177327896501200*n^6+13366490088236204288268005*n^5+ 29897256738382817163793095*n^4+46614938682290426666595684*n^3+ 47961251767169841273081724*n^2+29132822812375819152462000*n+ 7862213219288178184810240)/(5*n+29)/(5*n+26)/(5*n+28)/(5*n+32)/(867854188437500 *n^9+26829430849375000*n^8+363661907087684375*n^7+2834562449639356250*n^6+ 13989945678367514750*n^5+45297341145204823800*n^4+96112202161703680955*n^3+ 128698331012658749102*n^2+98532511354603021268*n+32800850147761197000)*a(n+3)+1 /2*(n+5)*(n+3)*(n+2)*(9568092427523437500*n^12+477588231237304687500*n^11+ 10837762604845613984375*n^10+147805031394254703984375*n^9+ 1348794796698283316931250*n^8+8673220295006896444401250*n^7+ 40280621903655384611691375*n^6+136068830808267352020126175*n^5+ 331618407143960283002624700*n^4+568266922663224901303099900*n^3+ 649402918821264417834453808*n^2+443924423723812442424378624*n+ 137111426600931170174988800)/(5*n+29)/(5*n+26)/(5*n+28)/(5*n+32)/( 867854188437500*n^9+26829430849375000*n^8+363661907087684375*n^7+ 2834562449639356250*n^6+13989945678367514750*n^5+45297341145204823800*n^4+ 96112202161703680955*n^3+128698331012658749102*n^2+98532511354603021268*n+ 32800850147761197000)*a(n+4)+5*(n+3)*(650890641328125000*n^13+ 35418003208242187500*n^12+881806707098378906250*n^11+13294162709921287734375*n^ 10+135333472953670897968750*n^9+981778744545730790700000*n^8+ 5218344820064235594526250*n^7+20560543953708540730913375*n^6+ 60000360400760329808928750*n^5+127961947308546587734925030*n^4+ 193646346396462395283270120*n^3+196668921099611087561121320*n^2+ 119976354401197336405751584*n+33149500241078950417006080)/(5*n+29)/(5*n+26)/(5* n+28)/(5*n+32)/(867854188437500*n^9+26829430849375000*n^8+363661907087684375*n^ 7+2834562449639356250*n^6+13989945678367514750*n^5+45297341145204823800*n^4+ 96112202161703680955*n^3+128698331012658749102*n^2+98532511354603021268*n+ 32800850147761197000)*a(n+5)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 4, 6, 7], then the first, 30, terms are [0, 1, 0, 0, 5, 0, 7, 568, 0, 5040, 286440, 16632, 7999992, 390233844, 98378280, 25050305280, 1122043802880, 745955578368, 138306576071664, 5872688639737920, 8098761907716480, 1236538421743584960, 50702731424831566080, 125446404225368274048, 16787965973682391488000, 673132692028074002524800, 2707197254855513317497600, 329353411249690291271808000, 13038560264415447717627672000, 79288938268475267794785523200] The enumerating sequence satisfies the recurrence 131769/100*n*(n-1)*(n+5)*(n+4)*(2144950159755*n^10+98733253844550*n^9+ 2031194076470550*n^8+24590832294003570*n^7+193999971164226291*n^6+ 1042043543538890160*n^5+3859370518993263520*n^4+9732482963407082920*n^3+ 15995395830516833804*n^2+15474859018927638848*n+6694914159437500992)*(n+3)^2*(n +2)^2*(n+1)^2/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/(2144950159755*n^10+ 77283752247000*n^9+1239117549058575*n^8+11638282801472370*n^7+70894425457983051 *n^6+292604188989411624*n^5+828666873374155885*n^4+1590405360849950370*n^3+ 1980742267915544534*n^2+1447005486080522964*n+471638727716994864)*a(n)+1089/50* n*(n+5)*(n+4)*(n+1)*(450439533548550*n^11+21409642607678325*n^10+ 457550030399048400*n^9+5799711921738406500*n^8+48410695643917762380*n^7+ 279162787204221425187*n^6+1133696567014888335678*n^5+3238584029422870037484*n^4 +6368865462355016217352*n^3+8198041789229076491944*n^2+6203823828489998484776*n +2085568113480823528544)*(n+3)^2*(n+2)^2/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/( 2144950159755*n^10+77283752247000*n^9+1239117549058575*n^8+11638282801472370*n^ 7+70894425457983051*n^6+292604188989411624*n^5+828666873374155885*n^4+ 1590405360849950370*n^3+1980742267915544534*n^2+1447005486080522964*n+ 471638727716994864)*a(n+1)+9/50*(n+5)*(n+4)*(n+2)*(n+1)*(45387145380415800*n^12 +2270744232872341200*n^11+51474619936582679925*n^10+698707706396259250950*n^9+ 6321283302229817623290*n^8+40131995626791001071678*n^7+183221079910215293712727 *n^6+605736394704740694635612*n^5+1438499320650197603836482*n^4+ 2392188381787436386811960*n^3+2643854858439460662474520*n^2+ 1743954817540045954334336*n+519594180628192385872000)*(n+3)^2/(5*n+29)/(5*n+31) /(5*n+28)/(5*n+32)/(2144950159755*n^10+77283752247000*n^9+1239117549058575*n^8+ 11638282801472370*n^7+70894425457983051*n^6+292604188989411624*n^5+ 828666873374155885*n^4+1590405360849950370*n^3+1980742267915544534*n^2+ 1447005486080522964*n+471638727716994864)*a(n+2)-3/50*(n+5)*(n+4)*(n+3)*(n+2)*( 57162921757470750*n^13+3059963128138288125*n^12+74888754891907456500*n^11+ 1109054804867025789975*n^10+11083398502082460482610*n^9+78887090504207709047829 *n^8+411156582133022276762620*n^7+1587577302798116522322385*n^6+ 4537042867771586674329160*n^5+9466778977763941872714118*n^4+ 13996353846564564481299352*n^3+13856690235428925381583648*n^2+ 8210900855172986432885568*n+2190976725065557679028480)/(5*n+29)/(5*n+31)/(5*n+ 28)/(5*n+32)/(2144950159755*n^10+77283752247000*n^9+1239117549058575*n^8+ 11638282801472370*n^7+70894425457983051*n^6+292604188989411624*n^5+ 828666873374155885*n^4+1590405360849950370*n^3+1980742267915544534*n^2+ 1447005486080522964*n+471638727716994864)*a(n+3)-3/25*(n+5)*(n+3)*(n+2)*( 70719006767122350*n^13+3962425446926037000*n^12+101689834036651905975*n^11+ 1582229000304756315525*n^10+16647361890799058661720*n^9+ 125030989307382132368448*n^8+689372006303438252901315*n^7+ 2824010836848347031088281*n^6+8591247193437542408590232*n^5+ 19160615099312197719968386*n^4+30433605682886665028477832*n^3+ 32581061597365482046585400*n^2+21058140409352334765369856*n+ 6201810270813095126398080)/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/(2144950159755*n ^10+77283752247000*n^9+1239117549058575*n^8+11638282801472370*n^7+ 70894425457983051*n^6+292604188989411624*n^5+828666873374155885*n^4+ 1590405360849950370*n^3+1980742267915544534*n^2+1447005486080522964*n+ 471638727716994864)*a(n+4)-3/25*(n+3)*(52782933531251040*n^14+ 3194980446809826480*n^13+89095401516601456635*n^12+1516629776154519050505*n^11+ 17600514394884738961128*n^10+147255602856237112924128*n^9+ 915665132556857957310671*n^8+4297507885325632681227757*n^7+ 15293088564600459790550758*n^6+41053102824424431990003546*n^5+ 81819316800553496120690104*n^4+117394372813584425289102048*n^3+ 114650300938443651467192144*n^2+68254459396884843584879936*n+ 18705390502614925891392000)/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/(2144950159755* n^10+77283752247000*n^9+1239117549058575*n^8+11638282801472370*n^7+ 70894425457983051*n^6+292604188989411624*n^5+828666873374155885*n^4+ 1590405360849950370*n^3+1980742267915544534*n^2+1447005486080522964*n+ 471638727716994864)*a(n+5)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 5, 6, 7], then the first, 30, terms are [0, 1, 0, 0, 0, 6, 7, 8, 0, 3150, 13860, 44352, 72072, 12696684, 122972850, 776936160, 3087564480, 202029635808, 3340899145584, 34924471021440, 245771676390240, 9347776167890160, 220994370120003480, 3414846339374678208, 36828268141006742400, 1050494421685637728800, 30768879013208834097600, 645846237834683453568000, 9782325011308943004912000, 249755075665637605566703200] The enumerating sequence satisfies the recurrence -17821/100*n*(n-1)*(n+5)*(n+4)*(1711723582265000*n^10+80271248187400000*n^9+ 1684307655641436875*n^8+20821262197012611250*n^7+167911853695849865400*n^6+ 922947213679998521950*n^5+3501510801382859403665*n^4+9053181844130240865760*n^3 +15266222022565792678788*n^2+15161421522007740595472*n+6734527524919818036480)* (n+3)^2*(n+2)^2*(n+1)^2/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/(1711723582265000*n ^10+63154012364750000*n^9+1038893983156761875*n^8+10031159056755716250*n^7+ 62940309779255869150*n^6+268084191855125320800*n^5+784855182100737212415*n^4+ 1559334027679181891350*n^3+2012048337957140186248*n^2+1522724448092154371616*n+ 513406108680363691776)*a(n)+3/25*n*(n+5)*(n+4)*(n+1)*(5109494893061025000*n^11+ 247273918178980537500*n^10+5386321140578696596875*n^9+69663996276844912750625*n ^8+593969309928636499590000*n^7+3502572793126090017946200*n^6+ 14562704409343367701281225*n^5+42644293809976976181373655*n^4+ 86085963728035502810376100*n^3+113931439989292784165436148*n^2+ 88818953718428779570996112*n+30836958807880947397024000)*(n+3)^2*(n+2)^2/(5*n+ 29)/(5*n+31)/(5*n+28)/(5*n+32)/(1711723582265000*n^10+63154012364750000*n^9+ 1038893983156761875*n^8+10031159056755716250*n^7+62940309779255869150*n^6+ 268084191855125320800*n^5+784855182100737212415*n^4+1559334027679181891350*n^3+ 2012048337957140186248*n^2+1522724448092154371616*n+513406108680363691776)*a(n+ 1)-1/50*(n+5)*(n+4)*(n+2)*(n+1)*(21747448112676825000*n^12+ 1106836000671624300000*n^11+25552767004638834584375*n^10+ 353685189858432083225625*n^9+3267539967997409297650250*n^8+ 21218073067692725948135950*n^7+99266098220402677792780675*n^6+ 337018600067124760747494805*n^5+823942608709809375872959940*n^4+ 1414540959923252524983435268*n^3+1618964239062906356975618480*n^2+ 1109561175852989063536728192*n+344625561805389729316945920)*(n+3)^2/(5*n+29)/(5 *n+31)/(5*n+28)/(5*n+32)/(1711723582265000*n^10+63154012364750000*n^9+ 1038893983156761875*n^8+10031159056755716250*n^7+62940309779255869150*n^6+ 268084191855125320800*n^5+784855182100737212415*n^4+1559334027679181891350*n^3+ 2012048337957140186248*n^2+1522724448092154371616*n+513406108680363691776)*a(n+ 2)-3/50*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(3526150579465900000*n^12+ 188278750032572350000*n^11+4580649020894696412500*n^10+67133142837002695895625* n^9+660002392019547031703625*n^8+4584748056805067788997850*n^7+ 23071103380248267128618150*n^6+84726524530933429711306865*n^5+ 225334980021030491333823145*n^4+423205446488034607987096844*n^3+ 532726184787090146881016724*n^2+403512283224938101724551312*n+ 139072234476084657400408320)/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/( 1711723582265000*n^10+63154012364750000*n^9+1038893983156761875*n^8+ 10031159056755716250*n^7+62940309779255869150*n^6+268084191855125320800*n^5+ 784855182100737212415*n^4+1559334027679181891350*n^3+2012048337957140186248*n^2 +1522724448092154371616*n+513406108680363691776)*a(n+3)+1/100*(n+5)*(n+3)*(n+2) *(117929196200147175000*n^13+6709579605872394750000*n^12+ 175026953466169148403125*n^11+2771221849370102736086250*n^10+ 29706177207196635296253125*n^9+227611314124810507384821850*n^8+ 1282149785103753570466527475*n^7+5374798216281048814482535990*n^6+ 16762615972360142519019946655*n^5+38401775833618370440377944494*n^4+ 62793556801694838701652194492*n^3+69376892585114705945715733416*n^2+ 46401591782114009972082659808*n+14183041083654805907245678080)/(5*n+29)/(5*n+31 )/(5*n+28)/(5*n+32)/(1711723582265000*n^10+63154012364750000*n^9+ 1038893983156761875*n^8+10031159056755716250*n^7+62940309779255869150*n^6+ 268084191855125320800*n^5+784855182100737212415*n^4+1559334027679181891350*n^3+ 2012048337957140186248*n^2+1522724448092154371616*n+513406108680363691776)*a(n+ 4)-3/50*(n+3)*(29896964087840490000*n^14+1835523600114815505000*n^13+ 51964154356911388843750*n^12+898875312861523881469375*n^11+ 10610714849071129760869525*n^10+90391556169527893169138350*n^9+ 572886303926897848698144340*n^8+2743167251018255842805095055*n^7+ 9968552410663100132422452953*n^6+27347666982026866066254823692*n^5+ 55730866310137538547657316280*n^4+81771859443072289925824111440*n^3+ 81624542204758088128149095312*n^2+49592325021709070666136784448*n+ 13829739511075120821673313280)/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/( 1711723582265000*n^10+63154012364750000*n^9+1038893983156761875*n^8+ 10031159056755716250*n^7+62940309779255869150*n^6+268084191855125320800*n^5+ 784855182100737212415*n^4+1559334027679181891350*n^3+2012048337957140186248*n^2 +1522724448092154371616*n+513406108680363691776)*a(n+5)+a(n+6) = 0 The asymptotics is CONST*n!*.8599320672^n/n^(5/2) where the CONST. is roughly, 0.4943 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3, 4, 5], then the first, 30, terms are [0, 1, 3, 16, 125, 1296, 16800, 261800, 4770360, 99568350, 2343123090, 61391484000, 1772641770900, 55931625750000, 1914804950058000, 70694359020885600, 2800031571437100000, 118434095757029052000, 5328242311045658004000, 254063027768147357760000, 12798914705149964598207000, 679272893430871268529000000, 37882701538869908538777960000, 2214894861537348248166668760000, 135475541925970857068362905000000, 8652128840223391250445366148500000, 575926446752465718087490383571500000, 39891608064418517208147503108952000000, 2870840148186465684315784192664169000000, 214358126491547865053337201555964500000000] The enumerating sequence satisfies the recurrence -1/3*n*(n-1)*(n+3)*(n+2)*(232*n^3+2400*n^2+8207*n+9300)*(n+1)^2/(3*n+13)/(3*n+ 14)/(232*n^3+1704*n^2+4103*n+3261)*a(n)-2/3*n*(n+1)*(n+2)*(n+3)*(80*n^2+343*n+ 51)/(3*n+13)/(3*n+14)/(232*n^3+1704*n^2+4103*n+3261)*a(n+1)+2/3*(n+3)*(n+1)*( 3016*n^5+43264*n^4+242671*n^3+663969*n^2+883674*n+455868)/(3*n+13)/(3*n+14)/( 232*n^3+1704*n^2+4103*n+3261)*a(n+2)-2/9*(n+2)*(28768*n^5+455824*n^4+2846852*n^ 3+8746991*n^2+13197471*n+7805826)/(3*n+13)/(3*n+14)/(232*n^3+1704*n^2+4103*n+ 3261)*a(n+3)+a(n+4) = 0 The asymptotics is CONST*n!