**This is a partial answer.**\
Using the group law of elliptic curves, we have 

$b_{10}=\frac{\left(n^{102}+133370 n^{100}+235431945 n^{98}-53960558412 n^{96}+37353015835420 n^{94}+1715285459850920 n^{92}+154018873425409932 n^{90}+4528460790052228992 n^{88}-10283455511101671494 n^{86}+424389497721502142548 n^{84}+21198683938074571284634 n^{82}+261033154046082383403576 n^{80}+2270064982003402818480028 n^{78}+15317757618354379225002184 n^{76}+84218098664615395090065004 n^{74}+345159573021345914583822624 n^{72}+1129951624304798874757797847 n^{70}+3131305476125014022272232326 n^{68}+7635830330877839055871498111 n^{66}+16446578072159313295259200380 n^{64}+31726949494863946397847804664 n^{62}+54769575000613973884608095824 n^{60}+84669779180983943334398860888 n^{58}+117776609406232459328970393984 n^{56}+146631440164415842271230869964 n^{54}+164805637943248865339648114968 n^{52}+165741869909257891183520432332 n^{50}+150306415334481456543373768912 n^{48}+121703129810992794049207385688 n^{46}+88124674373459097943143748048 n^{44}+56481814917222827023209545464 n^{42}+31767118902941960531617938880 n^{40}+15407922021660173371718373183 n^{38}+6306682290739982829447936934 n^{36}+2063724483753154237664688919 n^{34}+509509860910406920886925292 n^{32}+72423661674195782496620652 n^{30}-2085965471015373600109304 n^{28}-3102450405774431050519652 n^{26}-511897235486864832178688 n^{24}+72673456634449307534874 n^{22}+33312001105932688908244 n^{20}+3846262561102571988538 n^{18}+294258428803841831032 n^{16}+9380206793092459404 n^{14}+156074955019903848 n^{12}-7836007732260580 n^{10}-12400565501408 n^{8}+607383986505 n^{6}+2125016730 n^{4}-948799 n^{2}+100\right) n}{100 n^{102}-948799 n^{100}+2125016730 n^{98}+607383986505 n^{96}-12400565501408 n^{94}-7836007732260580 n^{92}+156074955019903848 n^{90}+9380206793092459404 n^{88}+294258428803841831032 n^{86}+3846262561102571988538 n^{84}+33312001105932688908244 n^{82}+72673456634449307534874 n^{80}-511897235486864832178688 n^{78}-3102450405774431050519652 n^{76}-2085965471015373600109304 n^{74}+72423661674195782496620652 n^{72}+509509860910406920886925292 n^{70}+2063724483753154237664688919 n^{68}+6306682290739982829447936934 n^{66}+15407922021660173371718373183 n^{64}+31767118902941960531617938880 n^{62}+56481814917222827023209545464 n^{60}+88124674373459097943143748048 n^{58}+121703129810992794049207385688 n^{56}+150306415334481456543373768912 n^{54}+165741869909257891183520432332 n^{52}+164805637943248865339648114968 n^{50}+146631440164415842271230869964 n^{48}+117776609406232459328970393984 n^{46}+84669779180983943334398860888 n^{44}+54769575000613973884608095824 n^{42}+31726949494863946397847804664 n^{40}+16446578072159313295259200380 n^{38}+7635830330877839055871498111 n^{36}+3131305476125014022272232326 n^{34}+1129951624304798874757797847 n^{32}+345159573021345914583822624 n^{30}+84218098664615395090065004 n^{28}+15317757618354379225002184 n^{26}+2270064982003402818480028 n^{24}+261033154046082383403576 n^{22}+21198683938074571284634 n^{20}+424389497721502142548 n^{18}-10283455511101671494 n^{16}+4528460790052228992 n^{14}+154018873425409932 n^{12}+1715285459850920 n^{10}+37353015835420 n^{8}-53960558412 n^{6}+235431945 n^{4}+133370 n^{2}+1}
$
