Bernoulli numbers are mysterious, showing up in lots of places, e.g., the zeta function, the gamma function, sums of powers (Faulhaber's formula), and the Taylor expansion of the tangent function.
Bernoulli numbers of odd index are all zero except for bernoulli[1] which is +1/2 or -1/2 depending on convention. odd index are omitted below so as not to take sides on this Holy War.
the values below were computed using the bernfrac function in Pari/GP. the bernvec function can speed up computation but was not needed. we give both the traditional improper fraction form and each number broken into an integer part and a proper fraction.
bernoulli[0] = 1
bernoulli[2] = 1 / 6
bernoulli[4] = -1 / 30
bernoulli[6] = 1 / 42
bernoulli[8] = -1 / 30
bernoulli[10] = 5 / 66
bernoulli[12] = -691 / 2730
bernoulli[14] = 7 / 6 = 1 + 1/6
bernoulli[16] = -3617 / 510 = -7 - 47/510
bernoulli[18] = 43867 / 798 = 54 + 775/798
bernoulli[20] = -174611 / 330 = -529 - 41/330
bernoulli[22] = 854513 / 138 = 6192 + 17/138
bernoulli[24] = -236364091 / 2730 = -86580 - 691/2730
bernoulli[26] = 8553103 / 6 = 1425517 + 1/6
bernoulli[28] = -23749461029 / 870 = -27298231 - 59/870
bernoulli[30] = 8615841276005 / 14322 = 601580873 + 12899/14322
bernoulli[32] = -7709321041217 / 510 = -15116315767 - 47/510
bernoulli[34] = 2577687858367 / 6 = 429614643061 + 1/6
bernoulli[36] = -26315271553053477373 / 1919190 = -13711655205088 - 638653/1919190
bernoulli[38] = 2929993913841559 / 6 = 488332318973593 + 1/6
bernoulli[40] = -261082718496449122051 / 13530 = -19296579341940068 - 2011/13530
bernoulli[42] = 1520097643918070802691 / 1806 = 841693047573682615 + 1/1806
bernoulli[44] = -27833269579301024235023 / 690 = -40338071854059455413 - 53/690
bernoulli[46] = 596451111593912163277961 / 282 = 2115074863808199160560 + 41/282
bernoulli[48] = -5609403368997817686249127547 / 46410 = -120866265222965259346027 - 14477/46410
bernoulli[50] = 495057205241079648212477525 / 66 = 7500866746076964366855720 + 5/66
bernoulli[52] = -801165718135489957347924991853 / 1590 = -503877810148106891413789303 - 83/1590
bernoulli[54] = 29149963634884862421418123812691 / 798 = 36528776484818123335110430842 + 775/798
bernoulli[56] = -2479392929313226753685415739663229 / 870 = -2849876930245088222626914643291 - 59/870
bernoulli[58] = 84483613348880041862046775994036021 / 354 = 238654274996836276446459819192192 + 53/354
bernoulli[60] = -1215233140483755572040304994079820246041491 / 56786730 = -21399949257225333665810744765191097 - 22298681/56786730
bernoulli[62] = 12300585434086858541953039857403386151 / 6 = 2050097572347809756992173309567231025 + 1/6
bernoulli[64] = -106783830147866529886385444979142647942017 / 510 = -209380059113463784090951852900279701847 - 47/510
bernoulli[66] = 1472600022126335654051619428551932342241899101 / 64722 = 22752696488463515559649260352769264581469 + 62483/64722
bernoulli[68] = -78773130858718728141909149208474606244347001 / 30 = -2625771028623957604730304973615820208144900 - 1/30
bernoulli[70] = 1505381347333367003803076567377857208511438160235 / 4686 = 321250821027180325182047923042649852435219411 + 289/4686
bernoulli[72] = -5827954961669944110438277244641067365282488301844260429 / 140100870 = -41598278166794710913917074495262358936689603011 - 48540859/140100870
bernoulli[74] = 34152417289221168014330073731472635186688307783087 / 6 = 5692069548203528002388345621912105864448051297181 + 1/6
bernoulli[76] = -24655088825935372707687196040585199904365267828865801 / 30 = -821836294197845756922906534686173330145508927628860 - 1/30
bernoulli[78] = 414846365575400828295179035549542073492199375372400483487 / 3318 = 125029043271669930167323398297028955241771963644484775 + 37/3318
bernoulli[80] = -4603784299479457646935574969019046849794257872751288919656867 / 230010 = -20015583233248370274925329198813298768724220132825915915 - 47717/230010
bernoulli[82] = 1677014149185145836823154509786269900207736027570253414881613 / 498 = 3367498291536437423339667690333875301621959894719384367232 + 77/498
bernoulli[84] = -2024576195935290360231131160111731009989917391198090877281083932477 / 3404310 = -594709705031354477186604968440515408405790715651069049904704 - 1058237/3404310