*2.707993250^n/n^(5/2) where the CONST. is roughly, 0.4227 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3, 4, 6], then the first, 30, terms are [0, 1, 3, 16, 125, 1290, 16597, 256256, 4621176, 95387040, 2218694940, 57434969592, 1638057797376, 51039880299408, 1725225807024720, 62880521636412480, 2458423098423932640, 102634631431670257920, 4557142107015390756864, 214444428806791542078720, 10660771293768635901301440, 558319540686410451903514560, 30724595752920635260299084480, 1772519106078751915851015329280, 106974195010911154785148059369600, 6740788915439817202711788308467200, 442705578076954625223581477464377600, 30253905336348183170767811253501772800, 2148094995617119036512795441993587865600, 158242248044101311882029401677583829760000] The enumerating sequence satisfies the recurrence -10907/576*n*(n-1)*(n+4)*(n+3)*(441590648*n^5+8013999802*n^4+57430115207*n^3+ 203120108291*n^2+354488307272*n+244089552858)*(n+1)^2*(n+2)^2/(4*n+17)/(2*n+11) /(4*n+19)/(441590648*n^5+5806046562*n^4+29790022479*n^3+74497855002*n^2+ 90690390343*n+42863647824)*a(n)-1/288*n*(n+4)*(n+3)*(n+1)*(808110885840*n^6+ 15877785966420*n^5+128127110602480*n^4+543682024710740*n^3+1280506066766843*n^2 +1591504609014520*n+821772505864053)*(n+2)^2/(4*n+17)/(2*n+11)/(4*n+19)/( 441590648*n^5+5806046562*n^4+29790022479*n^3+74497855002*n^2+90690390343*n+ 42863647824)*a(n+1)-1/144*(n+1)*(n+2)*(n+3)*(n+4)*(10260358706280*n^7+ 227246720224590*n^6+2117772726763305*n^5+10756312796349500*n^4+ 32122825271981014*n^3+56326380378980344*n^2+53591302645515431*n+ 21285387936749940)/(4*n+17)/(2*n+11)/(4*n+19)/(441590648*n^5+5806046562*n^4+ 29790022479*n^3+74497855002*n^2+90690390343*n+42863647824)*a(n+2)+1/36*(n+4)*(n +2)*(1890007973440*n^8+48474978953360*n^7+535624699632440*n^6+3328215144452320* n^5+12710486834009264*n^4+30520860720270772*n^3+44943189860149081*n^2+ 37041345547900538*n+13049435238269325)/(4*n+17)/(2*n+11)/(4*n+19)/(441590648*n^ 5+5806046562*n^4+29790022479*n^3+74497855002*n^2+90690390343*n+42863647824)*a(n +3)-1/6*(282618014720*n^9+8520376049920*n^8+112795874281360*n^7+860050658025980 *n^6+4159423061092066*n^5+13220581841248706*n^4+27589788684710501*n^3+ 36409705432404166*n^2+27530998495840695*n+9070828659708210)/(4*n+17)/(2*n+11)/( 4*n+19)/(441590648*n^5+5806046562*n^4+29790022479*n^3+74497855002*n^2+ 90690390343*n+42863647824)*a(n+4)+a(n+5) = 0 The asymptotics is CONST*n!*2.671181366^n/n^(5/2) where the CONST. is roughly, 0.4727 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3, 4, 7], then the first, 30, terms are [0, 1, 3, 16, 125, 1290, 16590, 255928, 4609584, 95001480, 2205772800, 56986498800, 1621747687560, 50415136936164, 1699978190290380, 61804004375393280, 2410037007966588000, 100345171564090252800, 4443281520673940904960, 208502651599244555192640, 10335960744370475757903360, 539750228382735170698497600, 29616140412103802268944232000, 1703537816589063555873829056000, 102505290908250522336526722240000, 6439817114175655530127461481027200, 421661196923481413241694330739760000, 28728119246480494710041556520456320000, 2033520110029559166948963174280336200000, 149341169725296963845810809263864030000000] The enumerating sequence satisfies the recurrence 11671/25*n*(n-1)*(n+6)*(n+4)*(96691080000000*n^8+3177933000000000*n^7+ 45342896146200000*n^6+366784095972120000*n^5+1839514398690351000*n^4+ 5856126906515911800*n^3+11554181730916647605*n^2+12913992958085897801*n+ 6258416564261558460)*(n+1)^2*(n+2)^2*(n+3)^2/(5*n+29)/(5*n+26)/(5*n+28)/(5*n+32 )/(96691080000000*n^8+2404404360000000*n^7+25804715386200000*n^6+ 156048611614920000*n^5+581278081622751000*n^4+1364865303071707800*n^3+ 1971160643222818205*n^2+1599287726382333991*n+557470387520827464)*a(n)-1/5*n*(n +6)*(n+4)*(n+1)*(838311663600000000*n^9+28810146605400000000*n^8+ 434050386843978000000*n^7+3757993653685035000000*n^6+20573295949957109130000*n^ 5+73700046152250399129000*n^4+172267498910454133752750*n^3+ 252318208705340055457365*n^2+208865825725763887631879*n+73731312435420777561956 )*(n+3)^2*(n+2)^2/(5*n+29)/(5*n+26)/(5*n+28)/(5*n+32)/(96691080000000*n^8+ 2404404360000000*n^7+25804715386200000*n^6+156048611614920000*n^5+ 581278081622751000*n^4+1364865303071707800*n^3+1971160643222818205*n^2+ 1599287726382333991*n+557470387520827464)*a(n+1)+(n+6)*(n+4)*(n+2)*(n+1)*( 565352744760000000*n^10+20842785230040000000*n^9+341251609594919400000*n^8+ 3264922341360904440000*n^7+20195183636729705337000*n^6+84288616759522253094600* n^5+240053483127591899980635*n^4+459820278979334656419417*n^3+ 565624803841965433688078*n^2+402236866776895178276260*n+ 125046051722299540879760)*(n+3)^2/(5*n+29)/(5*n+26)/(5*n+28)/(5*n+32)/( 96691080000000*n^8+2404404360000000*n^7+25804715386200000*n^6+ 156048611614920000*n^5+581278081622751000*n^4+1364865303071707800*n^3+ 1971160643222818205*n^2+1599287726382333991*n+557470387520827464)*a(n+2)-2*(n+6 )*(n+4)*(n+3)*(n+2)*(876988095600000000*n^11+35401263027000000000*n^10+ 642776749987794000000*n^9+6926674503030641400000*n^8+49201103351778283170000*n^ 7+241751644773753061221000*n^6+837932141581652636553350*n^5+ 2047222248864963717390845*n^4+3451967717183084494819050*n^3+ 3821538880607175245252646*n^2+2496423159781343960644687*n+ 727730096621987659925620)/(5*n+29)/(5*n+26)/(5*n+28)/(5*n+32)/(96691080000000*n ^8+2404404360000000*n^7+25804715386200000*n^6+156048611614920000*n^5+ 581278081622751000*n^4+1364865303071707800*n^3+1971160643222818205*n^2+ 1599287726382333991*n+557470387520827464)*a(n+3)+(n+6)*(n+3)*( 1124033805000000000*n^12+50431876785000000000*n^11+1027457167496175000000*n^10+ 12564234273715575000000*n^9+102668858037104349375000*n^8+ 590355814535086065975000*n^7+2448127012432596842653125*n^6+ 7372777178142990398231625*n^5+15993370239295530246676225*n^4+ 24352172340599412635365795*n^3+24682934765578663370096774*n^2+ 14936872684638271733740856*n+4075922728323420902336640)/(5*n+29)/(5*n+26)/(5*n+ 28)/(5*n+32)/(96691080000000*n^8+2404404360000000*n^7+25804715386200000*n^6+ 156048611614920000*n^5+581278081622751000*n^4+1364865303071707800*n^3+ 1971160643222818205*n^2+1599287726382333991*n+557470387520827464)*a(n+4)-(n+6)* (362591550000000000*n^13+18262600875000000000*n^12+420903431165250000000*n^11+ 5875449236554575000000*n^10+55402093219033061250000*n^9+ 372500168801932684125000*n^8+1836467220414939633993750*n^7+ 6718585097928859416350625*n^6+18229352185270995993863375*n^5+ 36207414571054831055301025*n^4+51134326561652137036922235*n^3+ 48583915900687075216500878*n^2+27807826804503466977344488*n+ 7233094349631229921423680)/(5*n+29)/(5*n+26)/(5*n+28)/(n+5)/(5*n+32)/( 96691080000000*n^8+2404404360000000*n^7+25804715386200000*n^6+ 156048611614920000*n^5+581278081622751000*n^4+1364865303071707800*n^3+ 1971160643222818205*n^2+1599287726382333991*n+557470387520827464)*a(n+5)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3, 5, 6], then the first, 30, terms are [0, 1, 3, 16, 120, 1176, 14287, 207816, 3527496, 68521950, 1500074730, 36556819992, 981781937136, 28814070933648, 917606722122090, 31516486253572320, 1161377441311127040, 45706655544676045920, 1913424088052442629664, 84902974111843588340640, 3980481440547396817992960, 196612920084880853820826560, 10205539029357513747825399480, 555386286573996453621408242880, 31620435769736797656865976784000, 1879797359822975248282375318970400, 116479864594747225541735357229237600, 7510597478263671022728992600478172800, 503179801180668540266430062589044025600, 34977314124568006611481211397482066832000] The enumerating sequence satisfies the recurrence -15163/144*n*(n-1)*(n+4)*(n+3)*(118940720068*n^6+2796149383704*n^5+ 26849888374975*n^4+134336447030544*n^3+367851199337875*n^2+520139142681474*n+ 294826296908160)*(n+1)^2*(n+2)^2/(2*n+11)/(4*n+21)/(118940720068*n^6+ 2082505063296*n^5+14653252257475*n^4+52519572966324*n^3+99763805460073*n^2+ 93313634195568*n+32374586245356)/(4*n+19)*a(n)-1/48*n*(n+4)*(n+3)*(n+1)*( 329465794588360*n^7+8239532484742620*n^6+85701910786706310*n^5+ 478834965450985585*n^4+1548345031384199678*n^3+2899687904823958001*n^2+ 2933469738569075046*n+1261514552753814480)*(n+2)^2/(2*n+11)/(4*n+21)/( 118940720068*n^6+2082505063296*n^5+14653252257475*n^4+52519572966324*n^3+ 99763805460073*n^2+93313634195568*n+32374586245356)/(4*n+19)*a(n+1)+1/48*(n+1)* (n+2)*(n+3)*(n+4)*(1722856330184980*n^8+47393649143692360*n^7+ 556516901332652545*n^6+3631566574736839945*n^5+14347224802466094832*n^4+ 34965032122441796311*n^3+50995390245970573113*n^2+40324474423247905674*n+ 13060204311625425360)/(2*n+11)/(4*n+21)/(118940720068*n^6+2082505063296*n^5+ 14653252257475*n^4+52519572966324*n^3+99763805460073*n^2+93313634195568*n+ 32374586245356)/(4*n+19)*a(n+2)-1/32*(n+4)*(n+2)*(379420897016920*n^9+ 11765373261642660*n^8+158897801131452770*n^7+1223127438138112435*n^6+ 5890349574869179862*n^5+18303782353920506110*n^4+36411057156358694830*n^3+ 44166599373454365595*n^2+29022515306860838898*n+7540187654807186400)/(2*n+11)/( 4*n+21)/(118940720068*n^6+2082505063296*n^5+14653252257475*n^4+52519572966324*n ^3+99763805460073*n^2+93313634195568*n+32374586245356)/(4*n+19)*a(n+3)-1/64*( 627412298358700*n^10+22278635579343000*n^9+350654096341828325*n^8+ 3216105368640339190*n^7+18995712387739140314*n^6+75301349504128933312*n^5+ 202216676776933017625*n^4+361638918539372785210*n^3+409665838867170721116*n^2+ 263066794064806934808*n+71704033993645683840)/(2*n+11)/(4*n+21)/(118940720068*n ^6+2082505063296*n^5+14653252257475*n^4+52519572966324*n^3+99763805460073*n^2+ 93313634195568*n+32374586245356)/(4*n+19)*a(n+4)+a(n+5) = 0 The asymptotics is CONST*n!*2.522039659^n/n^(5/2) where the CONST. is roughly, 0.5756 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3, 5, 7], then the first, 30, terms are [0, 1, 3, 16, 120, 1176, 14280, 207488, 3515904, 68141430, 1487628450, 36138868920, 967168402200, 28278463337124, 896968103211180, 30679628206106880, 1125678045809116800, 44105878770431718240, 1838065744110869713440, 81183744874210469531040, 3788326806457201506257760, 186235911473830068544293600, 9620651975313613981265719200, 521028662769836761341766124160, 29519942141510723318570442288000, 1746325866355399130506896974983200, 107675999194640087490275302766460000, 6908520590155987244926121289571920000, 460538862197393003747117987976143760000, 31853206434668669612635711846323263148000] The enumerating sequence satisfies the recurrence -52593/50*n*(n-1)*(n+5)*(n+4)*(406984375*n^8+13023500000*n^7+177757545625*n^6+ 1341854049875*n^5+6054652745950*n^4+16340629016765*n^3+24408386828682*n^2+ 15418011619032*n-384935022720)*(n+1)^2*(n+2)^2*(n+3)^2/(5*n+24)/(5*n+28)/(5*n+ 32)/(5*n+26)/(406984375*n^8+9767625000*n^7+97988608125*n^6+526011151125*n^5+ 1584412087200*n^4+2418438994215*n^3+700141002212*n^2-2864852371212*n-\ 2857249103760)*a(n)+3/5*n*(n+5)*(n+4)*(n+1)*(1367467500000*n^9+45810161250000*n ^8+662903793300000*n^7+5396243199126875*n^6+26884767898388875*n^5+ 82941028204128775*n^4+149459957079001445*n^3+123733888075007430*n^2-\ 18149858560900200*n-75543193259005248)*(n+3)^2*(n+2)^2/(5*n+24)/(5*n+28)/(5*n+ 32)/(5*n+26)/(406984375*n^8+9767625000*n^7+97988608125*n^6+526011151125*n^5+ 1584412087200*n^4+2418438994215*n^3+700141002212*n^2-2864852371212*n-\ 2857249103760)*a(n+1)+3/10*(n+5)*(n+4)*(n+2)*(n+1)*(2091899687500*n^10+ 75308388750000*n^9+1190207864778125*n^8+10847681516101250*n^7+62991293144507375 *n^6+243104280828013225*n^5+632199124621441030*n^4+1105183513219146405*n^3+ 1285926283514652282*n^2+970120035589757496*n+395944645058795520)*(n+3)^2/(5*n+ 24)/(5*n+28)/(5*n+32)/(5*n+26)/(406984375*n^8+9767625000*n^7+97988608125*n^6+ 526011151125*n^5+1584412087200*n^4+2418438994215*n^3+700141002212*n^2-\ 2864852371212*n-2857249103760)*a(n+2)-1/2*(n+5)*(n+4)*(n+3)*(n+2)*( 6145464062500*n^11+242745830468750*n^10+4266062785500000*n^9+43868848358690625* n^8+291756290845451250*n^7+1306944035610344625*n^6+3967782892994477825*n^5+ 7944869279074334950*n^4+9644471287974556905*n^3+5383489720325012514*n^2-\ 903188558545226856*n-1914563585591410176)/(5*n+24)/(5*n+28)/(5*n+32)/(5*n+26)/( 406984375*n^8+9767625000*n^7+97988608125*n^6+526011151125*n^5+1584412087200*n^4 +2418438994215*n^3+700141002212*n^2-2864852371212*n-2857249103760)*a(n+3)+1/2*( n+5)*(n+3)*(6623670703125*n^12+291441510937500*n^11+5771592329890625*n^10+ 67806090001996875*n^9+523933627262280000*n^8+2785646399939197875*n^7+ 10329372759112003050*n^6+26341214005644694500*n^5+43720947905051696980*n^4+ 39886243163498833434*n^3+4773020551851945060*n^2-25885976548318133328*n-\ 17672591137218716160)/(5*n+24)/(5*n+28)/(5*n+32)/(5*n+26)/(406984375*n^8+ 9767625000*n^7+97988608125*n^6+526011151125*n^5+1584412087200*n^4+2418438994215 *n^3+700141002212*n^2-2864852371212*n-2857249103760)*a(n+4)-(1526191406250*n^13 +75546474609375*n^12+1698092694531250*n^11+22886135759687500*n^10+ 205561896745921875*n^9+1292109188267962500*n^8+5795783994283128750*n^7+ 18494235571856152125*n^6+40700407533663474350*n^5+56449366018909070570*n^4+ 34730813378884700181*n^3-23477081669172313038*n^2-56749191542004412584*n-\ 30099889726783269120)/(5*n+24)/(5*n+28)/(5*n+32)/(5*n+26)/(406984375*n^8+ 9767625000*n^7+97988608125*n^6+526011151125*n^5+1584412087200*n^4+2418438994215 *n^3+700141002212*n^2-2864852371212*n-2857249103760)*a(n+5)+a(n+6) = 0 The asymptotics is CONST*n!