$b_{11}=\frac{\left(n^{120}+236252 n^{118}+734363634 n^{116}-312770699492 n^{114}+372591844872939 n^{112}+24846206675655560 n^{110}+4397460814156889380 n^{108}+197098278557698691880 n^{106}-3344010959849032904451 n^{104}+77041054916969931469604 n^{102}+6025814782162621289529574 n^{100}+103995465476911844397376836 n^{98}+1403483097587777368439021975 n^{96}+15498012900538452440256329232 n^{94}+142925176113346096459752929320 n^{92}+898285418497461503114875588304 n^{90}+4154689922196564345724488430173 n^{88}+16274187858368913956279733198300 n^{86}+61890000902737222242418970136450 n^{84}+225779196578375145753832632507996 n^{82}+756786678765469678606166591535567 n^{80}+2262534612561363902593953210725304 n^{78}+5917547652470211201971343014836380 n^{76}+13620277856356468776332219700867480 n^{74}+27459152125492247279736040270198329 n^{72}+49018178073029219641783948590200964 n^{70}+77697769677873785728455499712552550 n^{68}+110108815699791321232665610983560548 n^{66}+140147096559211503009874312621611515 n^{64}+160704499865273317527676592295669216 n^{62}+166800079319744449016824823686303024 n^{60}+156803394222544054034958059457136224 n^{58}+134108869765624608642875089743841427 n^{56}+104330906826393397189405252036072564 n^{54}+74132059482284441103456770631089478 n^{52}+48085116554380394968670681241369780 n^{50}+28566416464099272660681894365198097 n^{48}+15472531151366340592453782011285112 n^{46}+7634070524957506190629044185948700 n^{44}+3376267460065621992265053228120024 n^{42}+1319335874981699856700548139386471 n^{40}+442263567882306852924119639251404 n^{38}+122783505051121829165127645781602 n^{36}+27107460947533293092765040330156 n^{34}+4525830480756840714555835855797 n^{32}+455893058739248616779979806480 n^{30}+4694990023388238740422470184 n^{28}-6744049455656087942958073392 n^{26}-1138735253459131761691971889 n^{24}-36517193907554294094976812 n^{22}+6530567094465986775850822 n^{20}+533216700911799507392468 n^{18}+28783653421923421178997 n^{16}+559607160876545566536 n^{14}+7749092040428976292 n^{12}-163753085727869848 n^{10}-208299265226589 n^{8}+3950879704204 n^{6}+8107202130 n^{4}-2036308 n^{2}+121\right) n}{121 n^{120}-2036308 n^{118}+8107202130 n^{116}+3950879704204 n^{114}-208299265226589 n^{112}-163753085727869848 n^{110}+7749092040428976292 n^{108}+559607160876545566536 n^{106}+28783653421923421178997 n^{104}+533216700911799507392468 n^{102}+6530567094465986775850822 n^{100}-36517193907554294094976812 n^{98}-1138735253459131761691971889 n^{96}-6744049455656087942958073392 n^{94}+4694990023388238740422470184 n^{92}+455893058739248616779979806480 n^{90}+4525830480756840714555835855797 n^{88}+27107460947533293092765040330156 n^{86}+122783505051121829165127645781602 n^{84}+442263567882306852924119639251404 n^{82}+1319335874981699856700548139386471 n^{80}+3376267460065621992265053228120024 n^{78}+7634070524957506190629044185948700 n^{76}+15472531151366340592453782011285112 n^{74}+28566416464099272660681894365198097 n^{72}+48085116554380394968670681241369780 n^{70}+74132059482284441103456770631089478 n^{68}+104330906826393397189405252036072564 n^{66}+134108869765624608642875089743841427 n^{64}+156803394222544054034958059457136224 n^{62}+166800079319744449016824823686303024 n^{60}+160704499865273317527676592295669216 n^{58}+140147096559211503009874312621611515 n^{56}+110108815699791321232665610983560548 n^{54}+77697769677873785728455499712552550 n^{52}+49018178073029219641783948590200964 n^{50}+27459152125492247279736040270198329 n^{48}+13620277856356468776332219700867480 n^{46}+5917547652470211201971343014836380 n^{44}+2262534612561363902593953210725304 n^{42}+756786678765469678606166591535567 n^{40}+225779196578375145753832632507996 n^{38}+61890000902737222242418970136450 n^{36}+16274187858368913956279733198300 n^{34}+4154689922196564345724488430173 n^{32}+898285418497461503114875588304 n^{30}+142925176113346096459752929320 n^{28}+15498012900538452440256329232 n^{26}+1403483097587777368439021975 n^{24}+103995465476911844397376836 n^{22}+6025814782162621289529574 n^{20}+77041054916969931469604 n^{18}-3344010959849032904451 n^{16}+197098278557698691880 n^{14}+4397460814156889380 n^{12}+24846206675655560 n^{10}+372591844872939 n^{8}-312770699492 n^{6}+734363634 n^{4}+236252 n^{2}+1}$

