G is, {{1, 2}} The first 20 terms of the sequence enumerating spanning trees of Gx{1,...,N1\ } are 1, 4, 15, 56, 209, 780, 2911, 10864, 40545, 151316, 564719, 2107560, 7865521, 29354524, 109552575, 408855776, 1525870529, 5694626340, 21252634831, 79315912984 The generating function, in t, is t ------------ 2 t - 4 t + 1 and in Maple format t/(t^2-4*t+1) G is, {{1, 2}, {2, 3}} The first 20 terms of the sequence enumerating spanning trees of Gx{1,...,N1\ } are 1, 15, 192, 2415, 30305, 380160, 4768673, 59817135, 750331584, 9411975375, 118061508289, 1480934568960, 18576479568193, 233018797965135, 2922930580320960, 36664523428884015, 459910778352898337, 5769007865476035840, 72365017995700730081, 907729015392142395375 The generating function, in t, is t (t - 1) (t + 1) - ----------------------------- 4 3 2 t - 15 t + 32 t - 15 t + 1 and in Maple format -t*(t-1)*(t+1)/(t^4-15*t^3+32*t^2-15*t+1) G is, {{1, 2}, {2, 3}, {3, 4}} The first 20 terms of the sequence enumerating spanning trees of Gx{1,...,N1\ } are 1, 56, 2415, 100352, 4140081, 170537640, 7022359583, 289143013376, 11905151192865, 490179860527896, 20182531537581071, 830989874753525760, 34214941811800329425, 1408756312731277540744, 58003732850974438010175, 2388229243489107516194816, 98332273432993735566334273, 4048705133573796221764525560, 166700236720421701398973625263, 6863668260790646684044452390912 The generating function, in t, is 6 4 3 2 t (t - 49 t + 112 t - 49 t + 1) --------------------------------------------------------------------- 8 7 6 5 4 3 2 t - 56 t + 672 t - 2632 t + 4094 t - 2632 t + 672 t - 56 t + 1 and in Maple format t*(t^6-49*t^4+112*t^3-49*t^2+1)/(t^8-56*t^7+672*t^6-2632*t^5+4094*t^4-2632*t^3+ 672*t^2-56*t+1) G is, {{1, 2}, {2, 3}, {3, 4}, {4, 5}} The first 20 terms of the sequence enumerating spanning trees of Gx{1,...,N1\ } are 1, 209, 30305, 4140081, 557568000, 74795194705, 10021992194369, 1342421467113969, 179796299139278305, 24080189412483072000, 3225041354570508955681, 431926215138756947267505, 57847355494807961811035009, 7747424602888405489208931601, 1037602902862756514154816000000, 138964858389586339640223412108401, 18611389483734199394023624777573409, 2492600085599977923424220468405177105, 333830807688353225138019865387722924481, 44709541971379003103897461691112357888000 The generating function, in t, is 12 10 9 8 7 - t (t - 1) (t + 1) (t - 1439 t + 26752 t - 187328 t + 601502 t 6 5 4 3 2 / 16 - 896256 t + 601502 t - 187328 t + 26752 t - 1439 t + 1) / (t / 15 14 13 12 11 - 209 t + 11936 t - 274208 t + 3112032 t - 19456019 t 10 9 8 7 6 + 70651107 t - 152325888 t + 196664896 t - 152325888 t + 70651107 t 5 4 3 2 - 19456019 t + 3112032 t - 274208 t + 11936 t - 209 t + 1) and in Maple format -t*(t-1)*(t+1)*(t^12-1439*t^10+26752*t^9-187328*t^8+601502*t^7-896256*t^6+ 601502*t^5-187328*t^4+26752*t^3-1439*t^2+1)/(t^16-209*t^15+11936*t^14-274208*t^ 13+3112032*t^12-19456019*t^11+70651107*t^10-152325888*t^9+196664896*t^8-\ 152325888*t^7+70651107*t^6-19456019*t^5+3112032*t^4-274208*t^3+11936*t^2-209*t+ 1) G is, {{1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6}} The