The generating function of the , 4, component of the Graham TreeLine sequence for ulabeled trees with 1 to, 13, vertices are 3 12 5 2 30 15 10 7 6 3 [1, 1, 1, x + x, x + x + x , x + x + x + x + x + x , 60 34 21 18 16 12 10 9 8 7 4 105 65 x + x + x + x + x + x + x + x + x + x + x , x + x 41 38 35 33 24 23 21 20 19 17 14 + x + x + x + x + x + x + x + x + x + 2 x + 2 x 13 12 11 10 9 8 5 168 111 73 70 66 + x + x + x + x + 2 x + x + x , x + x + x + x + x 54 45 43 42 39 36 30 28 27 26 25 + x + x + x + x + 2 x + 2 x + 2 x + x + x + x + 2 x 24 23 22 21 20 19 18 17 16 15 + 2 x + x + 2 x + x + 2 x + 2 x + x + x + x + 2 x 14 13 12 11 10 9 6 252 175 120 117 + 3 x + 2 x + x + 2 x + 2 x + x + x , x + x + x + x 112 87 78 76 75 71 70 67 58 57 52 + x + x + x + x + 2 x + x + x + x + x + x + x 49 48 47 46 45 44 43 42 41 40 + 2 x + x + x + 2 x + x + x + 2 x + x + x + 3 x 39 37 33 32 31 30 29 28 27 + 3 x + 2 x + 2 x + x + 2 x + 3 x + 2 x + 3 x + 3 x 26 25 24 23 22 21 20 19 18 + 3 x + 3 x + 5 x + 4 x + x + 5 x + 2 x + 4 x + 2 x 17 16 15 14 13 12 11 10 7 360 + 3 x + 5 x + 5 x + 3 x + x + 3 x + 3 x + x + x , x 260 185 182 176 135 126 123 122 118 116 + x + x + x + x + x + x + x + x + x + x 113 110 92 90 86 83 80 79 77 76 75 + x + x + x + x + x + x + 5 x + x + x + 2 x + x 72 71 68 65 63 62 61 60 59 58 56 + 3 x + x + 2 x + x + x + x + x + x + 2 x + x + x 54 53 52 51 50 49 48 47 46 + 2 x + 2 x + 2 x + x + 6 x + x + 3 x + 3 x + 2 x 45 44 43 42 41 40 39 38 37 + 4 x + 5 x + x + 5 x + 5 x + 2 x + 2 x + 3 x + x 36 35 34 33 32 31 30 29 28 + 3 x + 3 x + 6 x + 3 x + 6 x + 7 x + 6 x + 8 x + 6 x 27 26 25 24 23 22 21 20 19 + 7 x + 9 x + 5 x + 9 x + 4 x + 7 x + 7 x + 7 x + 4 x 18 17 16 15 14 13 12 11 8 495 + 6 x + 7 x + 9 x + 4 x + x + 4 x + 3 x + x + x , x 369 271 268 261 201 192 189 187 183 180 + x + x + x + x + x + x + x + x + x + x 177 159 145 141 138 135 132 129 128 127 + x + x + x + x + x + x + x + x + x + 2 x 126 124 123 122 119 118 117 115 114 + x + 2 x + x + x + 2 x + x + x + x + 2 x 100 99 97 96 95 93 91 88 87 85 + x + x + x + x + x + 2 x + 3 x + 2 x + x + 4 x 84 83 82 81 80 79 78 77 76 + 3 x + 2 x + 3 x + 5 x + 3 x + x + 2 x + 5 x + 2 x 75 74 73 72 69 68 67 66 65 64 + x + x + 4 x + 2 x + 3 x + x + 2 x + 2 x + 2 x + x 63 62 61 60 59 58 57 56 55 + 5 x + 4 x + 2 x + 3 x + 2 x + 3 x + 6 x + 4 x + 5 x 54 53 52 51 50 49 48 47 46 + 6 x + 4 x + 7 x + 11 x + 5 x + 6 x + 12 x + 6 x + 10 x 45 44 43 42 41 40 39 38 37 + 11 x + 5 x + 8 x + 9 x + 8 x + 3 x + 8 x + 5 x + 14 x 36 35 34 33 32 31 30 29 + 7 x + 13 x + 12 x + 15 x + 16 x + 17 x + 16 x + 11 x 28 27 26 25 24 23 22 21 + 19 x + 13 x + 16 x + 13 x + 10 x + 15 x + 15 x + 13 x 20 19 18 17 16 15 14 13 12 + 9 x + 10 x + 13 x + 13 x + 5 x + x + 5 x + 4 x + x 9 660 505 381 378 370 288 279 276 273 269 + x , x + x + x + x + x + x + x + x + x + x 265 262 226 208 204 202 199 196 195 194 + x + x + x + x + x + x + x + x + x + x 193 192 190 189 188 187 184 182 181 178 + x + x + 2 x + x + x + x + 2 x + x + x + x 165 163 150 147 145 144 142 141 139 + x + x + 2 x + x + x + 2 x + x + 2 x + x 138 137 135 134 133 132 131 130 129 + 2 x + x + x + x + 2 x + x + x + 3 x + 5 x 128 127 125 124 123 121 120 119 + 2 x + 2 x + 3 x + 2 x + 4 x + x + 4 x + 3 x 118 116 115 114 111 108 105 104 103 + 2 x + x + 2 x + x + x + x + x + x + 2 x 102 101 100 99 98 97 96 95 94 + 2 x + x + 3 x + 2 x + 3 x + x + 6 x + x + 3 x 93 92 91 90 89 88 87 86 85 + 5 x + 2 x + 2 x + 7 x + 4 x + 5 x + 5 x + 7 x + 9 x 84 83 82 81 80 79 78 77 76 + 5 x + 5 x + 12 x + 6 x + x + 7 x + 10 x + 5 x + 3 x 75 74 73 72 71 70 69 68 67 + 2 x + 7 x + 5 x + 4 x + 3 x + 9 x + 5 x + 5 x + 7 x 66 65 64 63 62 61 60 59 + 8 x + 7 x + 9 x + 9 x + 7 x + 11 x + 13 x + 10 x 58 57 56 55 54 53 52 51 + 13 x + 15 x + 11 x + 15 x + 19 x + 18 x + 18 x + 18 x 50 49 48 47 46 45 44 43 + 19 x + 22 x + 22 x + 20 x + 20 x + 16 x + 22 x + 17 x 42 41 40 39 38 37 36 35 + 19 x + 14 x + 22 x + 22 x + 33 x + 26 x + 33 x + 30 x 34 33 32 31 30 29 28 27 + 36 x + 40 x + 30 x + 36 x + 28 x + 29 x + 36 x + 28 x 26 25 24 23 22 21 20 19 + 25 x + 20 x + 30 x + 31 x + 22 x + 15 x + 16 x + 19 x 18 17 16 15 14 13 10 + 19 x + 6 x + x + 7 x + 4 x + x + x ] and in Maple notation [1, 1, 1, x^3+x, x^12+x^5+x^2, x^30+x^15+x^10+x^7+x^6+x^3, x^60+x^34+x^21+x^18+ x^16+x^12+x^10+x^9+x^8+x^7+x^4, x^105+x^65+x^41+x^38+x^35+x^33+x^24+x^23+x^21+x ^20+x^19+2*x^17+2*x^14+x^13+x^12+x^11+x^10+2*x^9+x^8+x^5, x^168+x^111+x^73+x^70 +x^66+x^54+x^45+x^43+x^42+2*x^39+2*x^36+2*x^30+x^28+x^27+x^26+2*x^25+2*x^24+x^ 23+2*x^22+x^21+2*x^20+2*x^19+x^18+x^17+x^16+2*x^15+3*x^14+2*x^13+x^12+2*x^11+2* x^10+x^9+x^6, x^252+x^175+x^120+x^117+x^112+x^87+x^78+x^76+2*x^75+x^71+x^70+x^ 67+x^58+x^57+x^52+2*x^49+x^48+x^47+2*x^46+x^45+x^44+2*x^43+x^42+x^41+3*x^40+3*x ^39+2*x^37+2*x^33+x^32+2*x^31+3*x^30+2*x^29+3*x^28+3*x^27+3*x^26+3*x^25+5*x^24+ 4*x^23+x^22+5*x^21+2*x^20+4*x^19+2*x^18+3*x^17+5*x^16+5*x^15+3*x^14+x^13+3*x^12 +3*x^11+x^10+x^7, x^360+x^260+x^185+x^182+x^176+x^135+x^126+x^123+x^122+x^118+x ^116+x^113+x^110+x^92+x^90+x^86+x^83+5*x^80+x^79+x^77+2*x^76+x^75+3*x^72+x^71+2 *x^68+x^65+x^63+x^62+x^61+x^60+2*x^59+x^58+x^56+2*x^54+2*x^53+2*x^52+x^51+6*x^ 50+x^49+3*x^48+3*x^47+2*x^46+4*x^45+5*x^44+x^43+5*x^42+5*x^41+2*x^40+2*x^39+3*x ^38+x^37+3*x^36+3*x^35+6*x^34+3*x^33+6*x^32+7*x^31+6*x^30+8*x^29+6*x^28+7*x^27+ 9*x^26+5*x^25+9*x^24+4*x^23+7*x^22+7*x^21+7*x^20+4*x^19+6*x^18+7*x^17+9*x^16+4* x^15+x^14+4*x^13+3*x^12+x^11+x^8, x^495+x^369+x^271+x^268+x^261+x^201+x^192+x^ 189+x^187+x^183+x^180+x^177+x^159+x^145+x^141+x^138+x^135+x^132+x^129+x^128+2*x ^127+x^126+2*x^124+x^123+x^122+2*x^119+x^118+x^117+x^115+2*x^114+x^100+x^99+x^ 97+x^96+x^95+2*x^93+3*x^91+2*x^88+x^87+4*x^85+3*x^84+2*x^83+3*x^82+5*x^81+3*x^ 80+x^79+2*x^78+5*x^77+2*x^76+x^75+x^74+4*x^73+2*x^72+3*x^69+x^68+2*x^67+2*x^66+ 2*x^65+x^64+5*x^63+4*x^62+2*x^61+3*x^60+2*x^59+3*x^58+6*x^57+4*x^56+5*x^55+6*x^ 54+4*x^53+7*x^52+11*x^51+5*x^50+6*x^49+12*x^48+6*x^47+10*x^46+11*x^45+5*x^44+8* x^43+9*x^42+8*x^41+3*x^40+8*x^39+5*x^38+14*x^37+7*x^36+13*x^35+12*x^34+15*x^33+ 16*x^32+17*x^31+16*x^30+11*x^29+19*x^28+13*x^27+16*x^26+13*x^25+10*x^24+15*x^23 +15*x^22+13*x^21+9*x^20+10*x^19+13*x^18+13*x^17+5*x^16+x^15+5*x^14+4*x^13+x^12+ x^9, x^660+x^505+x^381+x^378+x^370+x^288+x^279+x^276+x^273+x^269+x^265+x^262+x^ 226+x^208+x^204+x^202+x^199+x^196+x^195+x^194+x^193+x^192+2*x^190+x^189+x^188+x ^187+2*x^184+x^182+x^181+x^178+x^165+x^163+2*x^150+x^147+x^145+2*x^144+x^142+2* x^141+x^139+2*x^138+x^137+x^135+x^134+2*x^133+x^132+x^131+3*x^130+5*x^129+2*x^ 128+2*x^127+3*x^125+2*x^124+4*x^123+x^121+4*x^120+3*x^119+2*x^118+x^116+2*x^115 +x^114+x^111+x^108+x^105+x^104+2*x^103+2*x^102+x^101+3*x^100+2*x^99+3*x^98+x^97 +6*x^96+x^95+3*x^94+5*x^93+2*x^92+2*x^91+7*x^90+4*x^89+5*x^88+5*x^87+7*x^86+9*x ^85+5*x^84+5*x^83+12*x^82+6*x^81+x^80+7*x^79+10*x^78+5*x^77+3*x^76+2*x^75+7*x^ 74+5*x^73+4*x^72+3*x^71+9*x^70+5*x^69+5*x^68+7*x^67+8*x^66+7*x^65+9*x^64+9*x^63 +7*x^62+11*x^61+13*x^60+10*x^59+13*x^58+15*x^57+11*x^56+15*x^55+19*x^54+18*x^53 +18*x^52+18*x^51+19*x^50+22*x^49+22*x^48+20*x^47+20*x^46+16*x^45+22*x^44+17*x^ 43+19*x^42+14*x^41+22*x^40+22*x^39+33*x^38+26*x^37+33*x^36+30*x^35+36*x^34+40*x ^33+30*x^32+36*x^31+28*x^30+29*x^29+36*x^28+28*x^27+25*x^26+20*x^25+30*x^24+31* x^23+22*x^22+15*x^21+16*x^20+19*x^19+19*x^18+6*x^17+x^16+7*x^15+4*x^14+x^13+x^ 10] ------------------- This took, 14466.185, seconds