All other $b_{m}$ can also be generated. Given,

$$(a^3-b)(b^3-a) = y^2$$

**First family ($a=n$)**

Denote $$E_1=\{(U,V): V^2 = -U^4+n^3U^3+nU-n^4\}\cup O.$$
It is birationally equivalent to Weierstrass form
$$E_2=\{(X,Y): Y^{2}+\left(3 n^{2}-1\right)XY + \left(2 n^{6}-10 n^{4}+8 n^{2}\right)Y
 = 
X^{3}+\left(\frac{3}{4} n^{4}-\frac{9}{2} n^{2}-\frac{1}{4}\right) X^{2}+4 n^{2} \left(n -1\right)^{2} \left(n +1\right)^{2} X +3 n^{10}-24 n^{8}+38 n^{6}-16 n^{4}-n^{2}
\}\cup O.$$
Let
$P=\left(3 n^{2}+1, -2 n^{6}-\frac{1}{2} n^{4}-5 n^{2}-\frac{1}{2}\right)\in E_2,$
then the $U$ corresponding to $[m-1]P$ is exactly $b_m$ where $m\geq 1$.

**Second family ($a=n^3$)**

Denote $$E_3=\{(U,V):V^{2} = n^{9} U^{3}-n^{12}-U^{4}+n^{3} U\}\cup O.$$
It is birationally equivalent to 
$$E_4=\{(X,Y):Y^{2} = 
X^{3}+\left(3 n^{10}-6 n^{2}\right) X^{2}+\left(3 n^{20}-15 n^{12}+12 n^{4}\right) X -9 n^{22}+18 n^{14}-9 n^{6}
\}\cup O.$$
Let $Q=(n^{12}-n^{10}+n^{6}+2 n^{2}+1, n^{18}+n^{12}+n^{6}+1) \in E_4,$ then the $U$ corresponding to $[m]Q$ is exactly $b_m$ where $m\geq 1$.

For example

$b_1=\frac{\left(n^{6}+n^{4}-2 n^{2}+1\right) n}{n^{6}-2 n^{4}+n^{2}+1}$

$b_2=\frac{n \left(n^{12}+8 n^{10}+10 n^{6}-4 n^{2}+1\right)}{n^{12}-4 n^{10}+10 n^{6}+8 n^{2}+1}$

$b_3=\frac{\left(n^{30}+17 n^{28}-18 n^{26}+101 n^{24}-172 n^{22}+80 n^{20}+282 n^{18}-82 n^{16}-244 n^{14}+282 n^{12}-28 n^{10}-64 n^{8}+101 n^{6}+9 n^{4}-10 n^{2}+1\right) n}{n^{30}-10 n^{28}+9 n^{26}+101 n^{24}-64 n^{22}-28 n^{20}+282 n^{18}-244 n^{16}-82 n^{14}+282 n^{12}+80 n^{10}-172 n^{8}+101 n^{6}-18 n^{4}+17 n^{2}+1}
$

$b_4=\frac{n \left(n^{48}+32 n^{46}+552 n^{42}-1088 n^{40}-16 n^{38}+5820 n^{36}+8160 n^{34}+544 n^{32}+6552 n^{30}+18560 n^{28}-4080 n^{26}+23302 n^{24}+8160 n^{22}-9280 n^{20}+6552 n^{18}-1088 n^{16}-4080 n^{14}+5820 n^{12}+32 n^{10}+544 n^{8}+552 n^{6}-16 n^{2}+1\right)}{n^{48}-16 n^{46}+552 n^{42}+544 n^{40}+32 n^{38}+5820 n^{36}-4080 n^{34}-1088 n^{32}+6552 n^{30}-9280 n^{28}+8160 n^{26}+23302 n^{24}-4080 n^{22}+18560 n^{20}+6552 n^{18}+544 n^{16}+8160 n^{14}+5820 n^{12}-16 n^{10}-1088 n^{8}+552 n^{6}+32 n^{2}+1}
$

$b_5=\frac{n \left(n^{78}+49 n^{76}-50 n^{74}+2093 n^{72}-9092 n^{70}+7024 n^{68}+71486 n^{66}+146362 n^{64}-221372 n^{62}+120846 n^{60}+1989884 n^{58}-1997248 n^{56}+2411691 n^{54}+2513095 n^{52}-3945398 n^{50}+4925063 n^{48}+1374136 n^{46}-4812512 n^{44}+9246036 n^{42}+3689036 n^{40}-10556008 n^{38}+9246036 n^{36}+2487832 n^{34}-5926208 n^{32}+4925063 n^{30}+3430735 n^{28}-4863038 n^{26}+2411691 n^{24}+1056332 n^{22}-1063696 n^{20}+120846 n^{18}+102298 n^{16}-177308 n^{14}+71486 n^{12}-3476 n^{10}+1408 n^{8}+2093 n^{6}+25 n^{4}-26 n^{2}+1\right)}{n^{78}-26 n^{76}+25 n^{74}+2093 n^{72}+1408 n^{70}-3476 n^{68}+71486 n^{66}-177308 n^{64}+102298 n^{62}+120846 n^{60}-1063696 n^{58}+1056332 n^{56}+2411691 n^{54}-4863038 n^{52}+3430735 n^{50}+4925063 n^{48}-5926208 n^{46}+2487832 n^{44}+9246036 n^{42}-10556008 n^{40}+3689036 n^{38}+9246036 n^{36}-4812512 n^{34}+1374136 n^{32}+4925063 n^{30}-3945398 n^{28}+2513095 n^{26}+2411691 n^{24}-1997248 n^{22}+1989884 n^{20}+120846 n^{18}-221372 n^{16}+146362 n^{14}+71486 n^{12}+7024 n^{10}-9092 n^{8}+2093 n^{6}-50 n^{4}+49 n^{2}+1}
$