*2.508789332^n/n^(5/2) where the CONST. is roughly, 0.6180 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3, 6, 7], then the first, 30, terms are [0, 1, 3, 16, 120, 1170, 14077, 201944, 3369240, 64156680, 1373927940, 32700690792, 856643755968, 24500160232548, 759759522582060, 25395694886802240, 910332862611812640, 34838070853251714048, 1417780154279980803312, 61142528199486411437760, 2785466443615859399235840, 133675015927529914529852160, 6740558366536414906806640320, 356313347992425386003196083328, 19703555839381141213807043923200, 1137618051481830702036013621660800, 68457139973369882315725022416137600, 4286497892558704640480939246991820800, 278864111935810663183332571405470825600, 18822658672552139030011132899322957987200] The enumerating sequence satisfies the recurrence -9619/20*n*(n-1)*(n+5)*(n+4)*(85501553705109*n^10+3738689768406420*n^9+ 72627090322536270*n^8+824703275574424440*n^7+6057190350282040887*n^6+ 30043382473365140100*n^5+101840464901026970730*n^4+232845752032104446360*n^3+ 343592274397763968404*n^2+295603686498014907280*n+112749726392521392000)*(n+1)^ 2*(n+2)^2*(n+3)^2/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/(85501553705109*n^10+ 2883674231355330*n^9+42826452323608395*n^8+368019198212152320*n^7+ 2021731336024018977*n^6+7401420619959298350*n^5+18247569881363431005*n^4+ 29875197497151497180*n^3+31105882708413867114*n^2+18693003858645169420*n+ 4991105664643288800)*a(n)-1/2*n*(n+5)*(n+4)*(n+1)*(408355420495600584*n^11+ 18468515464652462796*n^10+373282808684266703682*n^9+4443970502495509572879*n^8+ 34564609417501212332634*n^7+184044267788227679667444*n^6+ 682894776970065402351420*n^5+1760388011361157225455537*n^4+ 3077552772028640202805024*n^3+3456297482687884562393904*n^2+ 2226299542509322237835696*n+614893506552366997934400)*(n+3)^2*(n+2)^2/(5*n+29)/ (5*n+31)/(5*n+28)/(5*n+32)/(85501553705109*n^10+2883674231355330*n^9+ 42826452323608395*n^8+368019198212152320*n^7+2021731336024018977*n^6+ 7401420619959298350*n^5+18247569881363431005*n^4+29875197497151497180*n^3+ 31105882708413867114*n^2+18693003858645169420*n+4991105664643288800)*a(n+1)-1/ 10*(n+5)*(n+4)*(n+2)*(n+1)*(1966877741432327436*n^12+93872330398150595424*n^11+ 2018342063672745487425*n^10+25817891577859157178240*n^9+ 218518998068515713528258*n^8+1287258022964365639007322*n^7+ 5402835376963557535364295*n^6+16251470000116677354117870*n^5+ 34712963243134440424141826*n^4+51288383481832513113078344*n^3+ 49761573327859860264363160*n^2+28559521239236938695834400*n+ 7407537533629078440000000)*(n+3)^2/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/( 85501553705109*n^10+2883674231355330*n^9+42826452323608395*n^8+ 368019198212152320*n^7+2021731336024018977*n^6+7401420619959298350*n^5+ 18247569881363431005*n^4+29875197497151497180*n^3+31105882708413867114*n^2+ 18693003858645169420*n+4991105664643288800)*a(n+2)+1/10*(n+5)*(n+4)*(n+3)*(n+2) *(2639774969091535266*n^13+135226620177966325575*n^12+3150866343254380960554*n^ 11+44171711519778139115385*n^10+415196929057028236892220*n^9+ 2759602706042898950113959*n^8+13322183344246038710771550*n^7+ 47207890492341474097975271*n^6+122500826369207193773064754*n^5+ 229200588884496152923267106*n^4+299266792061952414019883176*n^3+ 256586089453977321655660304*n^2+128120479030881789861570880*n+ 27588709155882571898304000)/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/(85501553705109 *n^10+2883674231355330*n^9+42826452323608395*n^8+368019198212152320*n^7+ 2021731336024018977*n^6+7401420619959298350*n^5+18247569881363431005*n^4+ 29875197497151497180*n^3+31105882708413867114*n^2+18693003858645169420*n+ 4991105664643288800)*a(n+3)+1/5*(n+5)*(n+3)*(1032431260989191175*n^14+ 57533854085377815600*n^13+1469568079879428304800*n^12+22786883824805662988430*n ^11+239428719567954523989324*n^10+1801824325696541275785744*n^9+ 10006432303812545162689548*n^8+41626050202774636183043046*n^7+ 130262662609449282167073169*n^6+305109210525578090602843836*n^5+ 526922186801613947048459872*n^4+652180169672520410736837264*n^3+ 549788177548346255402008912*n^2+285372013600953227447753280*n+ 69927131710706558976576000)/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/(85501553705109 *n^10+2883674231355330*n^9+42826452323608395*n^8+368019198212152320*n^7+ 2021731336024018977*n^6+7401420619959298350*n^5+18247569881363431005*n^4+ 29875197497151497180*n^3+31105882708413867114*n^2+18693003858645169420*n+ 4991105664643288800)*a(n+4)-1/25*(5991435874331808066*n^15+ 366835074632118116235*n^14+10363842205032717149220*n^13+ 179090701398255874853385*n^12+2115164763682854423146892*n^11+ 18069332819759124366292575*n^10+115228548364377397373756160*n^9+ 557958401384864130803532855*n^8+2066023145001644456885417218*n^7+ 5843365761100202891496057370*n^6+12507400439638628003115441820*n^5+ 19881222059994283811318393260*n^4+22715278871957106668031659824*n^3+ 17632794112413070621197986320*n^2+8346274601142137629686868800*n+ 1830591776239093542957120000)/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/( 85501553705109*n^10+2883674231355330*n^9+42826452323608395*n^8+ 368019198212152320*n^7+2021731336024018977*n^6+7401420619959298350*n^5+ 18247569881363431005*n^4+29875197497151497180*n^3+31105882708413867114*n^2+ 18693003858645169420*n+4991105664643288800)*a(n+5)+a(n+6) = 0 The asymptotics is CONST*n!*2.448187650^n/n^(5/2) where the CONST. is roughly, 0.7585 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 4, 5, 6], then the first, 30, terms are [0, 1, 3, 12, 65, 486, 4837, 59696, 864864, 14301630, 266215950, 5523032592, 126504053676, 3170678655144, 86294023705890, 2534061282392640, 79862125607103840, 2688774918837527520, 96316341287605639200, 3657755587924468910880, 146792375960884573079160, 6207495303488785033487760, 275884767091222912221679080, 12856365964639670981663675520, 626845241773556916855386241600, 31915832470477622262451251396000, 1693861083030585372146128242957600, 93553247689139192116048940986732800, 5368930106469634989895116870855691200, 319706453954835402512415118889192064000] The enumerating sequence satisfies the recurrence -6401/48*n*(n-1)*(n+4)*(n+3)*(2997981860*n^6+64707294480*n^5+564189827465*n^4+ 2528406972960*n^3+6113813225537*n^2+7578477407490*n+3844342700784)*(n+1)^2*(n+2 )^2/(2*n+11)/(4*n+21)/(2997981860*n^6+46719403320*n^5+285623082965*n^4+ 858760970700*n^3+1311628054547*n^2+984861146676*n+353752060716)/(4*n+19)*a(n)+1 /24*n*(n+4)*(n+3)*(n+1)*(24643410889200*n^7+568859076959400*n^6+ 5416160250494800*n^5+27337900697438765*n^4+78135754028518460*n^3+ 124551239869086859*n^2+100004767840917516*n+29411013570240816)*(n+2)^2/(2*n+11) /(4*n+21)/(2997981860*n^6+46719403320*n^5+285623082965*n^4+858760970700*n^3+ 1311628054547*n^2+984861146676*n+353752060716)/(4*n+19)*a(n+1)-1/48*(n+1)*(n+2) *(n+3)*(n+4)*(67979238675500*n^8+1739154857036000*n^7+18810993273800425*n^6+ 111635490990770145*n^5+394329311168044410*n^4+839691773259867277*n^3+ 1036497236180954397*n^2+662539225439611278*n+161902060919574192)/(2*n+11)/(4*n+ 21)/(2997981860*n^6+46719403320*n^5+285623082965*n^4+858760970700*n^3+ 1311628054547*n^2+984861146676*n+353752060716)/(4*n+19)*a(n+2)+1/144*(n+4)*(n+2 )*(156974330189600*n^9+4565381415394800*n^8+57396489156390900*n^7+ 407468350922643765*n^6+1789059316039629810*n^5+4996979322100289424*n^4+ 8779940722863366076*n^3+9216673017606987171*n^2+5145754405605777558*n+ 1149516531917365248)/(2*n+11)/(4*n+21)/(2997981860*n^6+46719403320*n^5+ 285623082965*n^4+858760970700*n^3+1311628054547*n^2+984861146676*n+353752060716 )/(4*n+19)*a(n+3)-1/192*(94451418499300*n^10+3172020334584000*n^9+ 46851036692819025*n^8+399320084075502630*n^7+2164974951313101870*n^6+ 7755941075849188068*n^5+18455457780588635009*n^4+28544614307643192846*n^3+ 27222456024927518940*n^2+14455606630508579880*n+3384883437351538752)/(2*n+11)/( 4*n+21)/(2997981860*n^6+46719403320*n^5+285623082965*n^4+858760970700*n^3+ 1311628054547*n^2+984861146676*n+353752060716)/(4*n+19)*a(n+4)+a(n+5) = 0 The asymptotics is CONST*n!*2.161768135^n/n^(5/2) where the CONST. is roughly, 0.5280 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 4, 5, 7], then the first, 30, terms are [0, 1, 3, 12, 65, 486, 4830, 59368, 854784, 14034510, 259392210, 5344859520, 121606184700, 3027371351004, 81826990425180, 2386108848483360, 74670475587708000, 2496245031436796640, 88787017717485927840, 3347895311407100694240, 133401012678840705959160, 5600957683373507760505200, 247148037228839078051319600, 11434723733462597224236740160, 553529809570796162186250168000, 27980524493960988881348288071200, 1474323102893912219199918526860000, 80841726632781796105404124784640000, 4605993163024527177747763943063040000, 272297143899022769891608574428446348000] The enumerating sequence satisfies the recurrence 339347/50*n*(n-1)*(n+5)*(n+4)*(1735708376875000*n^9+68498279733125000*n^8+ 1190583263801478125*n^7+11954410805321384375*n^6+76356381515892087625*n^5+ 321443845813109285225*n^4+890889879209157595910*n^3+1565355254028417737884*n^2+ 1579524265758456585224*n+695827001618385725920)*(n+1)^2*(n+2)^2*(n+3)^2/(5*n+29 )/(5*n+28)/(5*n+32)/(5*n+26)/(1735708376875000*n^9+52876904341250000*n^8+ 705082527503978125*n^7+5392480287581037500*n^6+26014960814226072000*n^5+ 81883866406250378350*n^4+167570405080867878135*n^3+213954255991629510254*n^2+ 153565172808326110036*n+46686165089282636520)*a(n)-1/5*n*(n+5)*(n+4)*(n+1)*( 101226512539350000000*n^10+4146659442844875000000*n^9+75400509967616424000000*n ^8+800355095410892519346875*n^7+5483074474286239303511250*n^6+ 25280355373691001401081750*n^5+79234932389032890737392100*n^4+ 166123594446422045919379375*n^3+221922819941211613416832250*n^2+ 169421751651491409140294640*n+55548380674879003380669088)*(n+3)^2*(n+2)^2/(5*n+ 29)/(5*n+28)/(5*n+32)/(5*n+26)/(1735708376875000*n^9+52876904341250000*n^8+ 705082527503978125*n^7+5392480287581037500*n^6+26014960814226072000*n^5+ 81883866406250378350*n^4+167570405080867878135*n^3+213954255991629510254*n^2+ 153565172808326110036*n+46686165089282636520)*a(n+1)+1/5*(n+5)*(n+4)*(n+2)*(n+1 )*(50248757510531250000*n^11+2184020228316093750000*n^10+ 42540761541443454531250*n^9+489563950816854019890625*n^8+ 3692949193251538473515625*n^7+19136901396627511295509375*n^6+ 69348253369638497202717625*n^5+175181166592243538780592800*n^4+ 300995142797071301904909500*n^3+332906894945324598877278496*n^2+ 211241523793285626540387488*n+57298861272343243543098880)*(n+3)^2/(5*n+29)/(5*n +28)/(5*n+32)/(5*n+26)/(1735708376875000*n^9+52876904341250000*n^8+ 705082527503978125*n^7+5392480287581037500*n^6+26014960814226072000*n^5+ 81883866406250378350*n^4+167570405080867878135*n^3+213954255991629510254*n^2+ 153565172808326110036*n+46686165089282636520)*a(n+2)+(n+5)*(n+4)*(n+3)*(n+2)*( 2186992554862500000*n^12+102710276625206250000*n^11+2190468406961709937500*n^10 +28040326287546718890625*n^9+239853570983657410501250*n^8+ 1443548576829356478047250*n^7+6263973360050124565270600*n^6+ 19730202508336148801759165*n^5+44724036407308031476128810*n^4+ 71051420185526833009669352*n^3+74942425794659262416612824*n^2+ 46984319517390664290927152*n+13181949471285856179761856)/(5*n+29)/(5*n+28)/(5*n +32)/(5*n+26)/(1735708376875000*n^9+52876904341250000*n^8+705082527503978125*n^ 7+5392480287581037500*n^6+26014960814226072000*n^5+81883866406250378350*n^4+ 167570405080867878135*n^3+213954255991629510254*n^2+153565172808326110036*n+ 46686165089282636520)*a(n+3)-1/2*(n+5)*(n+3)*(13408347211359375000*n^13+ 690049377474703125000*n^12+16229114117690693515625*n^11+ 230807439567363427578125*n^10+2212914138968209884137500*n^9+ 15090873480240074418766250*n^8+75225971979426157837506625*n^7+ 277175305814709031020954025*n^6+753659052583756463098907650*n^5+ 1490558477909280217514885000*n^4+2078708243862129763385911504*n^3+ 1928859163884346619910342688*n^2+1062080620922137128182619072*n+ 260162909978263986780908800)/(5*n+29)/(5*n+28)/(5*n+32)/(5*n+26)/( 1735708376875000*n^9+52876904341250000*n^8+705082527503978125*n^7+ 5392480287581037500*n^6+26014960814226072000*n^5+81883866406250378350*n^4+ 167570405080867878135*n^3+213954255991629510254*n^2+153565172808326110036*n+ 46686165089282636520)*a(n+4)-2*(644670525984375000*n^12+30759029365717578125*n^ 11+667140827068558515625*n^10+8697359667609832234375*n^9+ 75906825442693157768750*n^8+467283946927968288434375*n^7+ 2081003280899104604257375*n^6+6757457441404991096020875*n^5+ 15886531691549044295976990*n^4+26384559611269445096962906*n^3+ 29399096864707667667414212*n^2+19739289049628698264941936*n+ 6038500268364095499609600)/(5*n+29)/(5*n+28)/(5*n+32)/(5*n+26)/( 1735708376875000*n^9+52876904341250000*n^8+705082527503978125*n^7+ 5392480287581037500*n^6+26014960814226072000*n^5+81883866406250378350*n^4+ 167570405080867878135*n^3+213954255991629510254*n^2+153565172808326110036*n+ 46686165089282636520)*a(n+5)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 4, 6, 7], then the first, 30, terms are [0, 1, 3, 12, 65, 480, 4627, 54664, 757008, 12000240, 214672920, 4283532792, 94342536288, 2272888141524, 59452982168580, 1678016392852800, 50832270021973440, 1645100393412123648, 56647980596187188208, 2067991466894395922880, 79779416402257850678400, 3243110331554482317888960, 138559308962231319802960320, 6207160829194231810504368768, 290941321637876110614863049600, 14240480326184038407806012688000, 726561579601713253930433639337600, 38577336078916531361183199261580800, 2128345712858710789818098753840011200, 121839752120380310059552566257379379200] The enumerating sequence satisfies the recurrence 7093/4*n*(n-1)*(n+5)*(n+4)*(467170144794639*n^10+21870224922224250*n^9+ 457561841946603210*n^8+5632082009948598210*n^7+45150078803851387107*n^6+ 246202692448198170780*n^5+924317827980826151720*n^4+2357461105038760327600*n^3+ 3905726049802760581124*n^2+3791243781603350803360*n+1634883080534988232000)*(n+ 1)^2*(n+2)^2*(n+3)^2/(467170144794639*n^10+17198523474277860*n^9+ 281752474162343715*n^8+2702854954200488850*n^7+16798243145656126707*n^6+ 70590400148712637260*n^5+202804463775874010465*n^4+392495587195692324290*n^3+ 488245664072125620674*n^2+350972914265607981940*n+109973534809337625600)/(5*n+ 32)/(5*n+31)/(5*n+29)/(5*n+28)*a(n)-n*(n+5)*(n+4)*(n+1)*(4919301624687548670*n^ 11+237672420868052675505*n^10+5164215624597267024345*n^9+ 66560064080451245036415*n^8+564883375827907412876718*n^7+ 3310842713242277321050905*n^6+13656049072896649742483967*n^5+ 39570119042635514083808865*n^4+78765733971532448763200852*n^3+ 102284134473511807822570410*n^2+77692477942127069040975748*n+ 