Maple program for A000022, A000200, A000598, A000602, A000678, based on Rains-Sloane paper E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1. kernelopts(printbytes=false): with(gfun): read format; N := 61; # First A000598 G000598 := 0: i := 0: while i<(N+1) do G000598 := series(1+z*(G000598^3/6+subs(z=z^2,G000598)*G000598/2+subs(z=z^3,G000598)/3) +O(z^(N+1)),z,N+1): t[ i ] := G000598: i := i+1: od: A000598 := n->coeff(G000598,z,n); # print A000598 for n from 0 to N-1 do lprint(n,A000598(n)); od; # Then A000678 i := 0: while icoeff(G000678,z,n); # print A000678 for n from 0 to N-1 do lprint(n,A000678(n)); od; # Then A000022 i := 1: while i<(N+1) do Tb := t[ i ]-t[ i-1 ]: Ts := t[ i ]-1: Q2 := series(Tb*Ts+O(z^(N+1)),z,200): q2[ i ] := Q2: i := i+1; od: q2[ 0 ] := 0: q[ -1 ] := 0: for i from 0 to N do c[ i ] := series(q[ i ]-q[ i-1 ]-q2[ i ]+O(z^(N+1)),z,200); od: # erase height information: i := 'i': cent := series(sum(c[ i ],i=0..N),z,200): G000022 := cent: A000022 := n->coeff(G000022,z,n); # print A000022 for n from 0 to N-1 do lprint(n,A000022(n)); od; # Then A000200 for i from 1 to N do tt := t[ i ]-t[ i-1 ]; b[ i ] := series((tt^2+subs(z=z^2,tt))/2+O(z^(N+1)),z,200): od: i := 'i': bicent := series(sum(b[ i ],i=1..N),z,200): G000200 := bicent: A000200 := n->coeff(G000200,z,n); # print A000200 for n from 0 to N-1 do lprint(n,A000200(n)); od; A000602 := proc(n) if n=0 then RETURN(1) else A000022(n)+A000200(n); fi; end; # print A000602 for n from 0 to N-1 do lprint(n,A000602(n)); od; THE RESULTS: # print A000598 0, 1 1, 1 2, 1 3, 2 4, 4 5, 8 6, 17 7, 39 8, 89 9, 211 10, 507 11, 1238 12, 3057 13, 7639 14, 19241 15, 48865 16, 124906 17, 321198 18, 830219 19, 2156010 20, 5622109 21, 14715813 22, 38649152 23, 101821927 24, 269010485 25, 712566567 26, 1891993344 27, 5034704828 28, 13425117806 29, 35866550869 30, 95991365288 31, 257332864506 32, 690928354105 33, 1857821351559 34, 5002305607153 35, 13486440075669 36, 36404382430278 37, 98380779170283 38, 266158552000477 39, 720807976831447 40, 1954002050661819 41, 5301950692017063 42, 14398991611139217 43, 39137768751465752 44, 106465954658531465 45, 289841389106439413 46, 789642117549095761 47, 2152814945971655556 48, 5873225808361331954 49, 16033495247557039074 50, 43797554941937577760 51, 119710153806425838814 52, 327387061440387339008 53, 895843085951178143691 54, 2452637333887058055384 55, 6718266278929859733622 56, 18411771398697833823000 57, 50482497809955343038577 58, 138479602828722198112039 59, 380035071896650874765695 60, 1043393111652308730560544 # print A000678 0, 0 1, 1 2, 1 3, 2 4, 4 5, 9 6, 18 7, 42 8, 96 9, 229 10, 549 11, 1347 12, 3326 13, 8330 14, 21000 15, 53407 16, 136639 17, 351757 18, 909962 19, 2365146 20, 6172068 21, 16166991 22, 42488077 23, 112004630 24, 296080425 25, 784688263 26, 2084521232 27, 5549613097 28, 14804572332 29, 39568107511 30, 105938822149 31, 284103144805 32, 763067158047 33, 2052459438451 34, 5528079077194 35, 14908290599141 36, 40253559153599 37, 108811562245870 38, 294451568992057 39, 797620980258275 40, 2162722316575299 41, 5869562580635247 42, 15943815991204954 43, 43345348850358244 44, 117934205327801553 45, 321120920694833012 46, 875012884317194451 47, 