$b_6=\frac{n \left(n^{108}+72 n^{106}+6234 n^{102}-31104 n^{100}-36 n^{98}+569433 n^{96}+3063744 n^{94}+15552 n^{92}-1574800 n^{90}+64105344 n^{88}-1531872 n^{86}+165039924 n^{84}+242797536 n^{82}-32052672 n^{80}+963577944 n^{78}-594241920 n^{76}-121398768 n^{74}+4018389060 n^{72}+20450880 n^{70}+297120960 n^{68}+8815738128 n^{66}-329795712 n^{64}-10225440 n^{62}+12135907374 n^{60}+1187302320 n^{58}+164897856 n^{56}+16524170140 n^{54}-329795712 n^{52}-593651160 n^{50}+12135907374 n^{48}+20450880 n^{46}+164897856 n^{44}+8815738128 n^{42}-594241920 n^{40}-10225440 n^{38}+4018389060 n^{36}+242797536 n^{34}+297120960 n^{32}+963577944 n^{30}+64105344 n^{28}-121398768 n^{26}+165039924 n^{24}+3063744 n^{22}-32052672 n^{20}-1574800 n^{18}-31104 n^{16}-1531872 n^{14}+569433 n^{12}+72 n^{10}+15552 n^{8}+6234 n^{6}-36 n^{2}+1\right)}{n^{108}-36 n^{106}+6234 n^{102}+15552 n^{100}+72 n^{98}+569433 n^{96}-1531872 n^{94}-31104 n^{92}-1574800 n^{90}-32052672 n^{88}+3063744 n^{86}+165039924 n^{84}-121398768 n^{82}+64105344 n^{80}+963577944 n^{78}+297120960 n^{76}+242797536 n^{74}+4018389060 n^{72}-10225440 n^{70}-594241920 n^{68}+8815738128 n^{66}+164897856 n^{64}+20450880 n^{62}+12135907374 n^{60}-593651160 n^{58}-329795712 n^{56}+16524170140 n^{54}+164897856 n^{52}+1187302320 n^{50}+12135907374 n^{48}-10225440 n^{46}-329795712 n^{44}+8815738128 n^{42}+297120960 n^{40}+20450880 n^{38}+4018389060 n^{36}-121398768 n^{34}-594241920 n^{32}+963577944 n^{30}-32052672 n^{28}+242797536 n^{26}+165039924 n^{24}-1531872 n^{22}+64105344 n^{20}-1574800 n^{18}+15552 n^{16}+3063744 n^{14}+569433 n^{12}-36 n^{10}-31104 n^{8}+6234 n^{6}+72 n^{2}+1}
$