26015215452422904429315600)*(n+3)^2*(n+2)^2/(467170144794639*n^10+ 17198523474277860*n^9+281752474162343715*n^8+2702854954200488850*n^7+ 16798243145656126707*n^6+70590400148712637260*n^5+202804463775874010465*n^4+ 392495587195692324290*n^3+488245664072125620674*n^2+350972914265607981940*n+ 109973534809337625600)/(5*n+32)/(5*n+31)/(5*n+29)/(5*n+28)*a(n+1)+2*(n+5)*(n+4) *(n+2)*(n+1)*(5552784341029079154*n^12+282160630789673752116*n^11+ 6500859398410502977155*n^10+89744691980365398507600*n^9+ 826231712839750862299908*n^8+5340003846866709696455094*n^7+ 24820854457152503657693622*n^6+83508574018974898455343200*n^5+ 201571811394199495728162917*n^4+339871061819697174898408068*n^3+ 379187239520899721326794094*n^2+250663411937565559592280672*n+ 73978080479396956765514400)*(n+3)^2/(467170144794639*n^10+17198523474277860*n^9 +281752474162343715*n^8+2702854954200488850*n^7+16798243145656126707*n^6+ 70590400148712637260*n^5+202804463775874010465*n^4+392495587195692324290*n^3+ 488245664072125620674*n^2+350972914265607981940*n+109973534809337625600)/(5*n+ 32)/(5*n+31)/(5*n+29)/(5*n+28)*a(n+2)-1/2*(n+5)*(n+4)*(n+3)*(n+2)*( 25977463071450696234*n^13+1410946700061081867255*n^12+35036827168040783105166*n ^11+526476327516038924933715*n^10+5338568321677958686601886*n^9+ 38556682018949192619189771*n^8+203925153513281644321177382*n^7+ 799126116056705017953090361*n^6+2318177670013430291289983692*n^5+ 4911234330301757700478209730*n^4+7375741427055970108644117592*n^3+ 7422279686628836022755900368*n^2+4474835400923960339092835648*n+ 1216571822916378904455897600)/(467170144794639*n^10+17198523474277860*n^9+ 281752474162343715*n^8+2702854954200488850*n^7+16798243145656126707*n^6+ 70590400148712637260*n^5+202804463775874010465*n^4+392495587195692324290*n^3+ 488245664072125620674*n^2+350972914265607981940*n+109973534809337625600)/(5*n+ 32)/(5*n+31)/(5*n+29)/(5*n+28)*a(n+3)+1/5*(n+5)*(n+3)*(47145409502240583963*n^ 14+2772822402502991644806*n^13+75109180585043744229627*n^12+ 1241572367896632393388884*n^11+13986955964173573213991937*n^10+ 113549078439043380001568646*n^9+684695217001015995789432537*n^8+ 3113451867973925906932905924*n^7+10720571742330918642992485876*n^6+ 27792130855356734197431263748*n^5+53337179243192805893312347996*n^4+ 73376983915172027950973655672*n^3+68277301251778265237111450464*n^2+ 38367186567246308950319425920*n+9788559299869662837989952000)/(467170144794639* n^10+17198523474277860*n^9+281752474162343715*n^8+2702854954200488850*n^7+ 16798243145656126707*n^6+70590400148712637260*n^5+202804463775874010465*n^4+ 392495587195692324290*n^3+488245664072125620674*n^2+350972914265607981940*n+ 109973534809337625600)/(5*n+32)/(5*n+31)/(5*n+29)/(5*n+28)*a(n+4)-1/25*( 78285569843816835786*n^15+5034878543383601095755*n^14+150037298837932857696675* n^13+2747241612434505511916055*n^12+34554767798267395102855467*n^11+ 316130170342834998552956895*n^10+2172208276002519199140769095*n^9+ 11409286588547730981605487825*n^8+46156216637367095705419785163*n^7+ 143713110713354886111660152310*n^6+341275902256117479982186311430*n^5+ 606316753464014205416049884520*n^4+778986254906187444127773456784*n^3+ 681988194928589189942934650640*n^2+362879186555574270535888545600*n+ 88149924332074846818066240000)/(467170144794639*n^10+17198523474277860*n^9+ 281752474162343715*n^8+2702854954200488850*n^7+16798243145656126707*n^6+ 70590400148712637260*n^5+202804463775874010465*n^4+392495587195692324290*n^3+ 488245664072125620674*n^2+350972914265607981940*n+109973534809337625600)/(5*n+ 32)/(5*n+31)/(5*n+29)/(5*n+28)*a(n+5)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 5, 6, 7], then the first, 30, terms are [0, 1, 3, 12, 60, 366, 2737, 25544, 298368, 4236750, 69847470, 1285781112, 25821523728, 559754839644, 13052071182150, 327053970603360, 8797300386114240, 253410500593137888, 7788053788647180048, 254250666254593008000, 8779350194959326222240, 319458306634652179154160, 12212620258784280344329320, 489330234939055709983912128, 20508299135441190202553424000, 897487706796925781608188717600, 40943474562891798462931043277600, 1944066827604698105974012729180800, 95928218148222901651066288323091200, 4912035957345918104975581567456399200] The enumerating sequence satisfies the recurrence 1695*n*(n-1)*(n+5)*(n+4)*(1369378865812*n^10+63812051431760*n^9+ 1331044445968095*n^8+16384612413404490*n^7+132033748211639676*n^6+ 729324324105375390*n^5+2802599560602241805*n^4+7409589155837937880*n^3+ 12901470766522966692*n^2+13331694980004090000*n+6175495271645049600)*(n+1)^2*(n +2)^2*(n+3)^2/(5*n+32)/(5*n+31)/(5*n+29)/(1369378865812*n^10+50118262773640*n^9 +818358032043795*n^8+7869165233305650*n^7+49538063046467586*n^6+ 214355441549035440*n^5+648443091079988255*n^4+1355878022601245270*n^3+ 1866520416746475672*n^2+1506146349320356320*n+525874876394492160)/(5*n+28)*a(n) -15*n*(n+5)*(n+4)*(n+1)*(1027034149359000*n^11+49399589797858500*n^10+ 1070026133312449170*n^9+13783940319290599535*n^8+117425855340119560210*n^7+ 695291744654238565352*n^6+2922047898368406860116*n^5+8716494617126218522097*n^4 +18061832419715576779264*n^3+24673059856898880241236*n^2+ 19870184550929248332240*n+7078866656760317664000)*(n+3)^2*(n+2)^2/(5*n+32)/(5*n +31)/(5*n+29)/(1369378865812*n^10+50118262773640*n^9+818358032043795*n^8+ 7869165233305650*n^7+49538063046467586*n^6+214355441549035440*n^5+ 648443091079988255*n^4+1355878022601245270*n^3+1866520416746475672*n^2+ 1506146349320356320*n+525874876394492160)/(5*n+28)*a(n+1)+3*(n+5)*(n+4)*(n+2)*( n+1)*(11002959186799420*n^12+556741670001389280*n^11+12785953961816106075*n^10+ 176301838280549682265*n^9+1626738540247615062786*n^8+10591180105312576146598*n^ 7+49937812366126331833967*n^6+171924644616373439641885*n^5+ 428932499396894432658744*n^4+755461464020275768017732*n^3+ 889255950996128117827008*n^2+625251998520144121114560*n+ 197210215836911354112000)*(n+3)^2/(5*n+32)/(5*n+31)/(5*n+29)/(1369378865812*n^ 10+50118262773640*n^9+818358032043795*n^8+7869165233305650*n^7+ 49538063046467586*n^6+214355441549035440*n^5+648443091079988255*n^4+ 1355878022601245270*n^3+1866520416746475672*n^2+1506146349320356320*n+ 525874876394492160)/(5*n+28)*a(n+2)-1/2*(n+5)*(n+4)*(n+3)*(n+2)*( 70821536182065016*n^13+3831393197313251300*n^12+94838831193147378938*n^11+ 1422791875606247754379*n^10+14443014341781591201810*n^9+ 104853305568281112553407*n^8+560643725582100694690238*n^7+ 2237656389522768506021053*n^6+6671292051953120184728614*n^5+ 14675542261716684207739697*n^4+23133624476859279573462792*n^3+ 24689978716462838864146404*n^2+15925262594488726523321232*n+ 4657263410668750530405120)/(5*n+32)/(5*n+31)/(5*n+29)/(1369378865812*n^10+ 50118262773640*n^9+818358032043795*n^8+7869165233305650*n^7+49538063046467586*n ^6+214355441549035440*n^5+648443091079988255*n^4+1355878022601245270*n^3+ 1866520416746475672*n^2+1506146349320356320*n+525874876394492160)/(5*n+28)*a(n+ 3)+1/20*(n+5)*(n+3)*(419132636353407900*n^14+24560865278216836800*n^13+ 663323315218769911925*n^12+10947153973513884302940*n^11+ 123404802235999502827201*n^10+1005927263851102270973148*n^9+ 6120238467369865475746935*n^8+28262880653586438543762324*n^7+ 99641807599512739924162495*n^6+267066833506585727092540212*n^5+ 535726389679440593879827400*n^4+779195042567303239097667456*n^3+ 774940226945682474389988144*n^2+469629527190553478781720000*n+ 129881076797310485357491200)/(5*n+32)/(5*n+31)/(5*n+29)/(1369378865812*n^10+ 50118262773640*n^9+818358032043795*n^8+7869165233305650*n^7+49538063046467586*n ^6+214355441549035440*n^5+648443091079988255*n^4+1355878022601245270*n^3+ 1866520416746475672*n^2+1506146349320356320*n+525874876394492160)/(5*n+28)*a(n+ 4)-1/50*(328511251150567176*n^15+21057330409511286060*n^14+ 625813935780516819590*n^13+11441748229944293848745*n^12+ 143976910951830631061002*n^11+1321608564291335862073285*n^10+ 9149159826970413227028270*n^9+48685219646175959856126195*n^8+ 200973144262886179573408678*n^7+644156683064193656240710075*n^6+ 1590888102116457036978650420*n^5+2972778019948138079949413960*n^4+ 4063562299314930955877629344*n^3+3825020634749691851740967280*n^2+ 2205547055078623778545686720*n+582204025145493328240051200)/(5*n+32)/(5*n+31)/( 5*n+29)/(1369378865812*n^10+50118262773640*n^9+818358032043795*n^8+ 7869165233305650*n^7+49538063046467586*n^6+214355441549035440*n^5+ 648443091079988255*n^4+1355878022601245270*n^3+1866520416746475672*n^2+ 1506146349320356320*n+525874876394492160)/(5*n+28)*a(n+5)+a(n+6) = 0 The asymptotics is CONST*n!*1.859932067^n/n^(5/2) where the CONST. is roughly, 0.7270 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 3, 4, 5, 6], then the first, 30, terms are [0, 1, 0, 4, 5, 96, 427, 6440, 56952, 890190, 11891880, 210272832, 3683179500, 75296885664, 1605918844530, 37835092495200, 943203720338880, 25322925098671200, 719842030821897984, 21749710027301736480, 693337132064381084280, 23307087763113699504960, 823011224755921750949400, 30477497830390123363730880, 1180675637555807945593641600, 47762839129388517621072986400, 2013943419851560007919405897600, 88369401092592109774973008320000, 4028930022025698923813273928734400, 190589555323840196305880053523568000] The enumerating sequence satisfies the recurrence -17135/576*n*(n-1)*(n+4)*(n+3)*(101931542*n^6+2525909476*n^5+25358833650*n^4+ 130888030885*n^3+360989177083*n^2+490580831104*n+240659227560)*(n+1)^2*(n+2)^2/ (2*n+11)/(4*n+21)/(4*n+19)/(101931542*n^6+1914320224*n^5+14258259400*n^4+ 52673160205*n^3+96747964698*n^2+71849193121*n+3114398370)*a(n)-25/576*n*(n+4)*( n+3)*(n+1)*(287446948440*n^7+7554235144980*n^6+82246372032784*n^5+ 476663729915216*n^4+1566099201672579*n^3+2846142757261894*n^2+2517150219714803* n+721891951062504)*(n+2)^2/(2*n+11)/(4*n+21)/(4*n+19)/(101931542*n^6+1914320224 *n^5+14258259400*n^4+52673160205*n^3+96747964698*n^2+71849193121*n+3114398370)* a(n+1)-5/576*(n+1)*(n+2)*(n+3)*(n+4)*(1925894554548*n^8+55428111857736*n^7+ 678298511424786*n^6+4581588476223630*n^5+18503836971992087*n^4+ 45007730390684994*n^3+62312851459880359*n^2+41351209815156780*n+ 7066420909833600)/(2*n+11)/(4*n+21)/(4*n+19)/(101931542*n^6+1914320224*n^5+ 14258259400*n^4+52673160205*n^3+96747964698*n^2+71849193121*n+3114398370)*a(n+2 )-5/576*(n+4)*(n+2)*(555730766984*n^9+17939239215532*n^8+252842349887816*n^7+ 2031743094792438*n^6+10172009465168015*n^5+32424794877614073*n^4+ 63949485725490279*n^3+70292798480714717*n^2+30421118502043746*n-\ 4276887626264400)/(2*n+11)/(4*n+21)/(4*n+19)/(101931542*n^6+1914320224*n^5+ 14258259400*n^4+52673160205*n^3+96747964698*n^2+71849193121*n+3114398370)*a(n+3 )+5/192*(n+2)*(218847020674*n^9+7611597851712*n^8+115383519210982*n^7+ 996780934626151*n^6+5377955160218822*n^5+18629407523662868*n^4+ 40822333421238968*n^3+53026488956959749*n^2+34617615787649034*n+ 6738362643131280)/(2*n+11)/(4*n+21)/(4*n+19)/(101931542*n^6+1914320224*n^5+ 14258259400*n^4+52673160205*n^3+96747964698*n^2+71849193121*n+3114398370)*a(n+4 )+a(n+5) = 0 The asymptotics is CONST*n!*1.716653699^n/n^(5/2) where the CONST. is roughly, 0.3211 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 3, 4, 5, 7], then the first, 30, terms are [0, 1, 0, 4, 5, 96, 420, 6448, 55440, 887670, 11513040, 207789120, 3552368820, 73603854324, 1542407466600, 36591311877120, 901120845424800, 24248703781542240, 683381086443095040, 20635880533512884640, 653566199444265907800, 21920837282516847074400, 769946895188846957181600, 28422765499717847641743360, 1095899910366825271411704000, 44172917469809225675409031200, 1854389450166114724288826400000, 81054045425679720234119959680000, 3679725030166918342689420060720000, 173376052999011038624918081559948000] The enumerating sequence satisfies the recurrence -177874/25*n*(n-1)*(n+5)*(n+4)*(682163305937500*n^9+27487848183812500*n^8+ 489452107811128125*n^7+5055951277714729375*n^6+33402641549070426125*n^5+ 146443232324504785725*n^4+426356949478865452450*n^3+795574675133064058296*n^2+ 864272808438855168016*n+416966825799709373760)*(n+1)^2*(n+2)^2*(n+3)^2/(5*n+29) /(5*n+26)/(5*n+28)/(5*n+32)/(682163305937500*n^9+21348378430375000*n^8+ 294107201354378125*n^7+2342144554484832500*n^6+11892061225070365500*n^5+ 39976666767802861350*n^4+89140216110050717675*n^3+127442681924612404046*n^2+ 106311278829328854624*n+39545638645268647440)*a(n)-8/5*n*(n+5)*(n+4)*(n+1)*( 9925476101390625000*n^10+414836405226557812500*n^9+7723406900291659875000*n^8+ 84322028673819221796875*n^7+597644888859193919077500*n^6+ 2872446379213051781623875*n^5+9478170671195645601029800*n^4+ 21194211152280407399812450*n^3+30723398789627046348842648*n^2+ 26055394629255555878035560*n+9806834460565101168541152)*(n+3)^2*(n+2)^2/(5*n+29 )/(5*n+26)/(5*n+28)/(5*n+32)/(682163305937500*n^9+21348378430375000*n^8+ 294107201354378125*n^7+2342144554484832500*n^6+11892061225070365500*n^5+ 39976666767802861350*n^4+89140216110050717675*n^3+127442681924612404046*n^2+ 106311278829328854624*n+39545638645268647440)*a(n+1)-2/5*(n+5)*(n+4)*(n+2)*(n+1 )*(40755846713235937500*n^11+1805284876594821562500*n^10+ 35991702223630270609375*n^9+426295021363845912556250*n^8+ 3333121716930909009763125*n^7+18067680013616570972050125*n^6+ 69312030595098169280708250*n^5+188297991062765124489090045*n^4+ 355340189987513999189695430*n^3+444192047532407028708646168*n^2+ 331579401989072524003770864*n+112201763099444199472904640)*(n+3)^2/(5*n+29)/(5* n+26)/(5*n+28)/(5*n+32)/(682163305937500*n^9+21348378430375000*n^8+ 294107201354378125*n^7+2342144554484832500*n^6+11892061225070365500*n^5+ 39976666767802861350*n^4+89140216110050717675*n^3+127442681924612404046*n^2+ 106311278829328854624*n+39545638645268647440)*a(n+2)-2*(n+5)*(n+4)*(n+3)*(n+2)* (1139212720915625000*n^12+54448801873834062500*n^11+1188026373368945843750*n^10 +15661084928111389446875*n^9+139061034525661823594375*n^8+ 877272698968283649450750*n^7+4037472458843819219372250*n^6+ 13680567751816510917315545*n^5+33931063557453582363610345*n^4+ 60184357039434234498957930*n^3+72589037995530417389359064*n^2+ 53528370523772565465106608*n+18267095141593798032402624)/(5*n+29)/(5*n+26)/(5*n +28)/(5*n+32)/(682163305937500*n^9+21348378430375000*n^8+294107201354378125*n^7 +2342144554484832500*n^6+11892061225070365500*n^5+39976666767802861350*n^4+ 89140216110050717675*n^3+127442681924612404046*n^2+106311278829328854624*n+ 39545638645268647440)*a(n+3)+1/2*(n+5)*(n+3)*(10078962845226562500*n^13+ 527080511058548437500*n^12+12614208929871752421875*n^11+ 182890795878261929796875*n^10+1792013180628178383012500*n^9+ 12528526483954269481571250*n^8+64289410579348821959244375*n^7+ 245133665428484178400591275*n^6+694464953371773572979164950*n^5+ 1443631560232423954269970300*n^4+2140240278447840136492116232*n^3+ 2142769571369702402423484192*n^2+1298357959708370919542301888*n+ 359502214822197482941367040)/(5*n+29)/(5*n+26)/(5*n+28)/(5*n+32)/( 682163305937500*n^9+21348378430375000*n^8+294107201354378125*n^7+ 2342144554484832500*n^6+11892061225070365500*n^5+39976666767802861350*n^4+ 89140216110050717675*n^3+127442681924612404046*n^2+106311278829328854624*n+ 39545638645268647440)*a(n+4)+5*(n+3)*(511622479453125000*n^13+ 28034412089929687500*n^12+703720829262374218750*n^11+10712681654972758828125*n^ 10+110320000051734445312500*n^9+811468513581589303425000*n^8+ 4385814470467637872015000*n^7+17635557339487171541077375*n^6+ 52764842449484017268093550*n^5+116050272482991738365498560*n^4+ 182461238573886132814712944*n^3+194342037798346937259250048*n^2+ 125808423575392004351862144*n+37429506276741647203935744)/(5*n+29)/(5*n+26)/(5* n+28)/(5*n+32)/(682163305937500*n^9+21348378430375000*n^8+294107201354378125*n^ 7+2342144554484832500*n^6+11892061225070365500*n^5+39976666767802861350*n^4+ 89140216110050717675*n^3+127442681924612404046*n^2+106311278829328854624*n+ 39545638645268647440)*a(n+5)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 3, 4, 6, 7], then the first, 30, terms are [0, 1, 0, 4, 5, 90, 427, 5608, 54432, 738360, 10639860, 168831432, 3074783712, 58512121668, 1256979283560, 28272350346240, 696374667468480, 18089674720685568, 503431565475606768, 14800588796378024640, 460311086060855789760, 15083057039425411196160, 519127832894042964271680, 18744416794228694721280128, 707689535878152607286707200, 27911478824017640432409628800, 1147065800952020717488258137600, 49064565578645370368736209088000, 2180348621084434758821158461566400, 100537058964599057953404596187043200] The enumerating sequence satisfies the recurrence 658629/100*n*(n-1)*(n+5)*(n+4)*(227209380174035563*n^10+10468968165362743630*n^ 9+215199370831819440570*n^8+2597397332901289237656*n^7+20371753613481986485461* n^6+108406766693546351571582*n^5+396028835703982124783210*n^4+ 979661346188688333451364*n^3+1568395957409501597366196*n^2+ 1465000970073134674268688*n+605021824531327560842880)*(n+1)^2*(n+2)^2*(n+3)^2/( 5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/(227209380174035563*n^10+8196874363622388000 *n^9+131203079451386348235*n^8+1225420094578908216216*n^7+ 7383875310409983161139*n^6+29932257462031466713176*n^5+82454979726163243500005* n^4+151888468123816441168524*n^3+178207150294614270452298*n^2+ 119433197774448813178764*n+34356848582069251680960)*a(n)+27/50*n*(n+5)*(n+4)*(n +1)*(2303903114964720608820*n^11+109611191869225301321430*n^10+ 2343128842791693763446390*n^9+29687991468422810846663885*n^8+ 247550298226445749651334118*n^7+1425307053368233691470326534*n^6+ 5777553807955118384618831208*n^5+16474349661163526767847248675*n^4+ 32356367608175865808076651712*n^3+41654002906221914807301233076*n^2+ 31611367155284819658248871432*n+10709493144630580565200962720)*(n+3)^2*(n+2)^2/ (5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/(227209380174035563*n^10+ 8196874363622388000*n^9+131203079451386348235*n^8+1225420094578908216216*n^7+ 7383875310409983161139*n^6+29932257462031466713176*n^5+82454979726163243500005* n^4+151888468123816441168524*n^3+178207150294614270452298*n^2+ 119433197774448813178764*n+34356848582069251680960)*a(n+1)-9/50*(n+5)*(n+4)*(n+ 2)*(n+1)*(124965159095719559650*n^12+6257793127332387235100*n^11+ 139153671131851386030755*n^10+1785884105291627408372790*n^9+ 14292225532133427290425724*n^8+70182992213317010809793546*n^7+ 170105111307234175462220425*n^6-193087311684901072750539780*n^5-\ 3086429575369034311536664314*n^4-11353298516913068714549117296*n^3-\ 22101039112919770372788867000*n^2-23138639102334015418793476800*n-\ 10276035086041837712217648000)*(n+3)^2/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/( 227209380174035563*n^10+8196874363622388000*n^9+131203079451386348235*n^8+ 1225420094578908216216*n^7+7383875310409983161139*n^6+29932257462031466713176*n ^5+82454979726163243500005*n^4+151888468123816441168524*n^3+ 178207150294614270452298*n^2+119433197774448813178764*n+34356848582069251680960 )*a(n+2)-3/50*(n+5)*(n+4)*(n+3)*(n+2)*(461235041753292192890*n^13+ 24711268188836061015575*n^12+595736696154497947177390*n^11+ 8495566799067615898970935*n^10+79066756796721028902463158*n^9+ 497319703010542846801150683*n^8+2089389265323918910894352782*n^7+ 5323213401570538073262660517*n^6+4412673725524058684392471292*n^5-\ 21301464774990378408548895598*n^4-91969094761224870214827793032*n^3-\ 171056197175904885482621954112*n^2-164905766723338685758000846080*n-\ 66969599766796005021815040000)/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/( 227209380174035563*n^10+8196874363622388000*n^9+131203079451386348235*n^8+ 1225420094578908216216*n^7+7383875310409983161139*n^6+29932257462031466713176*n ^5+82454979726163243500005*n^4+151888468123816441168524*n^3+ 178207150294614270452298*n^2+119433197774448813178764*n+34356848582069251680960 )*a(n+3)-3/25*(n+5)*(n+3)*(2310719396369941675710*n^14+134198038998178402825620 *n^13+3582568747628037998983090*n^12+58229975160348172147025685*n^11+ 643232969202061698339731201*n^10+5103365526459492083111854464*n^9+ 29954433394741671835087912068*n^8+131944513327644975463890411261*n^7+ 437498737325127059446169048435*n^6+1084169019682679957920891314546*n^5+ 1970552561683684196701418718472*n^4+2536726063928962153508411517864*n^3+ 2173331151494597589004093716624*n^2+1099088742942994586854632730560*n+ 243761931335022570536493888000)/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/( 227209380174035563*n^10+8196874363622388000*n^9+131203079451386348235*n^8+ 1225420094578908216216*n^7+7383875310409983161139*n^6+29932257462031466713176*n ^5+82454979726163243500005*n^4+151888468123816441168524*n^3+ 178207150294614270452298*n^2+119433197774448813178764*n+34356848582069251680960 )*a(n+4)-3/25*(n+3)*(2751051175147222596804*n^14+166648508585846827525698*n^13+ 4647474347088970419216721*n^12+79049284556474813203529085*n^11+ 915662568822289672537838588*n^10+7636383910460096601823971594*n^9+ 47251313143799230732490306493*n^8+220191627187037553317048162445*n^7+ 775803891875713492289172797402*n^6+2054303226742051693029115597818*n^5+ 4018971826555932185991455484136*n^4+5623789396605841941189875071920*n^3+ 5310140669699208861459505166256*n^2+3020804699444928864267764610240*n+ 778618485769971400794859968000)/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/( 227209380174035563*n^10+8196874363622388000*n^9+131203079451386348235*n^8+ 1225420094578908216216*n^7+7383875310409983161139*n^6+29932257462031466713176*n ^5+82454979726163243500005*n^4+151888468123816441168524*n^3+ 178207150294614270452298*n^2+119433197774448813178764*n+34356848582069251680960 )*a(n+5)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 3, 5, 6, 7], then the first, 30, terms are [0, 1, 0, 4, 0, 96, 7, 5888, 1512, 686070, 388080, 130189752, 130882752, 36604876308, 57772644930, 14305251280320, 32692676496480, 7420453117485408, 23199752172765168, 4934376416182973760, 20234943881040208320, 4093464142687135376160, 21315353571113374052280, 4144997065835342324095488, 26709916028243647384785600, 5032130510785705130132128800, 39295975535518282954150377600, 7215660842382395697887440848000, 67107550629222390177341177126400, 12066704916291271344113435243023200] The enumerating sequence satisfies the recurrence -68221/100*n*(n-1)*(n+5)*(n+4)*(1165947087567619112*n^10+52843418602084722240*n ^9+1068626778189524988530*n^8+12692734141629453521789*n^7+ 98017886447429575429520*n^6+513983364987737365549550*n^5+ 1852514350438810712560896*n^4+4529063877231259509831881*n^3+ 7183735802886221192901138*n^2+6670797026052721064072584*n+ 2751764964164940365748160)*(n+1)^2*(n+2)^2*(n+3)^2/(5*n+28)/(5*n+29)/(5*n+31)/( 5*n+32)/(1165947087567619112*n^10+41183947726408531120*n^9+ 645503629711305348410*n^8+5906169335280189320749*n^7+34896298971144183801197*n^ 6+138944815776560672558335*n^5+377010579872329917201711*n^4+ 687183925295283282773052*n^3+803589158023730583298842*n^2+ 543033212535356793746320*n+160512950830729461549312)*a(n)+1/100*n*(n+5)*(n+4)*( n+1)*(428415597855845966513280*n^11+20059409127933779289635520*n^10+ 421801972418958805577479200*n^9+5253573053963353876053457000*n^8+ 43023788490226991872849375753*n^7+242991780552826881040995085920*n^6+ 964560246656437487888860950832*n^5+2687004812195240439962453674714*n^4+ 5138533044497092952147785021079*n^3+6410006760524594461350043752998*n^2+ 4680472106823412777689614781464*n+1509556281873559790144555063360)*(n+3)^2*(n+2 )^2/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/(1165947087567619112*n^10+ 41183947726408531120*n^9+645503629711305348410*n^8+5906169335280189320749*n^7+ 34896298971144183801197*n^6+138944815776560672558335*n^5+ 377010579872329917201711*n^4+687183925295283282773052*n^3+ 803589158023730583298842*n^2+543033212535356793746320*n+ 160512950830729461549312)*a(n+1)-1/100*(n+5)*(n+4)*(n+2)*(n+1)*( 743885901339016669647120*n^12+36690173107672140314930880*n^11+ 819570699860166953800550700*n^10+10957014639812916105065819990*n^9+ 97582816221538555438077366593*n^8+609472255696795448116752736518*n^7+ 2735136058884659364896840884474*n^6+8878477371294569140672971407254*n^5+ 20668249713755466428937920117561*n^4+33608859603704740874618238834542*n^3+ 36183203325517006328713449315472*n^2+23113000208742759732504188546656*n+ 6608244574131501641050923413760)*(n+3)^2/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/( 1165947087567619112*n^10+41183947726408531120*n^9+645503629711305348410*n^8+ 5906169335280189320749*n^7+34896298971144183801197*n^6+138944815776560672558335 *n^5+377010579872329917201711*n^4+687183925295283282773052*n^3+ 803589158023730583298842*n^2+543033212535356793746320*n+ 160512950830729461549312)*a(n+2)-1/100*(n+5)*(n+4)*(n+3)*(n+2)*( 326628417111192818035680*n^13+17253268415521960223581200*n^12+ 415637978260537773532350200*n^11+6042340999193212609289258920*n^10+ 59096838047819825916340510247*n^9+410289813523722274011210309561*n^8+ 2078280899701004182709201960368*n^7+7768289956007114632072595649250*n^6+ 21399555119589131892737421719903*n^5+42845260314348840994255474583757*n^4+ 60495130155395150320151149056098*n^3+56920961483914069476180780687472*n^2+ 31906924953527024213751098559264*n+8021473510448986153888784759040)/(5*n+28)/(5 *n+29)/(5*n+31)/(5*n+32)/(1165947087567619112*n^10+41183947726408531120*n^9+ 645503629711305348410*n^8+5906169335280189320749*n^7+34896298971144183801197*n^ 6+138944815776560672558335*n^5+377010579872329917201711*n^4+ 687183925295283282773052*n^3+803589158023730583298842*n^2+ 543033212535356793746320*n+160512950830729461549312)*a(n+3)+1/100*(n+5)*(n+3)*( 178057609477888952689080*n^14+10206673985552035789150560*n^13+ 268832758654481799709627910*n^12+4309991648434928753480894675*n^11+ 46959728536364851597498397547*n^10+367581124565984413737686773762*n^9+ 2130011212722690243469916473064*n^8+9273864481549375138588328561779*n^7+ 30454732121724307964665246157567*n^6+74976194159245828173681717475960*n^5+ 136021406506880961968522174236096*n^4+176035463292146936474750511934976*n^3+ 153310875275660732791557715160336*n^2+80218179366303985290102173125568*n+ 18962455613436065174974818362880)/(5*n+28)/(5*n+29)/(5*n+31)/(5*n+32)/( 1165947087567619112*n^10+41183947726408531120*n^9+645503629711305348410*n^8+ 5906169335280189320749*n^7+34896298971144183801197*n^6+138944815776560672558335 *n^5+377010579872329917201711*n^4+687183925295283282773052*n^3+ 803589158023730583298842*n^2+543033212535356793746320*n+ 160512950830729461549312)*a(n+4)+1/50*(n+3)*(26352736073203327169424*n^14+ 1576481620305767136025128*n^13+43416857174440458148576156*n^12+ 729329248381045418492015060*n^11+8344898464746884863426960935*n^10+ 68764552712231428416962815360*n^9+420613026156617558664402369302*n^8+ 1938871916860885652423668990660*n^7+6763579984741156072952253295575*n^6+ 17754184061940255457707485935200*n^5+34488368953292195221979154668752*n^4+ 48021805240432914439043969961760*n^3+45245023696982622938454588527216*n^2+ 25773626836980882309197914380352*n+6681405624275222718520523512320)/(5*n+28)/(5 *n+29)/(5*n+31)/(5*n+32)/(1165947087567619112*n^10+41183947726408531120*n^9+ 645503629711305348410*n^8+5906169335280189320749*n^7+34896298971144183801197*n^ 6+138944815776560672558335*n^5+377010579872329917201711*n^4+ 687183925295283282773052*n^3+803589158023730583298842*n^2+ 543033212535356793746320*n+160512950830729461549312)*a(n+5)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 4, 5, 6, 7], then the first, 30, terms are [0, 1, 0, 0, 5, 6, 7, 568, 2520, 8190, 300300, 2816352, 17909892, 