bernoulli[86] = 660714619417678653573847847426261496277830686653388931761996983 / 6 = 110119103236279775595641307904376916046305114442231488626999497 + 1/6
bernoulli[88] = -1311426488674017507995511424019311843345750275572028644296919890574047 / 61410 = -21355259545253501188658385019041065678973298739163469211804590304 - 5407/61410
bernoulli[90] = 1179057279021082799884123351249215083775254949669647116231545215727922535 / 272118 = 4332889698664119241961661305937920621845136851180910914498655788032 + 230759/272118
bernoulli[92] = -1295585948207537527989427828538576749659341483719435143023316326829946247 / 1410 = -918855282416693282262005552155018971389603889162719959591004487113437 - 77/1410
bernoulli[94] = 1220813806579744469607301679413201203958508415202696621436215105284649447 / 6 = 203468967763290744934550279902200200659751402533782770239369184214108241 + 1/6
bernoulli[96] = -211600449597266513097597728109824233673043954389060234150638733420050668349987259 / 4501770 = -47003833958035731078575255535006060654596737369759057915139763564120483354 - 1450679/4501770
bernoulli[98] = 67908260672905495624051117546403605607342195728504487509073961249992947058239 / 6 = 11318043445484249270675186257733934267890365954750747918178993541665491176373 + 1/6
bernoulli[100] = -94598037819122125295227433069493721872702841533066936133385696204311395415197247711 / 33330 = -2838224957069370695926415633648176473828468092801288212822853171446486511107028 - 4471/33330
bernoulli[102] = 3204019410860907078243020782116241775491817197152717450679002501086861530836678158791 / 4326 = 740642489796788506297508271409209841768797317880887066731161003487485328441210855 + 61/4326
bernoulli[104] = -319533631363830011287103352796174274671189606078272738327103470162849568365549721224053 / 1590 = -200964548027566044834656196727153631868672708225328766243461301989213565009779698883 - 83/1590
bernoulli[106] = 36373903172617414408151820151593427169231298640581690038930816378281879873386202346572901 / 642 = 56657170050805941445719346030519356961419468287510420621387564452152460861972277798400 + 101/642
bernoulli[108] = -3469342247847828789552088659323852541399766785760491146870005891371501266319724897592306597338057 / 209191710 = -16584511154136216915823713374319912301494962614725464727402466815589878137712650743149939 - 71532367/209191710
bernoulli[110] = 7645992940484742892248134246724347500528752413412307906683593870759797606269585779977930217515 / 1518 = 5036885995049237741928942191518015481244237426490321414152565132252831097674298932791785387 + 49/1518
bernoulli[112] = -2650879602155099713352597214685162014443151499192509896451788427680966756514875515366781203552600109 / 1671270 = -1586146823765818636936340157296643878274097841277896388047286451429731136509885006831200945121 - 226439/1671270
bernoulli[114] = 21737832319369163333310761086652991475721156679090831360806110114933605484234593650904188618562649 / 42 = 517567436175456269840732406825071225612408492359305508590621669403181082957966515497718776632444 + 1/42
bernoulli[116] = -309553916571842976912513458033841416869004128064329844245504045721008957524571968271388199595754752259 / 1770 = -174889218402171173396900258776181591451414761618265448726273472158762122895238400153326666438279521 - 89/1770
bernoulli[118] = 366963119969713111534947151585585006684606361080699204301059440676414485045806461889371776354517095799 / 6 = 61160519994952185255824525264264167780767726846783200716843240112735747507634410314895296059086182633 + 1/6
bernoulli[120] = -51507486535079109061843996857849983274095170353262675213092869167199297474922985358811329367077682677803282070131 / 2328255930 = -22122776912707834942288323456712932445573185054987780150566552693027736635002572659102528031391154956836 - 971032651/2328255930
bernoulli[122] = 49633666079262581912532637475990757438722790311060139770309311793150683214100431329033113678098037968564431 / 6 = 8272277679877096985422106245998459573120465051843356628384885298858447202350071888172185613016339661427405 + 1/6