$b_7=\frac{n \left(n^{150}+97 n^{148}-98 n^{146}+15705 n^{144}-123784 n^{142}+108128 n^{140}+3446348 n^{138}+24162948 n^{136}-27663384 n^{134}-54131940 n^{132}+1284695208 n^{130}-1216696704 n^{128}+6634254058 n^{126}-229454414 n^{124}-5801634356 n^{122}+81110865930 n^{120}-223888114072 n^{118}+145342483424 n^{116}+785626162028 n^{114}-1023645380268 n^{112}+167080903272 n^{110}+4270749389916 n^{108}-2214069073416 n^{106}-2065478795328 n^{104}+11467933754407 n^{102}+4703023132063 n^{100}-14705536010894 n^{98}+29106714141151 n^{96}+16623299156016 n^{94}-36970181795136 n^{92}+54069311943896 n^{90}+39190788849032 n^{88}-74153669206960 n^{86}+81131871039992 n^{84}+53330412573456 n^{82}-96765223483392 n^{80}+100555075829164 n^{78}+64281686810620 n^{76}-117821240784152 n^{74}+100555075829164 n^{72}+52483881399888 n^{70}-95918692309824 n^{68}+81131871039992 n^{66}+35216158023752 n^{64}-70179038381680 n^{62}+54069311943896 n^{60}+16621068832368 n^{58}-36967951471488 n^{56}+29106714141151 n^{54}+3131577515695 n^{52}-13134090394526 n^{50}+11467933754407 n^{48}-265305036072 n^{46}-4014242832672 n^{44}+4270749389916 n^{42}-307766370060 n^{40}-548798106936 n^{38}+785626162028 n^{36}-77870021944 n^{34}-675608704 n^{32}+81110865930 n^{30}+3907562866 n^{28}-9938651636 n^{26}+6634254058 n^{24}+623897544 n^{22}-555899040 n^{20}-54131940 n^{18}+13727076 n^{16}-17227512 n^{14}+3446348 n^{12}-53992 n^{10}+38336 n^{8}+15705 n^{6}+49 n^{4}-50 n^{2}+1\right)}{n^{150}-50 n^{148}+49 n^{146}+15705 n^{144}+38336 n^{142}-53992 n^{140}+3446348 n^{138}-17227512 n^{136}+13727076 n^{134}-54131940 n^{132}-555899040 n^{130}+623897544 n^{128}+6634254058 n^{126}-9938651636 n^{124}+3907562866 n^{122}+81110865930 n^{120}-675608704 n^{118}-77870021944 n^{116}+785626162028 n^{114}-548798106936 n^{112}-307766370060 n^{110}+4270749389916 n^{108}-4014242832672 n^{106}-265305036072 n^{104}+11467933754407 n^{102}-13134090394526 n^{100}+3131577515695 n^{98}+29106714141151 n^{96}-36967951471488 n^{94}+16621068832368 n^{92}+54069311943896 n^{90}-70179038381680 n^{88}+35216158023752 n^{86}+81131871039992 n^{84}-95918692309824 n^{82}+52483881399888 n^{80}+100555075829164 n^{78}-117821240784152 n^{76}+64281686810620 n^{74}+100555075829164 n^{72}-96765223483392 n^{70}+53330412573456 n^{68}+81131871039992 n^{66}-74153669206960 n^{64}+39190788849032 n^{62}+54069311943896 n^{60}-36970181795136 n^{58}+16623299156016 n^{56}+29106714141151 n^{54}-14705536010894 n^{52}+4703023132063 n^{50}+11467933754407 n^{48}-2065478795328 n^{46}-2214069073416 n^{44}+4270749389916 n^{42}+167080903272 n^{40}-1023645380268 n^{38}+785626162028 n^{36}+145342483424 n^{34}-223888114072 n^{32}+81110865930 n^{30}-5801634356 n^{28}-229454414 n^{26}+6634254058 n^{24}-1216696704 n^{22}+1284695208 n^{20}-54131940 n^{18}-27663384 n^{16}+24162948 n^{14}+3446348 n^{12}+108128 n^{10}-123784 n^{8}+15705 n^{6}-98 n^{4}+97 n^{2}+1}
$