2385963304276811923 48, 6510342288827238315 49, 17775536033527519517 50, 48563412946169091538 51, 132755572620358656409 52, 363114382908903499230 53, 993738020349673316314 54, 2721004621466290269856 55, 7454305911590424530084 56, 20431384311592156454085 57, 56026505140223290632010 58, 153704764139162989364304 59, 421863977181994671246360 60, 1158357123576323660235906 # print A000022 0, 0 1, 1 2, 0 3, 1 4, 1 5, 2 6, 2 7, 6 8, 9 9, 20 10, 37 11, 86 12, 181 13, 422 14, 943 15, 2223 16, 5225 17, 12613 18, 30513 19, 74883 20, 184484 21, 458561 22, 1145406 23, 2879870 24, 7274983 25, 18471060 26, 47089144 27, 120528657 28, 309576725 29, 797790928 30, 2062142876 31, 5345531935 32, 13893615154 33, 36201693122 34, 94550040702 35, 247489226192 36, 649164795179 37, 1706111387068 38, 4492268106137 39, 11849143694903 40, 31306272399249 41, 82844289232592 42, 219556833416379 43, 582711418814803 44, 1548650597699086 45, 4121164035477355 46, 10980644552977297 47, 29292322287537199 48, 78230449952046203 49, 209157421594847524 50, 559792108243652284 51, 1499747256886637180 52, 4021888564802046862 53, 10795592352734214301 54, 29003616992986723961 55, 77988882023425041312 56, 209881653036255078479 57, 565281325218726852976 58, 1523670273799826063416 59, 4110001603629684771638 60, 11094472600919142562161 # print A000200 0, 0 1, 0 2, 1 3, 0 4, 1 5, 1 6, 3 7, 3 8, 9 9, 15 10, 38 11, 73 12, 174 13, 380 14, 915 15, 2124 16, 5134 17, 12281 18, 30010 19, 73401 20, 181835 21, 452165 22, 1133252 23, 2851710 24, 7215262 25, 18326528 26, 46750268 27, 119687146 28, 307528889 29, 792716193 30, 2049703887 31, 5314775856 32, 13817638615 33, 36012395538 34, 94076195437 35, 246293726710 36, 646132792949 37, 1698379393093 38, 4472479368458 39, 11798335239066 40, 31175528748092 41, 82507166303190 42, 218686061352847 43, 580458289071624 44, 1542810414137770 45, 4105998336743848 46, 10941189533706121 47, 29189484334449811 48, 77961916522544436 49, 208454979170534748 50, 557951543503300986 51, 1494916923080733431 52, 4009193215733249729 53, 10762179560838416600 54, 28915563880161713792 55, 77756549834124657812 56, 209267918157156750893 57, 563658253142606014960 58, 1519373298107001119114 59, 4098613763234069144311 60, 11064261934851268512023 # print A000602 0, 1 1, 1 2, 1 3, 1 4, 2 5, 3 6, 5 7, 9 8, 18 9, 35 10, 75 11, 159 12, 355 13, 802 14, 1858 15, 4347 16, 10359 17, 24894 18, 60523 19, 148284 20, 366319 21, 910726 22, 2278658 23, 5731580 24, 14490245 25, 36797588 26, 93839412 27, 240215803 28, 617105614 29, 1590507121 30, 4111846763 31, 10660307791 32, 27711253769 33, 72214088660 34, 188626236139 35, 493782952902 36, 1295297588128 37, 3404490780161 38, 8964747474595 39, 23647478933969 40, 62481801147341 41, 165351455535782 42, 438242894769226 43, 1163169707886427 44, 3091461011836856 45, 8227162372221203 46, 21921834086683418 47, 58481806621987010 48, 156192366474590639 49, 417612400765382272 50, 1117743651746953270 51, 2994664179967370611 52, 8031081780535296591 53, 21557771913572630901 54, 57919180873148437753 55, 155745431857549699124 56, 419149571193411829372 57, 1128939578361332867936 58, 3043043571906827182530 59, 8208615366863753915949 60, 22158734535770411074184 Neil Sloane Apr 06 2005