463386924, 6943866930, 71737105440, 1634922569280, 32847878071008, 488685451638384, 11153303856414720, 270008469763152120, 5305324284813112560, 129914198343842192280, 3578070447871420108608, 87089508359821055088000, 2360374900591723946330400, 71996550262238197053057600, 2071174093258927441232448000, 62705260036448174354921712000, 2094296744534332979984013871200] The enumerating sequence satisfies the recurrence 71219/100*n*(n-1)*(n+5)*(n+4)*(124797744971643550*n^10+5603334992350975500*n^9+ 112139877270594138200*n^8+1316547284412409704325*n^7+10034694532852838798245*n^ 6+51846084375242811442315*n^5+183738894227953880558785*n^4+ 440602456353985648013144*n^3+683434833243842029292536*n^2+ 618415298289316498677800*n+247518133224260304393120)*(n+1)^2*(n+2)^2*(n+3)^2/(5 *n+32)/(5*n+31)/(5*n+29)/(124797744971643550*n^10+4355357542634540000*n^9+ 67325760863159318450*n^8+606172596575694490725*n^7+3514307492629169941070*n^6+ 13680148200936479442870*n^5+36119714116447477576010*n^4+63668858774783416864429 *n^3+71416723219687957076864*n^2+45787572641074443487800*n+ 12652830265974900011352)/(5*n+28)*a(n)-1/100*n*(n+5)*(n+4)*(n+1)*( 12145316540640350286000*n^11+563534536266557461089000*n^10+ 11729041665782010092312000*n^9+144393775448055874072981400*n^8+ 1166789299910260703409550225*n^7+6488281574082261335287341480*n^6+ 25288846289749779280802978800*n^5+68924309740311743830670079450*n^4+ 128338421148243493513502253503*n^3+154841088384071535496298835958*n^2+ 108298178396487643292176776984*n+32963898469369321057847430720)*(n+3)^2*(n+2)^2 /(5*n+32)/(5*n+31)/(5*n+29)/(124797744971643550*n^10+4355357542634540000*n^9+ 67325760863159318450*n^8+606172596575694490725*n^7+3514307492629169941070*n^6+ 13680148200936479442870*n^5+36119714116447477576010*n^4+63668858774783416864429 *n^3+71416723219687957076864*n^2+45787572641074443487800*n+ 12652830265974900011352)/(5*n+28)*a(n+1)-1/100*(n+5)*(n+4)*(n+2)*(n+1)*( 4152020975206580908500*n^12+203031039096343278519000*n^11+ 4510196095565404375303250*n^10+60218546313793245572220350*n^9+ 538650603433260091356760975*n^8+3404412532870583548105379970*n^7+ 15611328976580328616897264920*n^6+52421437967984615827771877370*n^5+ 128162589484897213891289123467*n^4+222879447792132304235848483246*n^3+ 262063260308656455699515963528*n^2+187190213561381902983059213664*n+ 61410150613579328295476538240)*(n+3)^2/(5*n+32)/(5*n+31)/(5*n+29)/( 124797744971643550*n^10+4355357542634540000*n^9+67325760863159318450*n^8+ 606172596575694490725*n^7+3514307492629169941070*n^6+13680148200936479442870*n^ 5+36119714116447477576010*n^4+63668858774783416864429*n^3+ 71416723219687957076864*n^2+45787572641074443487800*n+12652830265974900011352)/ (5*n+28)*a(n+2)-1/100*(n+5)*(n+4)*(n+3)*(n+2)*(20379471753869391715000*n^13+ 1067870642404934737012500*n^12+25502746686114324973696000*n^11+ 367245461147033555207691150*n^10+3554449003093992060723336875*n^9+ 24391209538541146807503573305*n^8+121935386751335083028809818690*n^7+ 448964547396183850839740585740*n^6+1215356224333344016257092549463*n^5+ 2383729081611083073039229467013*n^4+3283575422606588194801190006036*n^3+ 2997695137020253598852237550932*n^2+1618218002542465033459218482736*n+ 387709810144630566276647180160)/(5*n+32)/(5*n+31)/(5*n+29)/(124797744971643550* n^10+4355357542634540000*n^9+67325760863159318450*n^8+606172596575694490725*n^7 +3514307492629169941070*n^6+13680148200936479442870*n^5+36119714116447477576010 *n^4+63668858774783416864429*n^3+71416723219687957076864*n^2+ 45787572641074443487800*n+12652830265974900011352)/(5*n+28)*a(n+3)+1/100*(n+5)* (n+3)*(n+2)*(8403256157665618439250*n^13+461333123286609119685000*n^12+ 11578780544789324890604000*n^11+175829197702764334650997775*n^10+ 1801358427294997261784561200*n^9+13139022408439639931818272255*n^8+ 70139904776698149827720303220*n^7+277189513564018629997861574385*n^6+ 809969505325618552047960073678*n^5+1725716785183561322739802199869*n^4+ 2600494680859149836883365091692*n^3+2617293287599584387959926988796*n^2+ 1570686447048061956254288574000*n+421968780799902206898675897600)/(5*n+32)/(5*n +31)/(5*n+29)/(124797744971643550*n^10+4355357542634540000*n^9+ 67325760863159318450*n^8+606172596575694490725*n^7+3514307492629169941070*n^6+ 13680148200936479442870*n^5+36119714116447477576010*n^4+63668858774783416864429 *n^3+71416723219687957076864*n^2+45787572641074443487800*n+ 12652830265974900011352)/(5*n+28)*a(n+4)+1/50*(n+3)*(1884695544561760892100*n^ 14+111949650450629964936450*n^13+3062664115657867589113050*n^12+ 51136328134110914891147800*n^11+582026442455233166085073015*n^10+ 4776145213195270753048089295*n^9+29135891098160178958225680910*n^8+ 134211134664983748657375909378*n^7+469094654606785824648433060599*n^6+ 1238109643388202292862604842273*n^5+2429580474570372419489258388590*n^4+ 3438493348201227673768353682292*n^3+3319565537728799026315453536536*n^2+ 1958260941857285813357547708192*n+533083699551675022370836830720)/(5*n+32)/(5*n +31)/(5*n+29)/(124797744971643550*n^10+4355357542634540000*n^9+ 67325760863159318450*n^8+606172596575694490725*n^7+3514307492629169941070*n^6+ 13680148200936479442870*n^5+36119714116447477576010*n^4+63668858774783416864429 *n^3+71416723219687957076864*n^2+45787572641074443487800*n+ 12652830265974900011352)/(5*n+28)*a(n+5)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3, 4, 5, 6], then the first, 30, terms are [0, 1, 3, 16, 125, 1296, 16807, 262136, 4782456, 99976590, 2356983090, 61878485592, 1790573140356, 56627151108528, 1943277124558770, 71924499646839840, 2856075563106205440, 121122832073591668320, 5463859762091211665184, 261242523824583455874720, 13197150499740769020939480, 702378921917108770025852160, 39282754344098437935309226680, 2303350063648430795548902031680, 141294172216332539051646792393600, 9050078880561199154500515425536800, 604186411653551423427733377132357600, 41972773188218803408123455453897052800, 3029592468470398256728980332516847585600, 226887446467390337050816751123158151760000] The enumerating sequence satisfies the recurrence -1/8*n*(n-1)*(n+4)*(n+3)*(50965771*n^6+1226578858*n^5+12189436410*n^4+ 63999590228*n^3+187175660327*n^2+289049048502*n+184120412688)*(n+1)^2*(n+2)^2/( 2*n+11)/(4*n+21)/(50965771*n^6+920784232*n^5+6821028685*n^4+26488317748*n^3+ 56812206088*n^2+63765852556*n+29261257608)/(4*n+19)*a(n)-1/8*n*(n+4)*(n+3)*(n+1 )*(36375880*n^5+767226310*n^4+6324994361*n^3+25238331059*n^2+48035869026*n+ 33998367600)*(n+2)^2/(2*n+11)/(4*n+21)/(50965771*n^6+920784232*n^5+6821028685*n ^4+26488317748*n^3+56812206088*n^2+63765852556*n+29261257608)/(4*n+19)*a(n+1)-1 /16*(n+1)*(n+2)*(n+3)*(n+4)*(5351405955*n^8+150196403910*n^7+1815491652510*n^6+ 12332549764320*n^5+51443766600217*n^4+134798562100340*n^3+216426866014856*n^2+ 194403138371404*n+74669571935616)/(2*n+11)/(4*n+21)/(50965771*n^6+920784232*n^5 +6821028685*n^4+26488317748*n^3+56812206088*n^2+63765852556*n+29261257608)/(4*n +19)*a(n+2)+1/144*(n+4)*(n+2)*(359818343260*n^9+11358284311930*n^8+ 157468271817285*n^7+1257754288568145*n^6+6374903128163382*n^5+21249863593431384 *n^4+46554099857812847*n^3+64591908548157653*n^2+51462913133639490*n+ 17925565909852944)/(2*n+11)/(4*n+21)/(50965771*n^6+920784232*n^5+6821028685*n^4 +26488317748*n^3+56812206088*n^2+63765852556*n+29261257608)/(4*n+19)*a(n+3)-1/ 192*(1018550933435*n^10+36735789678350*n^9+590387800918890*n^8+5564748260194320 *n^7+34047337481385819*n^6+141208663589699550*n^5+401785004486010400*n^4+ 773911209471892564*n^3+965114086594793184*n^2+703155279765759408*n+ 227148568987132800)/(2*n+11)/(4*n+21)/(50965771*n^6+920784232*n^5+6821028685*n^ 4+26488317748*n^3+56812206088*n^2+63765852556*n+29261257608)/(4*n+19)*a(n+4)+a( n+5) = 0 The asymptotics is CONST*n!*2.716653699^n/n^(5/2) where the CONST. is roughly, 0.4039 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3, 4, 5, 7], then the first, 30, terms are [0, 1, 3, 16, 125, 1296, 16800, 261808, 4770864, 99591030, 2344047090, 61428379320, 1774136183820, 55994053159044, 1917516997976580, 70817387981820480, 2805869680155141600, 118724023582266342240, 5343305935294554907680, 254881245374203471863840, 12845331697434741693147960, 682019929167037917309453600, 38052101516423748865792480800, 2225766583839097484597850197760, 136200807868748777788771329192000, 8702363017770658052455774470535200, 579534801034946999396056919076060000, 40160111503228731677773903359687120000, 2891516013115052887889664871403551680000, 216004062149037528416842925558183106348000] The enumerating sequence satisfies the recurrence -6523/50*n*(n-1)*(n+5)*(n+4)*(1364326611875000*n^9+54095772363875000*n^8+ 946916815533428125*n^7+9602803338631236875*n^6+62165971375757971625*n^5+ 266379663333559172325*n^4+755390195087824758650*n^3+1366767484534644944032*n^2+ 1431513951058613795632*n+661141494626547071040)*(n+1)^2*(n+2)^2*(n+3)^2/(5*n+28 )/(5*n+29)/(5*n+26)/(5*n+32)/(1364326611875000*n^9+41816832857000000*n^8+ 563266394649928125*n^7+4374463820688240000*n^6+21576946370891791000*n^5+ 70064566902942882950*n^4+149702517465233532475*n^3+202837528336373548282*n^2+ 158051881234893802968*n+53927142941404470240)*a(n)-6/5*(n+5)*(n+4)*(n+1)*( 61394697534375000*n^10+2526401802675937500*n^9+46358383738173515625*n^8+ 499299405603534128125*n^7+3493572890112771266250*n^6+16580832865987322282875*n^ 5+54003979315707333070425*n^4+119018741539674630964290*n^3+ 169512234854726811657218*n^2+140446438632973337826940*n+51151553945619507961936 )*(n+3)^2*(n+2)^2*n/(5*n+28)/(5*n+29)/(5*n+26)/(5*n+32)/(1364326611875000*n^9+ 41816832857000000*n^8+563266394649928125*n^7+4374463820688240000*n^6+ 21576946370891791000*n^5+70064566902942882950*n^4+149702517465233532475*n^3+ 202837528336373548282*n^2+158051881234893802968*n+53927142941404470240)*a(n+1)+ 1/5*(n+5)*(n+4)*(n+2)*(n+1)*(15839831963868750000*n^11+691411245000063750000*n^ 10+13566066166032150218750*n^9+157851513231211130071875*n^8+ 1209558545893283396710000*n^7+6404698857403569140322000*n^6+ 23896078356221038368767000*n^5+62771184644728687916426545*n^4+ 113665535293100653602548230*n^3+134985988822409451173440688*n^2+ 94505615952466365156306864*n+29508474055982261745706560)*(n+3)^2/(5*n+28)/(5*n+ 29)/(5*n+26)/(5*n+32)/(1364326611875000*n^9+41816832857000000*n^8+ 563266394649928125*n^7+4374463820688240000*n^6+21576946370891791000*n^5+ 70064566902942882950*n^4+149702517465233532475*n^3+202837528336373548282*n^2+ 158051881234893802968*n+53927142941404470240)*a(n+2)-6*(n+5)*(n+4)*(n+3)*(n+2)* (1794089494615625000*n^12+84591611868112812500*n^11+1810742981224305484375*n^10 +23261477031058081896875*n^9+199671471408914856815000*n^8+ 1206074144780580782548625*n^7+5254528010893172460985250*n^6+ 16629975178244651219817730*n^5+37928577629774371962351055*n^4+ 60764138273094040298438370*n^3+64871847146621829805645068*n^2+ 41413164998058605737440664*n+11947281331739297816164128)/(5*n+28)/(5*n+29)/(5*n +26)/(5*n+32)/(1364326611875000*n^9+41816832857000000*n^8+563266394649928125*n^ 7+4374463820688240000*n^6+21576946370891791000*n^5+70064566902942882950*n^4+ 149702517465233532475*n^3+202837528336373548282*n^2+158051881234893802968*n+ 53927142941404470240)*a(n+3)-1/2*(n+5)*(n+3)*(5423198282203125000*n^13+ 280109074532840625000*n^12+6631494195973770546875*n^11+95259534147347935796875* n^10+926055221129971832162500*n^9+6431480834695217853601250*n^8+ 32816575359167933894976875*n^7+124504234676561716341819575*n^6+ 351045776862778100548339350*n^5+726089862026249900330812900*n^4+ 1070175521983395469497495016*n^3+1063625496320571081551598240*n^2+ 638421681361495737294369408*n+174635957963287938575439360)/(5*n+28)/(5*n+29)/(5 *n+26)/(5*n+32)/(1364326611875000*n^9+41816832857000000*n^8+563266394649928125* n^7+4374463820688240000*n^6+21576946370891791000*n^5+70064566902942882950*n^4+ 149702517465233532475*n^3+202837528336373548282*n^2+158051881234893802968*n+ 53927142941404470240)*a(n+4)-6*(381744807687500000*n^12+18503351879762890625*n^ 11+407253741900243281250*n^10+5379705106448083437500*n^9+ 47479586400973315837500*n^8+294789548726787783281875*n^7+ 1319493829509977204155000*n^6+4287349423708365530126500*n^5+ 10029088174867504621258010*n^4+16458213706104259625849196*n^3+ 17968915239528551609425552*n^2+11706902493748301227729536*n+ 3437930210846827536852480)/(5*n+28)/(5*n+29)/(5*n+26)/(5*n+32)/( 1364326611875000*n^9+41816832857000000*n^8+563266394649928125*n^7+ 4374463820688240000*n^6+21576946370891791000*n^5+70064566902942882950*n^4+ 149702517465233532475*n^3+202837528336373548282*n^2+158051881234893802968*n+ 53927142941404470240)*a(n+5)+a(n+6) = 0 The asymptotics is CONST*n!