$b_8=\frac{n \left(n^{192}+128 n^{190}+34976 n^{186}-316672 n^{184}-64 n^{182}+16598640 n^{180}+183271808 n^{178}+158336 n^{176}-802597152 n^{174}+15783441920 n^{172}-91635904 n^{170}+169138570616 n^{168}+51637997184 n^{166}-7891720960 n^{164}+3341392321312 n^{162}-12315000076032 n^{160}-25818998592 n^{158}+68913458089296 n^{156}-3394489735296 n^{154}+6157500038016 n^{152}+698084705941344 n^{150}+928291810704384 n^{148}+1697244867648 n^{146}+2377304988780828 n^{144}+11074118982856320 n^{142}-464145905352192 n^{140}+11302556844978208 n^{138}+57145559892452096 n^{136}-5537059491428160 n^{134}+57930167190755696 n^{132}+194825874649326464 n^{130}-28572779946226048 n^{128}+191829548714416992 n^{126}+408517766483552768 n^{124}-97412937324663232 n^{122}+512207488884484680 n^{120}+746736636788616320 n^{118}-204258883241776384 n^{116}+1022547154958318240 n^{114}+1094659815751805184 n^{112}-373368318394308160 n^{110}+1693928781347738320 n^{108}+1353209721461361024 n^{106}-547329907875902592 n^{104}+2186162763283367904 n^{102}+1489207748984260608 n^{100}-676604860730680512 n^{98}+2476945507038563910 n^{96}+1353209721461361024 n^{94}-744603874492130304 n^{92}+2186162763283367904 n^{90}+1094659815751805184 n^{88}-676604860730680512 n^{86}+1693928781347738320 n^{84}+746736636788616320 n^{82}-547329907875902592 n^{80}+1022547154958318240 n^{78}+408517766483552768 n^{76}-373368318394308160 n^{74}+512207488884484680 n^{72}+194825874649326464 n^{70}-204258883241776384 n^{68}+191829548714416992 n^{66}+57145559892452096 n^{64}-97412937324663232 n^{62}+57930167190755696 n^{60}+11074118982856320 n^{58}-28572779946226048 n^{56}+11302556844978208 n^{54}+928291810704384 n^{52}-5537059491428160 n^{50}+2377304988780828 n^{48}-3394489735296 n^{46}-464145905352192 n^{44}+698084705941344 n^{42}-12315000076032 n^{40}+1697244867648 n^{38}+68913458089296 n^{36}+51637997184 n^{34}+6157500038016 n^{32}+3341392321312 n^{30}+15783441920 n^{28}-25818998592 n^{26}+169138570616 n^{24}+183271808 n^{22}-7891720960 n^{20}-802597152 n^{18}-316672 n^{16}-91635904 n^{14}+16598640 n^{12}+128 n^{10}+158336 n^{8}+34976 n^{6}-64 n^{2}+1\right)}{n^{192}-64 n^{190}+34976 n^{186}+158336 n^{184}+128 n^{182}+16598640 n^{180}-91635904 n^{178}-316672 n^{176}-802597152 n^{174}-7891720960 n^{172}+183271808 n^{170}+169138570616 n^{168}-25818998592 n^{166}+15783441920 n^{164}+3341392321312 n^{162}+6157500038016 n^{160}+51637997184 n^{158}+68913458089296 n^{156}+1697244867648 n^{154}-12315000076032 n^{152}+698084705941344 n^{150}-464145905352192 n^{148}-3394489735296 n^{146}+2377304988780828 n^{144}-5537059491428160 n^{142}+928291810704384 n^{140}+11302556844978208 n^{138}-28572779946226048 n^{136}+11074118982856320 n^{134}+57930167190755696 n^{132}-97412937324663232 n^{130}+57145559892452096 n^{128}+191829548714416992 n^{126}-204258883241776384 n^{124}+194825874649326464 n^{122}+512207488884484680 n^{120}-373368318394308160 n^{118}+408517766483552768 n^{116}+1022547154958318240 n^{114}-547329907875902592 n^{112}+746736636788616320 n^{110}+1693928781347738320 n^{108}-676604860730680512 n^{106}+1094659815751805184 n^{104}+2186162763283367904 n^{102}-744603874492130304 n^{100}+1353209721461361024 n^{98}+2476945507038563910 n^{96}-676604860730680512 n^{94}+1489207748984260608 n^{92}+2186162763283367904 n^{90}-547329907875902592 n^{88}+1353209721461361024 n^{86}+1693928781347738320 n^{84}-373368318394308160 n^{82}+1094659815751805184 n^{80}+1022547154958318240 n^{78}-204258883241776384 n^{76}+746736636788616320 n^{74}+512207488884484680 n^{72}-97412937324663232 n^{70}+408517766483552768 n^{68}+191829548714416992 n^{66}-28572779946226048 n^{64}+194825874649326464 n^{62}+57930167190755696 n^{60}-5537059491428160 n^{58}+57145559892452096 n^{56}+11302556844978208 n^{54}-464145905352192 n^{52}+11074118982856320 n^{50}+2377304988780828 n^{48}+1697244867648 n^{46}+928291810704384 n^{44}+698084705941344 n^{42}+6157500038016 n^{40}-3394489735296 n^{38}+68913458089296 n^{36}-25818998592 n^{34}-12315000076032 n^{32}+3341392321312 n^{30}-7891720960 n^{28}+51637997184 n^{26}+169138570616 n^{24}-91635904 n^{22}+15783441920 n^{20}-802597152 n^{18}+158336 n^{16}+183271808 n^{14}+16598640 n^{12}-64 n^{10}-316672 n^{8}+34976 n^{6}+128 n^{2}+1}
$