bernoulli[124] = -95876775334247128750774903107542444620578830013297336819553512729358593354435944413631943610268472689094609001 / 30 = -3195892511141570958359163436918081487352627667109911227318450424311953111814531480454398120342282422969820300 - 1/30
bernoulli[126] = 5556330281949274850616324408918951380525567307126747246796782304333594286400508981287241419934529638692081513802696639 / 4357878 = 1275008222338779298231002430292667986695719179638977329516058573538220731833362242193847881912832263475958141508 + 4096615/4357878
bernoulli[128] = -267754707742548082886954405585282394779291459592551740629978686063357792734863530145362663093519862048495908453718017 / 510 = -525009230867741338994028246245651754469198940377552432607801345222270181833065745383064045281411494212737075399447 - 47/510
bernoulli[130] = 1928215175136130915645299522271596435307611010164728458783733020528548622403504078595174411693893882739334735142562418015 / 8646 = 223018178942416252098692981988387281437382721508758785424905507810380363451712245962893177387681457638137258286208931 + 589/8646
bernoulli[132] = -410951945846993378209020486523571938123258077870477502433469747962650070754704863812646392801863686694106805747335370312946831 / 4206930 = -97684521930955204438633513398980239301166902674985678971000170661895983711329844759158434488299944780185742512315481910 - 1310531/4206930
bernoulli[134] = 264590171870717725633635737248879015151254525593168688411918554840667765591690540727987316391252434348664694639349484190167 / 6 = 44098361978452954272272622874813169191875754265528114735319759140111294265281756787997886065208739058110782439891580698361 + 1/6
bernoulli[136] = -84290226343367405131287578060366193649336612397547435767189206912230442242628212786558235455817749737691517685781164837036649737 / 4110 = -20508570886464088839729337727583015486456596690400835953087398275481859426430222089186918602388746894815454424764273682977287 - 167/4110
bernoulli[138] = 2694866548990880936043851683724113040849078494664282483862150893060478501559546243423633375693325757795709438325907154973590288136429 / 274386 = 9821443327979127710757296960209752104149185799072410705583196274811683181939115856580267855114057414721266530821204999429964677 + 273107/274386
bernoulli[140] = -3289490986435898803930699548851884006880537476931130981307467085162504802973618096693859598125274741604181467826651144393874696601946049 / 679470 = -4841260079820888050878919670996341276113054994232462038511585625800263150652152555217830953721687111431235327279572526224667309229 - 117419/679470
bernoulli[142] = 14731853280888589565870080442453214239804217023990642676194878997407546061581643106569966189211748270209483494554402556608073385149191 / 6 = 2455308880148098260978346740408869039967369503998440446032479832901257676930273851094994364868624711701580582425733759434678897524865 + 1/6
bernoulli[144] = -3050244698373607565035155836901726357405007104256566761884191852434851033744761276392695669329626855965183503295793517411526056244431024612640493 / 2381714790 = -1280692680408474754878251327860178572181183417119632011809521429908427882645327686944705780380037383050883058628440358894326745245777738 - 965295473/2381714790
bernoulli[146] = 4120570026280114871526113315907864026165545608808541153973817680034790262683524284855810008621905238290240143481403022987037271683989824863 / 6 = 686761671046685811921018885984644004360924268134756858995636280005798377113920714142635001436984206381706690580233837164506211947331637477 + 1/6
bernoulli[148] = -1691737145614018979865561095112166189607682852147301400816480675916957871178648433284821493606361235973346584667336181793937950344828557898347149 / 4470 = -378464685819691046949789954163795568144895492650402997945521404008267980129451551070429864341467838025357177777927557448308266296382227717751 - 179/4470
bernoulli[150] = 463365579389162741443284425811806264982233725425295799852299807325379315501572305760030594769688296308375193913787703707693010224101613904227979066275 / 2162622 = 214261012506652915508713231351482720966601526029650951415596348934478293248460575061213006604801160955717270014726431021090606783849241293313384 + 1933427/2162622