*2.709489118^n/n^(5/2) where the CONST. is roughly, 0.4178 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3, 4, 6, 7], then the first, 30, terms are [0, 1, 3, 16, 125, 1290, 16597, 256264, 4621680, 95409720, 2219618940, 57471837192, 1639548318408, 51101956573668, 1727911423978740, 63001723596651840, 2464139724344538240, 102916587203294312448, 4571681490285181704432, 215227770102974850661440, 10704826607119503013995840, 560903148244748672578575360, 30882408727759059877060374720, 1782547741590804886439796582528, 107636441703898319775176562748800, 6786181096923023038706657334940800, 445931424075652823691683147345097600, 30491338787468386990178892265290700800, 2166176233178078934692707918580715825600, 159665471829528862785317587304470738915200] The enumerating sequence satisfies the recurrence 39821/20*n*(n-1)*(n+5)*(n+4)*(681628140522106689*n^10+30970508413583325270*n^9+ 626203982371566043020*n^8+7415779267855623927840*n^7+56927711899083964171347*n^ 6+295802104823777945896830*n^5+1052829178574822842875580*n^4+ 2532352409302094622476460*n^3+3935319050643356244223364*n^2+ 3563353606479817603023600*n+1425259443425807586912000)*(n+1)^2*(n+2)^2*(n+3)^2/ (5*n+32)/(5*n+31)/(5*n+29)/(681628140522106689*n^10+24154227008362258380*n^9+ 378142672972810916595*n^8+3439290334909442490720*n^7+20092587733267088963037*n^ 6+78630288810134493042660*n^5+208257195182653504919305*n^4+ 367507083950920912100780*n^3+411849606279359679460574*n^2+ 263073012834817422971260*n+72007399771623347682000)/(5*n+28)*a(n)-1/2*n*(n+5)*( n+4)*(n+1)*(1124686431861476036850*n^11+52788368530204700750775*n^10+ 1108150806396559916356848*n^9+13707046244277603770664063*n^8+ 110731115796592827478950582*n^7+611496952380755958212161965*n^6+ 2345627105319151385798828700*n^5+6213062882217543560868886413*n^4+ 11040219772536865466134283500*n^3+12360087553103659189096582824*n^2+ 7651322438996292900745759880*n+1877859944262183408624501600)*(n+3)^2*(n+2)^2/(5 *n+32)/(5*n+31)/(5*n+29)/(681628140522106689*n^10+24154227008362258380*n^9+ 378142672972810916595*n^8+3439290334909442490720*n^7+20092587733267088963037*n^ 6+78630288810134493042660*n^5+208257195182653504919305*n^4+ 367507083950920912100780*n^3+411849606279359679460574*n^2+ 263073012834817422971260*n+72007399771623347682000)/(5*n+28)*a(n+1)+1/5*(n+5)*( n+4)*(n+2)*(n+1)*(57422399442003833801427*n^12+2838738138053515405926318*n^11+ 63396929670261830503282125*n^10+845172135585860684316385650*n^9+ 7485520204408442282481037551*n^8+46364469285189329387804777604*n^7+ 205754022529425401555344567575*n^6+658537250610889361726762441820*n^5+ 1507128000870864105981811244762*n^4+2402519985416969321553774045308*n^3+ 2528927019885097258440408689860*n^2+1575940408028975997148610572000*n+ 438969425634708598751957640000)*(n+3)^2/(5*n+32)/(5*n+31)/(5*n+29)/( 681628140522106689*n^10+24154227008362258380*n^9+378142672972810916595*n^8+ 3439290334909442490720*n^7+20092587733267088963037*n^6+78630288810134493042660* n^5+208257195182653504919305*n^4+367507083950920912100780*n^3+ 411849606279359679460574*n^2+263073012834817422971260*n+72007399771623347682000 )/(5*n+28)*a(n+2)-1/10*(n+5)*(n+4)*(n+3)*(n+2)*(151797223637992115426922*n^13+ 8035549459973120236680375*n^12+193826836845339793788293388*n^11+ 2819227427499361651724215245*n^10+27570833169492449117455526214*n^9+ 191314913520795953296793222691*n^8+968370620599599491337364622636*n^7+ 3617153318038858204666712386779*n^6+9961447753243205330755959147904*n^5+ 19955068541274121598355040532814*n^4+28228726606619601233809260586936*n^3+ 26664886977525499221924008766096*n^2+15048016003177440443749496384000*n+ 3823151272459257836385858528000)/(5*n+32)/(5*n+31)/(5*n+29)/(681628140522106689 *n^10+24154227008362258380*n^9+378142672972810916595*n^8+3439290334909442490720 *n^7+20092587733267088963037*n^6+78630288810134493042660*n^5+ 208257195182653504919305*n^4+367507083950920912100780*n^3+ 411849606279359679460574*n^2+263073012834817422971260*n+72007399771623347682000 )/(5*n+28)*a(n+3)+1/5*(n+5)*(n+3)*(52551484749832859401833*n^14+ 3018351104160027941163186*n^13+79615421931137607473176512*n^12+ 1277677405968531708898311924*n^11+13929603772069327598443012776*n^10+ 109073721562314898770496297014*n^9+632186791748321964531495358308*n^8+ 2753183824282630506049693610856*n^7+9045741490176561962578939186147*n^6+ 22291796138250970974622047280500*n^5+40515664042604796652976186053480*n^4+ 52596465050853460493607093834120*n^3+46031407230201756531951989695744*n^2+ 24264321220206962551002598737600*n+5798048573529383584128470496000)/(5*n+32)/(5 *n+31)/(5*n+29)/(681628140522106689*n^10+24154227008362258380*n^9+ 378142672972810916595*n^8+3439290334909442490720*n^7+20092587733267088963037*n^ 6+78630288810134493042660*n^5+208257195182653504919305*n^4+ 367507083950920912100780*n^3+411849606279359679460574*n^2+ 263073012834817422971260*n+72007399771623347682000)/(5*n+28)*a(n+4)-1/25*( 88662098750272505464986*n^15+5580044639518206296807235*n^14+ 162305211014346402185596245*n^13+2892801289040018781617775135*n^12+ 35312367481310784251675004537*n^11+312527304466583365556352140715*n^10+ 2070252306061595113762117257285*n^9+10443859891982939680494627882105*n^8+ 40417887060328078637958680167693*n^7+119871674929124820685305737202490*n^6+ 269906588765458717378552246670470*n^5+452463457850055780257064287595760*n^4+ 545710310220457603079821874214784*n^3+446091299592059174930877335732560*n^2+ 220390459694243924122818438408000*n+49420316998090786614903691200000)/(5*n+32)/ (5*n+31)/(5*n+29)/(681628140522106689*n^10+24154227008362258380*n^9+ 378142672972810916595*n^8+3439290334909442490720*n^7+20092587733267088963037*n^ 6+78630288810134493042660*n^5+208257195182653504919305*n^4+ 367507083950920912100780*n^3+411849606279359679460574*n^2+ 263073012834817422971260*n+72007399771623347682000)/(5*n+28)*a(n+5)+a(n+6) = 0 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3, 5, 6, 7], then the first, 30, terms are [0, 1, 3, 16, 120, 1176, 14287, 207824, 3528000, 68544630, 1500989490, 36592606512, 983189215008, 28870718744868, 919963755620910, 31618396677067200, 1165968656554141440, 45922441828964789088, 1924007828117906683152, 85444550038799232240960, 4009376126327905001148480, 198218959104473423567811360, 10298448648572435495351527320, 560974715861313519557269365888, 31969567174806734101988706547200, 1902428277864839155134670328762400, 118000338957844244347374857606277600, 7616371595985628220677714401440620800, 510791398282953214378809287801276505600, 35543365176590178155757815603053892335200] The enumerating sequence satisfies the recurrence -31176/5*n*(n-1)*(n+5)*(n+4)*(145743385945952389*n^10+6643118146772163480*n^9+ 134897199026727049015*n^8+1605457889008002576740*n^7+12387473526787310923147*n^ 6+64663243239557990396420*n^5+230909533339895570900085*n^4+ 555935636362014954371360*n^3+861619395961968315187764*n^2+ 773912344661290320555600*n+304678291444577777145600)*(n+1)^2*(n+2)^2*(n+3)^2/( 145743385945952389*n^10+5185684287312639590*n^9+81667588073345435200*n^8+ 747943343764469783220*n^7+4398974063199438527757*n^6+17299080155740376820690*n^ 5+45850771085061443413550*n^4+80357780550065079158980*n^3+ 88279269524190710590304*n^2+54051061761186047729520*n+13606411945623607094400)/ (5*n+32)/(5*n+31)/(5*n+29)/(5*n+28)*a(n)+36*n*(n+5)*(n+4)*(n+1)*( 1748920631351428668*n^11+82340798708293104762*n^10+1744870565248428009060*n^9+ 21905980057256028691051*n^8+180161097976866075099350*n^7+ 1011013814453910962541254*n^6+3897265534831633320004614*n^5+ 10069790658490875357121953*n^4+16215791507147915885742940*n^3+ 13226066926059516212172540*n^2+679014364520267349338928*n-\ 4738118014570981668299520)*(n+3)^2*(n+2)^2/(145743385945952389*n^10+ 5185684287312639590*n^9+81667588073345435200*n^8+747943343764469783220*n^7+ 4398974063199438527757*n^6+17299080155740376820690*n^5+45850771085061443413550* n^4+80357780550065079158980*n^3+88279269524190710590304*n^2+ 54051061761186047729520*n+13606411945623607094400)/(5*n+32)/(5*n+31)/(5*n+29)/( 5*n+28)*a(n+1)+6/5*(n+5)*(n+4)*(n+2)*(n+1)*(1948297583325491536152*n^12+ 96598393719352247545248*n^11+2164822006682888128288735*n^10+ 28961063100851063324818350*n^9+257233044618723516111386056*n^8+ 1595477163898163035897699244*n^7+7072496678855500031039501125*n^6+ 22525388698970516413962182540*n^5+51017188833408556795174815072*n^4+ 79859989971133006684534949618*n^3+81653052315587224974238921860*n^2+ 48679214508983965093058103600*n+12696247911010304481274848000)*(n+3)^2/( 145743385945952389*n^10+5185684287312639590*n^9+81667588073345435200*n^8+ 747943343764469783220*n^7+4398974063199438527757*n^6+17299080155740376820690*n^ 5+45850771085061443413550*n^4+80357780550065079158980*n^3+ 88279269524190710590304*n^2+54051061761186047729520*n+13606411945623607094400)/ (5*n+32)/(5*n+31)/(5*n+29)/(5*n+28)*a(n+2)-1/5*(n+5)*(n+4)*(n+3)*(n+2)*( 12307737456363787346272*n^13+653306072181344066656080*n^12+ 15809824227933679172398798*n^11+230732056703788599191702520*n^10+ 2263232030985205521166588045*n^9+15735933998542286332756459503*n^8+ 79663886831272055946230275070*n^7+296766547900127138849858216802*n^6+ 811588866825667080644472331013*n^5+1604451462373905152032770467487*n^4+ 2219847929008566192876586213922*n^3+2024252604926610941244700271208*n^2+ 1081648633263238436861768921280*n+252527327664647468127794496000)/( 145743385945952389*n^10+5185684287312639590*n^9+81667588073345435200*n^8+ 747943343764469783220*n^7+4398974063199438527757*n^6+17299080155740376820690*n^ 5+45850771085061443413550*n^4+80357780550065079158980*n^3+ 88279269524190710590304*n^2+54051061761186047729520*n+13606411945623607094400)/ (5*n+32)/(5*n+31)/(5*n+29)/(5*n+28)*a(n+3)+1/20*(n+5)*(n+3)*( 25190141083512464962371*n^14+1450471590372103543268172*n^13+ 38374632870592583988686381*n^12+617834133618249402643567892*n^11+ 6756829193770831328522625319*n^10+53044895661914940008338673452*n^9+ 307906727504579683244998019295*n^8+1340559333120163264821907540860*n^7+ 4391345170857491349034617193642*n^6+10746971642927986147676891874616*n^5+ 19288151998504074602733537353024*n^4+24526084564741238337816696323648*n^3+ 20779641667694270514567519511968*n^2+10421575353647843303599712069760*n+ 2306944665272571952027954636800)/(145743385945952389*n^10+5185684287312639590*n ^9+81667588073345435200*n^8+747943343764469783220*n^7+4398974063199438527757*n^ 6+17299080155740376820690*n^5+45850771085061443413550*n^4+ 80357780550065079158980*n^3+88279269524190710590304*n^2+54051061761186047729520 *n+13606411945623607094400)/(5*n+32)/(5*n+31)/(5*n+29)/(5*n+28)*a(n+4)-1/50*( 24032792855715657041322*n^15+1516010771141460211748175*n^14+ 44223353911323083802385010*n^13+790836741835236540424256425*n^12+ 9688530200426407399253877414*n^11+86057754964099996506325973195*n^10+ 571959307253041942683087894430*n^9+2892851605873009274445398310675*n^8+ 11209753052473730055335753182456*n^7+33219263044636956697986275711210*n^6+ 74500372519278558293556183768360*n^5+123807674254631097276127812426400*n^4+ 146996233112746716062133322519808*n^3+117058505875741619185579232533920*n^2+ 55441475343513299413458123523200*n+11614637551850613547329523968000)/( 145743385945952389*n^10+5185684287312639590*n^9+81667588073345435200*n^8+ 747943343764469783220*n^7+4398974063199438527757*n^6+17299080155740376820690*n^ 5+45850771085061443413550*n^4+80357780550065079158980*n^3+ 88279269524190710590304*n^2+54051061761186047729520*n+13606411945623607094400)/ (5*n+32)/(5*n+31)/(5*n+29)/(5*n+28)*a(n+5)+a(n+6) = 0 The asymptotics is CONST*n!*2.525081333^n/n^(5/2) where the CONST. is roughly, 0.5614 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 4, 5, 6, 7], then the first, 30, terms are [0, 1, 3, 12, 65, 486, 4837, 59704, 865368, 14321790, 266890470, 5544127512, 127158179148, 3191466322044, 86982627161430, 2558004325836960, 80737481298714240, 2722409801450827488, 97673113623540703248, 3715136986636872276480, 149333256655959711174840, 6325126593772632447962160, 281570042756979449978024520, 13142810265803015283545387328, 641868944739733893884022921600, 32735004468885269965835394573600, 1740235138055445432371516623677600, 96275604890587772698073073269020800, 5534461539447978743174671438227091200, 330120095277120803725249744713675727200] The enumerating sequence satisfies the recurrence 3982*n*(n-1)*(n+5)*(n+4)*(2495954899432871*n^10+111396644840460350*n^9+ 2211514933096418945*n^8+25680783768338032090*n^7+192844931023384358463*n^6+ 976527134423150398480*n^5+3368528306316081850445*n^4+7790676838797642366720*n^3 +11511103659462813694356*n^2+9750925345842843902480*n+3561916858064196211200)*( n+1)^2*(n+2)^2*(n+3)^2/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/(2495954899432871*n^ 10+86437095846131640*n^9+1321263100006754990*n^8+11699428929891308610*n^7+ 66168695134000097803*n^6+248316167779784381280*n^5+621033386254996336800*n^4+ 1008976736514230167590*n^3+996373599833458052176*n^2+515252381189426741280*n+ 92686266277656806160)*a(n)-2*n*(n+5)*(n+4)*(n+1)*(12654491340124655970*n^11+ 583762726351320958455*n^10+12058218115292485725340*n^9+146953046916121994760975 *n^8+1171536784602452012278881*n^7+6398593318127468447851260*n^6+ 24352408976193897487456719*n^5+64316750223730066035745830*n^4+ 114873298587514658283740294*n^3+131082450838192668420849060*n^2+ 84941663917370171494467456*n+23169783375615862745424960)*(n+3)^2*(n+2)^2/(5*n+ 29)/(5*n+31)/(5*n+28)/(5*n+32)/(2495954899432871*n^10+86437095846131640*n^9+ 1321263100006754990*n^8+11699428929891308610*n^7+66168695134000097803*n^6+ 248316167779784381280*n^5+621033386254996336800*n^4+1008976736514230167590*n^3+ 996373599833458052176*n^2+515252381189426741280*n+92686266277656806160)*a(n+1)+ (n+5)*(n+4)*(n+2)*(n+1)*(37341981250415183031*n^12+1815973128459788028474*n^11+ 39872442302027397926562*n^10+521856074838116578793404*n^9+ 4526354593643775538896206*n^8+27350762501191634073907036*n^7+ 117751426949724110429540914*n^6+362730735783652616312204668*n^5+ 790059429616021436416343119*n^4+1179492391740290199973561122*n^3+ 1135522175468469673356264728*n^2+623885781275066652365413776*n+ 144035381198951903160855360)*(n+3)^2/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/( 2495954899432871*n^10+86437095846131640*n^9+1321263100006754990*n^8+ 11699428929891308610*n^7+66168695134000097803*n^6+248316167779784381280*n^5+ 621033386254996336800*n^4+1008976736514230167590*n^3+996373599833458052176*n^2+ 515252381189426741280*n+92686266277656806160)*a(n+2)-1/2*(n+5)*(n+4)*(n+3)*(n+2 )*(68918306683140434052*n^13+3592771457458344439590*n^12+ 85298693600461750207850*n^11+1219736916968884009196655*n^10+ 11704292226979262373918988*n^9+79455745504244514055011141*n^8+ 391801185831653522342012754*n^7+1417307190148415551276189581*n^6+ 