$b_9=\frac{n \left(n^{246}+161 n^{244}-162 n^{242}+70889 n^{240}-886072 n^{238}+815264 n^{236}+67168788 n^{234}+897625628 n^{232}-965202088 n^{230}-7533890684 n^{228}+162996922840 n^{226}-154980191872 n^{224}+2923225655062 n^{222}-3434476283314 n^{220}+588533092276 n^{218}+91286448190678 n^{216}-632953450113928 n^{214}+541330085779616 n^{212}+3976314344076004 n^{210}-1090394821561124 n^{208}-3154993135737352 n^{206}+73086884931770356 n^{204}+118326290318272616 n^{202}-189655330193395904 n^{200}+230955731693089813 n^{198}+3499332161619581501 n^{196}-3634216456765647658 n^{194}+2478107376147849869 n^{192}+30369365465676168480 n^{190}-31042735811777144704 n^{188}+43287562760568065040 n^{186}+147750037408130553904 n^{184}-176351158630260450336 n^{182}+328253205970334800336 n^{180}+275447499691377442144 n^{178}-522531661942750613504 n^{176}+1763647576429316154568 n^{174}-497677701223797023832 n^{172}-1030988005482559690256 n^{170}+7252352429902643052360 n^{168}-4674303703745530099360 n^{166}-2056503577151626242432 n^{164}+23909615760761991579152 n^{162}-17596976765173123462288 n^{160}-4780826928162619993888 n^{158}+61196562643270347939664 n^{156}-41580211741817672931296 n^{154}-14626439830635910902016 n^{152}+130461830738443908472114 n^{150}-78776027855432042069230 n^{148}-35678636656434421276964 n^{146}+232029559140234705141058 n^{144}-121859057658065336028304 n^{142}-72231440481014089120576 n^{140}+354669816499191018773144 n^{138}-159393078928890377197176 n^{136}-120903215574438005338672 n^{134}+466538779722347253428664 n^{132}-182771131209529204580528 n^{130}-163457552653990651723008 n^{128}+535192237922122350672388 n^{126}-180050062369945581104492 n^{124}-193420299902930951001032 n^{122}+535192237922122350672388 n^{120}-158250521639756152254576 n^{118}-187978162223763704048960 n^{116}+466538779722347253428664 n^{114}-119123856427606180698040 n^{112}-161172438075722201837808 n^{110}+354669816499191018773144 n^{108}-78657685025559730470736 n^{106}-115432813113519694678144 n^{104}+232029559140234705141058 n^{102}-42459909690596997422606 n^{100}-71994754821269465923588 n^{98}+130461830738443908472114 n^{96}-18768619018460513573728 n^{94}-37438032553993070259584 n^{92}+61196562643270347939664 n^{90}-5408079356125052331280 n^{88}-16969724337210691124896 n^{86}+23909615760761991579152 n^{84}-482429047975491884064 n^{82}-6248378232921664457728 n^{80}+7252352429902643052360 n^{78}+497340563948659248808 n^{76}-2026006270655015962896 n^{74}+1763647576429316154568 n^{72}+340117714716622905568 n^{70}-587201876967996076928 n^{68}+328253205970334800336 n^{66}+109195186103636013744 n^{64}-137796307325765910176 n^{62}+43287562760568065040 n^{60}+18026147008239783584 n^{58}-18699517354340759808 n^{56}+2478107376147849869 n^{54}+1854221823390674381 n^{52}-1989106118536740538 n^{50}+230955731693089813 n^{48}+91096350745720840 n^{46}-162425390620844128 n^{44}+73086884931770356 n^{42}+1036451101533500 n^{40}-5281839058831976 n^{38}+3976314344076004 n^{36}-275439331891816 n^{34}+183815967557504 n^{32}+91286448190678 n^{30}-166317753650 n^{28}-2679625437388 n^{26}+2923225655062 n^{24}+78112991672 n^{22}-70096260704 n^{20}-7533890684 n^{18}+481883324 n^{16}-549459784 n^{14}+67168788 n^{12}-407512 n^{10}+336704 n^{8}+70889 n^{6}+81 n^{4}-82 n^{2}+1\right)}{n^{246}-82 n^{244}+81 n^{242}+70889 n^{240}+336704 n^{238}-407512 n^{236}+67168788 n^{234}-549459784 n^{232}+481883324 n^{230}-7533890684 n^{228}-70096260704 n^{226}+78112991672 n^{224}+2923225655062 n^{222}-2679625437388 n^{220}-166317753650 n^{218}+91286448190678 n^{216}+183815967557504 n^{214}-275439331891816 n^{212}+3976314344076004 n^{210}-5281839058831976 n^{208}+1036451101533500 n^{206}+73086884931770356 n^{204}-162425390620844128 n^{202}+91096350745720840 n^{200}+230955731693089813 n^{198}-1989106118536740538 n^{196}+1854221823390674381 n^{194}+2478107376147849869 n^{192}-18699517354340759808 n^{190}+18026147008239783584 n^{188}+43287562760568065040 n^{186}-137796307325765910176 n^{184}+109195186103636013744 n^{182}+328253205970334800336 n^{180}-587201876967996076928 n^{178}+340117714716622905568 n^{176}+1763647576429316154568 n^{174}-2026006270655015962896 n^{172}+497340563948659248808 n^{170}+7252352429902643052360 n^{168}-6248378232921664457728 n^{166}-482429047975491884064 n^{164}+23909615760761991579152 n^{162}-16969724337210691124896 n^{160}-5408079356125052331280 n^{158}+61196562643270347939664 n^{156}-37438032553993070259584 n^{154}-18768619018460513573728 n^{152}+130461830738443908472114 n^{150}-71994754821269465923588 n^{148}-42459909690596997422606 n^{146}+232029559140234705141058 n^{144}-115432813113519694678144 n^{142}-78657685025559730470736 n^{140}+354669816499191018773144 n^{138}-161172438075722201837808 n^{136}-119123856427606180698040 n^{134}+466538779722347253428664 n^{132}-187978162223763704048960 n^{130}-158250521639756152254576 n^{128}+535192237922122350672388 n^{126}-193420299902930951001032 n^{124}-180050062369945581104492 n^{122}+535192237922122350672388 n^{120}-163457552653990651723008 n^{118}-182771131209529204580528 n^{116}+466538779722347253428664 n^{114}-120903215574438005338672 n^{112}-159393078928890377197176 n^{110}+354669816499191018773144 n^{108}-72231440481014089120576 n^{106}-121859057658065336028304 n^{104}+232029559140234705141058 n^{102}-35678636656434421276964 n^{100}-78776027855432042069230 n^{98}+130461830738443908472114 n^{96}-14626439830635910902016 n^{94}-41580211741817672931296 n^{92}+61196562643270347939664 n^{90}-4780826928162619993888 n^{88}-17596976765173123462288 n^{86}+23909615760761991579152 n^{84}-2056503577151626242432 n^{82}-4674303703745530099360 n^{80}+7252352429902643052360 n^{78}-1030988005482559690256 n^{76}-497677701223797023832 n^{74}+1763647576429316154568 n^{72}-522531661942750613504 n^{70}+275447499691377442144 n^{68}+328253205970334800336 n^{66}-176351158630260450336 n^{64}+147750037408130553904 n^{62}+43287562760568065040 n^{60}-31042735811777144704 n^{58}+30369365465676168480 n^{56}+2478107376147849869 n^{54}-3634216456765647658 n^{52}+3499332161619581501 n^{50}+230955731693089813 n^{48}-189655330193395904 n^{46}+118326290318272616 n^{44}+73086884931770356 n^{42}-3154993135737352 n^{40}-1090394821561124 n^{38}+3976314344076004 n^{36}+541330085779616 n^{34}-632953450113928 n^{32}+91286448190678 n^{30}+588533092276 n^{28}-3434476283314 n^{26}+2923225655062 n^{24}-154980191872 n^{22}+162996922840 n^{20}-7533890684 n^{18}-965202088 n^{16}+897625628 n^{14}+67168788 n^{12}+815264 n^{10}-886072 n^{8}+70889 n^{6}-162 n^{4}+161 n^{2}+1}
$