bernoulli[152] = -3737018141155108502105892888491282165837489531488932951768507127182409731328472084456653639812530140212355374618917309552824925858430886313795805601 / 30 = -124567271371836950070196429616376072194582984382964431725616904239413657710949069481888454660417671340411845820630576985094164195281029543793193520 - 1/30
bernoulli[154] = 10259718682038021051027794238379184461025738652460569233992776489750881337506863808448685054322627708245455888249006715516690124228801409697850408284121 / 138 = 74345787551000152543679668394052061311780714872902675608643307896745516938455534843831051118279910929314897740934831271860073363976821809404713103508 + 17/138
bernoulli[156] = -81718086083262628510756459753673452313595710396116467582152090596092548699138346942995509488284650803976836337164670494733866559829768848363506624334818961419869 / 1794590070 = -45535795304641704894063333223321274876772114534277160901794185563554661092679704253013898315109171870084423423319549792635298912486331125393726615424111 - 522242099/1794590070
bernoulli[158] = 171672676901153210072183083506103395137513922274029564150500135265308148197358551999205867870374013289728260984269623579880772408522396975250682773558018919 / 6 = 28612112816858868345363847251017232522918987045671594025083355877551358032893091999867644645062335548288043497378270596646795401420399495875113795593003153 + 1/6
bernoulli[160] = -4240860794203310376065563492361156949989398087086373214710625778458441940477839981850928830420029285687066701804645453159767402961229305942765784122421197736180867 / 230010 = -18437723552033869727688202653628785487541402926335260270034458408149393245849484726102903484283419354319667413610910191555877583414761557944288440165302368315 - 47717/230010
bernoulli[162] = 1584451495144416428390934243279426140836596476080786316960222380784239380974799880364363647978168634590418215854419793716549388865905348534375629928732008786233507729 / 130074 = 12181154536221046699501316506599521355817430663167015060351971806696491081805740427482538001277493077712826666777525052789561241031300248584464458144840696728273 + 125527/130074
bernoulli[164] = -20538064609143216265571979586692646837805331023148645068133372383930344948316600591203926388540940814833173322793804325084945094828524860626092013547281335356200073083 / 2490 = -8248218718531412154848184572968934473014189165923150629772438708405761023420321522571857987365839684671957157748515793206805258967279060492406431143486480062730953 - 113/2490
bernoulli[166] = 5734032969370860921631095311392645731505222358555208498573088911303001784652122964703205752709194193095246308611264121678834250704468082648313788124754168671815815821441 / 1002 = 5722587793783294332965164981429786159186848661232743012547992925452097589473176611480245262184824544007231844921421279120593064575317447752808171781191785101612590640 + 161/1002
bernoulli[168] = -13844828515176396081238346585063517228531109156984345249260453934317772754836791258987516540324983611569758649525983347408589045734176589270143058509026392246407576578281097477 / 3404310 = -4066853052505910472676796938311586556021957212176430833050002477541050243613769386156817839833911603693482276739187485102293576593840334537731011132660184368170811876204 - 1058237/3404310
bernoulli[170] = 195334207626637530414976779238462234481410337350988427215139995707346979124686918267688171536352650572535330369818176979951931477427594872783018749894699157917782460035894085 / 66 = 2959609206464205006287526958158518704263792990166491321441515086474954229161923004055881386914434099583868641966942075453817143597387801102773011362040896332087613030846880 + 5/66
bernoulli[172] = -11443702211333328447187179942991846613008046506032421731755258148665287832264931024781365962633301701773088470841621804328201008020129996955549467573217659587609679405537739509973 / 5190 = -2204952256518945750903117522734459848363785453956150622688874402440325208528888444081188046750154470476510302666979153049749712527963390550202209551679703196071229172550624183 - 203/5190
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