3748806640993409607329862100*n^5+7128868617204396525818217765*n^4+ 9413197252649472096401186768*n^3+8093057666273195653968863028*n^2+ 3993764354751064387975287888*n+827136528645703330633754880)/(5*n+29)/(5*n+31)/( 5*n+28)/(5*n+32)/(2495954899432871*n^10+86437095846131640*n^9+ 1321263100006754990*n^8+11699428929891308610*n^7+66168695134000097803*n^6+ 248316167779784381280*n^5+621033386254996336800*n^4+1008976736514230167590*n^3+ 996373599833458052176*n^2+515252381189426741280*n+92686266277656806160)*a(n+3)+ 1/20*(n+5)*(n+3)*(426216746491855350573*n^14+24137026220793794953926*n^13+ 627394314171658410603987*n^12+9911357294721588737723134*n^11+ 106190296137740276115832123*n^10+815119079128872621567721484*n^9+ 4615051123611622457716422057*n^8+19538047491847232798501776734*n^7+ 61984431572183099921068611744*n^6+146120753502830529394105930442*n^5+ 250708789405520661074082069356*n^4+301379468453705451379365569992*n^3+ 237179558756726975280775604880*n^2+107160630083724296006111729088*n+ 20097270628226204062793057280)/(5*n+29)/(5*n+31)/(5*n+28)/(5*n+32)/( 2495954899432871*n^10+86437095846131640*n^9+1321263100006754990*n^8+ 11699428929891308610*n^7+66168695134000097803*n^6+248316167779784381280*n^5+ 621033386254996336800*n^4+1008976736514230167590*n^3+996373599833458052176*n^2+ 515252381189426741280*n+92686266277656806160)*a(n+4)-1/50*( 430297632752428094658*n^15+26734767350373175075815*n^14+ 767269173796850506551010*n^13+13480865333448712636771315*n^12+ 161999410889093634077159036*n^11+1408654088925286538734058825*n^10+ 9142630768016439721216947900*n^9+45018277854473182244568533445*n^8+ 169165691542806527454097248554*n^7+483650662312045564731293661980*n^6+ 1039240757169965551220591969170*n^5+1638640983292149955051739242540*n^4+ 1819367662854415332077473123752*n^3+1323740198625450729895025852880*n^2+ 549551488043975885097279433920*n+92406114330805160639749555200)/(5*n+29)/(5*n+ 31)/(5*n+28)/(5*n+32)/(2495954899432871*n^10+86437095846131640*n^9+ 1321263100006754990*n^8+11699428929891308610*n^7+66168695134000097803*n^6+ 248316167779784381280*n^5+621033386254996336800*n^4+1008976736514230167590*n^3+ 996373599833458052176*n^2+515252381189426741280*n+92686266277656806160)*a(n+5)+ a(n+6) = 0 The asymptotics is CONST*n!*2.165611317^n/n^(5/2) where the CONST. is roughly, 0.5156 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 3, 4, 5, 6, 7], then the first, 30, terms are [0, 1, 0, 4, 5, 96, 427, 6448, 56952, 892710, 11901120, 211132152, 3691179492, 75680392788, 1612083883410, 38065701273600, 948438285674880, 25505717972743008, 725037228721613808, 21934176380361685440, 699460900088589639000, 23536806530554190633760, 831581363439862153798680, 30822164982947470179535488, 1194818886417888980523475200, 48374813978557927867336730400, 2041218151172453330970189897600, 89637520514399242994349428688000, 4089816827216978655998632678646400, 193621936812308310032459261920591200] The enumerating sequence satisfies the recurrence -84901/100*n*(n-1)*(n+5)*(n+4)*(3529021386328*n^10+165756026358320*n^9+ 3492682551775565*n^8+43638558718665046*n^7+359987107422401311*n^6+ 2063763045292098415*n^5+8399258429248185046*n^4+24168189397782246399*n^3+ 47346049420505351794*n^2+57123589190060859776*n+32127193244878465440)*(n+1)^2*( n+2)^2*(n+3)^2/(3529021386328*n^10+130465812495040*n^9+2159684276935445*n^8+ 21240832687000686*n^7+139129896118491809*n^6+644655856882018539*n^5+ 2177243872764706136*n^4+5354303341164048635*n^3+9174784928433993482*n^2+ 9776902371970051812*n+4836638465747337528)/(5*n+32)/(5*n+31)/(5*n+29)/(5*n+28)* a(n)-1/50*n*(n+5)*(n+4)*(n+1)*(667620265865531040*n^11+32359155465265274160*n^ 10+707891409744864347100*n^9+9252091469206842287005*n^8+80620917434054704909859 *n^7+494706154185475048944552*n^6+2196910679087146258849604*n^5+ 7111520389650329904220263*n^4+16530894116030778891326525*n^3+ 26309418319032896910850616*n^2+25693177763013587978274156*n+ 11556807209692289807715120)*(n+3)^2*(n+2)^2/(3529021386328*n^10+130465812495040 *n^9+2159684276935445*n^8+21240832687000686*n^7+139129896118491809*n^6+ 644655856882018539*n^5+2177243872764706136*n^4+5354303341164048635*n^3+ 9174784928433993482*n^2+9776902371970051812*n+4836638465747337528)/(5*n+32)/(5* n+31)/(5*n+29)/(5*n+28)*a(n+1)-1/50*(n+5)*(n+4)*(n+2)*(n+1)*( 1113812084845911720*n^12+56770336598464813680*n^11+1316291588216814547855*n^10+ 18404251896069397248535*n^9+173480044321575049756151*n^8+ 1167511989270460020037239*n^7+5790322195972849871902815*n^6+ 21477269385352100213864877*n^5+59509959975013802682353381*n^4+ 120588891879059174406612629*n^3+169620343894823442693275158*n^2+ 148115017984382190811207800*n+60324081779308930381828800)*(n+3)^2/( 3529021386328*n^10+130465812495040*n^9+2159684276935445*n^8+21240832687000686*n ^7+139129896118491809*n^6+644655856882018539*n^5+2177243872764706136*n^4+ 5354303341164048635*n^3+9174784928433993482*n^2+9776902371970051812*n+ 4836638465747337528)/(5*n+32)/(5*n+31)/(5*n+29)/(5*n+28)*a(n+2)-1/50*(n+5)*(n+4 )*(n+3)*(n+2)*(726449052375618800*n^13+39569245918677313000*n^12+ 988648976421135934450*n^11+15035838801487311101330*n^10+ 155817718163553950763389*n^9+1167433940031516628640644*n^8+ 6546288114849042852137286*n^7+28018852098518827727136814*n^6+ 92205873104165949578864521*n^5+231730024894439744436633432*n^4+ 433327958027934438776593746*n^3+569966363328235978787583340*n^2+ 470167585692366073950610608*n+182385080032236371452702080)/(3529021386328*n^10+ 130465812495040*n^9+2159684276935445*n^8+21240832687000686*n^7+ 139129896118491809*n^6+644655856882018539*n^5+2177243872764706136*n^4+ 5354303341164048635*n^3+9174784928433993482*n^2+9776902371970051812*n+ 4836638465747337528)/(5*n+32)/(5*n+31)/(5*n+29)/(5*n+28)*a(n+3)+1/100*(n+5)*(n+ 3)*(154412330758781640*n^14+9105602902413671280*n^13+246560416363786995215*n^12 +4065441406177243541460*n^11+45601399077651684166718*n^10+ 367600319554706090180057*n^9+2187177979660765941734049*n^8+ 9664751473084036674770154*n^7+31213157976838747486980100*n^6+ 69820878125088907587793989*n^5+91658399964745193185105246*n^4+ 14547310689183354615610996*n^3-179613576354372175118509368*n^2-\ 299099471822193410507240736*n-167824712883003978542492160)/(3529021386328*n^10+ 130465812495040*n^9+2159684276935445*n^8+21240832687000686*n^7+ 139129896118491809*n^6+644655856882018539*n^5+2177243872764706136*n^4+ 5354303341164048635*n^3+9174784928433993482*n^2+9776902371970051812*n+ 4836638465747337528)/(5*n+32)/(5*n+31)/(5*n+29)/(5*n+28)*a(n+4)+1/50*(n+3)*( 317971884950925456*n^14+19545541818725767752*n^13+554695172391879461414*n^12+ 9645986830107009372375*n^11+115059654871217716223417*n^10+ 998730979153737194189267*n^9+6530431955996800991489206*n^8+ 32831428068356798516072085*n^7+128203317039591353971987741*n^6+ 388937373226736638563522693*n^5+905951147076748939084613830*n^4+ 1573882729465635347939745460*n^3+1925887810063916519185059416*n^2+ 1480573770923627235604639008*n+536729266641152408439559680)/(3529021386328*n^10 +130465812495040*n^9+2159684276935445*n^8+21240832687000686*n^7+ 139129896118491809*n^6+644655856882018539*n^5+2177243872764706136*n^4+ 5354303341164048635*n^3+9174784928433993482*n^2+9776902371970051812*n+ 4836638465747337528)/(5*n+32)/(5*n+31)/(5*n+29)/(5*n+28)*a(n+5)+a(n+6) = 0 The asymptotics is CONST*n!*1.718055200^n/n^(5/2) where the CONST. is roughly, 0.3179 ----------------------------------------------------------------- If the set of allowed vertex-degrees is, [1, 2, 3, 4, 5, 6, 7], then the first, 30, terms are [0, 1, 3, 16, 125, 1296, 16807, 262144, 4782960, 99999270, 2357907090, 61915380912, 1792067625348, 56689590625668, 1945990451394990, 72047639356492800, 2861922355599347040, 121413407403091291488, 5478970584718263224592, 262064161605831387042240, 13243816524986751335912280, 705144306778313792564734560, 39453528934372627138093749720, 2314326823627985803120470674688, 142027642670804447647235414068800, 9100969102986732139596924490394400, 607848518163638852971327674855237600, 42245791977530221343171410958757100800, 3050657336591657733024566079774602025600, 228567756261229579372769676961488876463200] The enumerating sequence satisfies the recurrence -1/5*n*(n-1)*(n+5)*(n+4)*(1764510693164*n^10+81592551285160*n^9+ 1688026349061785*n^8+20571918676496740*n^7+163518238373630502*n^6+ 885600156807868180*n^5+3308930931611357465*n^4+8420132865428877160*n^3+ 13961958936785883684*n^2+13618718943870698960*n+5932168704926707200)*(n+1)^2*(n +2)^2*(n+3)^2/(1764510693164*n^10+63947444353520*n^9+1033096368687725*n^8+ 9793298447088540*n^7+60296318349494302*n^6+251807547992309580*n^5+ 721938299717047425*n^4+1402175650744218760*n^3+1764314454395397784*n^2+ 1297583201735308800*n+423161125222107600)/(5*n+32)/(5*n+31)/(5*n+29)/(5*n+28)*a (n)-6*n*(n+5)*(n+4)*(n+1)*(42848729800*n^9+1755608733494*n^8+31474206317007*n^7 +322855718520960*n^6+2077013812947724*n^5+8617847471872136*n^4+ 22736814269401093*n^3+35798008044223170*n^2+28643438607023256*n+ 7088292515749760)*(n+3)^2*(n+2)^2/(1764510693164*n^10+63947444353520*n^9+ 1033096368687725*n^8+9793298447088540*n^7+60296318349494302*n^6+ 251807547992309580*n^5+721938299717047425*n^4+1402175650744218760*n^3+ 1764314454395397784*n^2+1297583201735308800*n+423161125222107600)/(5*n+32)/(5*n +31)/(5*n+29)/(5*n+28)*a(n+1)+1/5*(n+5)*(n+4)*(n+2)*(n+1)*(164099494464252*n^12 +8244505247376888*n^11+187972725803701495*n^10+2570454213248577495*n^9+ 23466799303453066321*n^8+150589120564285999124*n^7+696026189307453076055*n^6+ 2332963828446713159105*n^5+5623527957729868246177*n^4+9498468509239406972588*n^ 3+10660798803123519506500*n^2+7131459992251147822800*n+2147856608473069536000)* (n+3)^2/(1764510693164*n^10+63947444353520*n^9+1033096368687725*n^8+ 9793298447088540*n^7+60296318349494302*n^6+251807547992309580*n^5+ 721938299717047425*n^4+1402175650744218760*n^3+1764314454395397784*n^2+ 1297583201735308800*n+423161125222107600)/(5*n+32)/(5*n+31)/(5*n+29)/(5*n+28)*a (n+2)-3/5*(n+5)*(n+4)*(n+3)*(n+2)*(1019887180648792*n^13+54809648497688420*n^12 +1347793083157912258*n^11+20073708143150467985*n^10+201963665013507686449*n^9+ 1448996709636393922046*n^8+7623845821996845912616*n^7+29771697879871487300049*n ^6+86248904580731640250069*n^5+182981120363183833907644*n^4+ 276173010429686154182696*n^3+280633506812803875349856*n^2+ 171958694421583114571120*n+47948372595318828561600)/(1764510693164*n^10+ 63947444353520*n^9+1033096368687725*n^8+9793298447088540*n^7+60296318349494302* n^6+251807547992309580*n^5+721938299717047425*n^4+1402175650744218760*n^3+ 1764314454395397784*n^2+1297583201735308800*n+423161125222107600)/(5*n+32)/(5*n +31)/(5*n+29)/(5*n+28)*a(n+3)+1/20*(n+5)*(n+3)*(28315103093202708*n^14+ 1649096907591395016*n^13+44268814494634136039*n^12+725816157138282939958*n^11+ 8117847470634289480697*n^10+65499315663326671498832*n^9+ 393040509309211512830385*n^8+1781258465784306746587710*n^7+ 6124208488608730413736099*n^6+15889182885641969970615572*n^5+ 30607792750892402013453256*n^4+42427087052625439733538592*n^3+ 39980591233614293005128816*n^2+22910336978770118391049920*n+ 6018672868494404359449600)/(1764510693164*n^10+63947444353520*n^9+ 1033096368687725*n^8+9793298447088540*n^7+60296318349494302*n^6+ 251807547992309580*n^5+721938299717047425*n^4+1402175650744218760*n^3+ 1764314454395397784*n^2+1297583201735308800*n+423161125222107600)/(5*n+32)/(5*n +31)/(5*n+29)/(5*n+28)*a(n+4)-3/50*(57286604164262424*n^15+3651499342898596980* n^14+107916565303170319970*n^13+1961187535025327938695*n^12+ 24503586011255752246558*n^11+222896618109631054832275*n^10+ 1524533365117095134581810*n^9+7981052738760042275388285*n^8+ 32231527391530788288173762*n^7+100376590153617887096498685*n^6+ 238985628130242215110016820*n^5+426998930047320375479524520*n^4+ 553921043601017392314080256*n^3+492245083870922341453204560*n^2+ 267774663408516723723470400*n+67169841655372205387904000)/(1764510693164*n^10+ 63947444353520*n^9+1033096368687725*n^8+9793298447088540*n^7+60296318349494302* n^6+251807547992309580*n^5+721938299717047425*n^4+1402175650744218760*n^3+ 1764314454395397784*n^2+1297583201735308800*n+423161125222107600)/(5*n+32)/(5*n +31)/(5*n+29)/(5*n+28)*a(n+5)+a(n+6) = 0 The asymptotics is CONST*n!*2.718055200^n/n^(5/2) where the CONST. is roughly, 0.3999 ----------------------------------------------------------------- To sum up, here are all the sets followed by their growth constants [[{1, 2, 3}, 2.414213562], [{1, 2, 4}, 2.040041912], [{1, 2, 5}, 1.792804743], [{1, 2, 6}, 1.633119605], [{1, 2, 7}, 1.524148279], [{1, 3, 4}, 1.660324085], [ {1, 3, 5}, 1.505261323], [{1, 3, 6}, 1.442871802], [{1, 3, 7}, 1.421588928], [{ 1, 4, 5}, 1.141560030], [{1, 4, 6}, 1.072064682], [{1, 5, 6}, .8483939058], [{1 , 2, 3, 4}, 2.660324085], [{1, 2, 3, 5}, 2.505261323], [{1, 2, 3, 6}, 2.4428718\ 02], [{1, 2, 3, 7}, 2.421588928], [{1, 2, 4, 5}, 2.141560030], [{1, 2, 4, 6}, 2\ .072064682], [{1, 2, 5, 6}, 1.848393906], [{1, 3, 4, 5}, 1.707993250], [{1, 3, 4, 6}, 1.671181366], [{1, 3, 5, 7}, 1.508789333], [{1, 4, 5, 6}, 1.161768135], [{1, 5, 6, 7}, .8599320672], [{1, 2, 3, 4, 5}, 2.707993250], [{1, 2, 3, 4, 6}, 2.671181366], [{1, 2, 3, 5, 6}, 2.522039659], [{1, 2, 3, 5, 7}, 2.508789332], [ {1, 2, 3, 6, 7}, 2.448187650], [{1, 2, 4, 5, 6}, 2.161768135], [{1, 2, 5, 6, 7} , 1.859932067], [{1, 3, 4, 5, 6}, 1.716653699], [{1, 2, 3, 4, 5, 6}, 2.71665369\ 9], [{1, 2, 3, 4, 5, 7}, 2.709489118], [{1, 2, 3, 5, 6, 7}, 2.525081333], [{1, 2, 4, 5, 6, 7}, 2.165611317], [{1, 3, 4, 5, 6, 7}, 1.718055200], [{1, 2, 3, 4, 5, 6, 7}, 2.718055200]] ----------------------------------------------------- This took, 376.967, to generate