first 20 terms of the sequence enumerating spanning trees of Gx{1,...,N1\ } are 1, 780, 380160, 170537640, 74795194705, 32565539635200, 14143261515284447, 6136973985625588560, 2662079368040434932480, 1154617875754582889149500, 500769437567956298239402223, 217185579535490113365186969600, 94193702839904633186530210863025, 40851869157273984726590135085017940, 17717469746416280095776019395706656000, 7684070867169415429692559499446691755680, 3332583081296808509759455619848802299528513, 1445341907485491645328460310146924377335398400, 626845049313054375044367343971643549398400207439, 271862811296852944176805652529210910158678393501000 The generating function, in t, is 30 28 27 26 25 t (t - 33359 t + 3642600 t - 173371343 t + 4540320720 t 24 23 22 - 70164186331 t + 634164906960 t - 2844883304348 t 21 20 19 - 1842793012320 t + 104844096982372 t - 678752492380560 t 18 17 16 + 2471590551535210 t - 5926092273213840 t + 9869538714631398 t 15 14 13 - 11674018886109840 t + 9869538714631398 t - 5926092273213840 t 12 11 10 + 2471590551535210 t - 678752492380560 t + 104844096982372 t 9 8 7 6 - 1842793012320 t - 2844883304348 t + 634164906960 t - 70164186331 t 5 4 3 2 / 32 + 4540320720 t - 173371343 t + 3642600 t - 33359 t + 1) / (t / 31 30 29 28 27 - 780 t + 194881 t - 22377420 t + 1419219792 t - 55284715980 t 26 25 24 + 1410775106597 t - 24574215822780 t + 300429297446885 t 23 22 21 - 2629946465331120 t + 16741727755133760 t - 78475174345180080 t 20 19 + 273689714665707178 t - 716370537293731320 t 18 17 + 1417056251105102122 t - 2129255507292156360 t 16 15 + 2437932520099475424 t - 2129255507292156360 t 14 13 + 1417056251105102122 t - 716370537293731320 t 12 11 10 + 273689714665707178 t - 78475174345180080 t + 16741727755133760 t 9 8 7 - 2629946465331120 t + 300429297446885 t - 24574215822780 t 6 5 4 3 + 1410775106597 t - 55284715980 t + 1419219792 t - 22377420 t 2 + 194881 t - 780 t + 1) and in Maple format t*(t^30-33359*t^28+3642600*t^27-173371343*t^26+4540320720*t^25-70164186331*t^24 +634164906960*t^23-2844883304348*t^22-1842793012320*t^21+104844096982372*t^20-\ 678752492380560*t^19+2471590551535210*t^18-5926092273213840*t^17+ 9869538714631398*t^16-11674018886109840*t^15+9869538714631398*t^14-\ 5926092273213840*t^13+2471590551535210*t^12-678752492380560*t^11+ 104844096982372*t^10-1842793012320*t^9-2844883304348*t^8+634164906960*t^7-\ 70164186331*t^6+4540320720*t^5-173371343*t^4+3642600*t^3-33359*t^2+1)/(t^32-780 *t^31+194881*t^30-22377420*t^29+1419219792*t^28-55284715980*t^27+1410775106597* t^26-24574215822780*t^25+300429297446885*t^24-2629946465331120*t^23+ 16741727755133760*t^22-78475174345180080*t^21+273689714665707178*t^20-\ 716370537293731320*t^19+1417056251105102122*t^18-2129255507292156360*t^17+ 2437932520099475424*t^16-2129255507292156360*t^15+1417056251105102122*t^14-\ 716370537293731320*t^13+273689714665707178*t^12-78475174345180080*t^11+ 16741727755133760*t^10-2629946465331120*t^9+300429297446885*t^8-24574215822780* t^7+1410775106597*t^6-55284715980*t^5+1419219792*t^4-22377420*t^3+194881*t^2-\ 780*t+1)