and so on.

The sum of the coefficients in the denominator of $b_m$ is $2^{k(m)}$ where $k(m)=0, 4, 8, 16, 24, 36, 48, 64, 80, 100, 120, 144, 168, 196, 224$ (see https://oeis.org/A137932). 

The degree in the denominator of $b_m$ is $d(m)=6\beta(m)$ where
$\beta(m)=1, 2, 5, 8, 13, 18, 25, 32, 41, 50, 61, 72, 85, 98, 113$(see https://oeis.org/A000982).

I have verified this for $1\leq m \leq 15 $.

**Note**
In the first family($a=n$), I think it is nice to denote $b_1=\frac{n\times \color{red}{1}}{1}$,$b_2=\frac{n\,(4 + n^2 + 10n^4 + n^6)}{(1 + 10n^2 + n^4 + 4n^6)}\cdots$,then we also have :The sum of the coefficients in the denominator of $b_m$ is $2^{k(m)}$ where $k(m)=0, 4, 8, 16, 24, 36, 48, 64, 80, 100, 120, 144, 168, 196, 224, 256$ (see https://oeis.org/A137932).

The degree in the denominator of $b_m$ is $d(m)=2\alpha(m)$ where
$\alpha(m)=0, 3, 4, 9, 12, 19, 24, 33, 40, 51, 60, 73, 84, 99, 112, 129$(see https://oeis.org/A097063).

I have verified this for $1\leq m \leq 16 $.

It is worth noting

$$\alpha(m)-\beta(m)=(-1)^m$$
for  $1\leq m \leq 15 $. I think it is true for all $m\geq 1$.