Talk:Bernoulli numbers
Bernoulli numbers, the why and the why for
I added the Bernoulli numbers task for several reasons:
- they're an important sequence
- forces utilization of gihugic integers
- displaying the numbers can utilize some creative coding
The numerator can get quite large. Here is a list of the non-zero Bernoulli numbers from B0 to B200:
(Shown at four-fifth size.)
B(n) Bernoulli number expressed as a fraction (right justified and aligned on the solidi) ──── ──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────── 0 1/1 1 -1/2 2 1/6 4 -1/30 6 1/42 8 -1/30 10 5/66 12 -691/2730 14 7/6 16 -3617/510 18 43867/798 20 -174611/330 22 854513/138 24 -236364091/2730 26 8553103/6 28 -23749461029/870 30 8615841276005/14322 32 -7709321041217/510 34 2577687858367/6 36 -26315271553053477373/1919190 38 2929993913841559/6 40 -261082718496449122051/13530 42 1520097643918070802691/1806 44 -27833269579301024235023/690 46 596451111593912163277961/282 48 -5609403368997817686249127547/46410 50 495057205241079648212477525/66 52 -801165718135489957347924991853/1590 54 29149963634884862421418123812691/798 56 -2479392929313226753685415739663229/870 58 84483613348880041862046775994036021/354 60 -1215233140483755572040304994079820246041491/56786730 62 12300585434086858541953039857403386151/6 64 -106783830147866529886385444979142647942017/510 66 1472600022126335654051619428551932342241899101/64722 68 -78773130858718728141909149208474606244347001/30 70 1505381347333367003803076567377857208511438160235/4686 72 -5827954961669944110438277244641067365282488301844260429/140100870 74 34152417289221168014330073731472635186688307783087/6 76 -24655088825935372707687196040585199904365267828865801/30 78 414846365575400828295179035549542073492199375372400483487/3318 80 -4603784299479457646935574969019046849794257872751288919656867/230010 82 1677014149185145836823154509786269900207736027570253414881613/498 84 -2024576195935290360231131160111731009989917391198090877281083932477/3404310 86 660714619417678653573847847426261496277830686653388931761996983/6 88 -1311426488674017507995511424019311843345750275572028644296919890574047/61410 90 1179057279021082799884123351249215083775254949669647116231545215727922535/272118 92 -1295585948207537527989427828538576749659341483719435143023316326829946247/1410 94 1220813806579744469607301679413201203958508415202696621436215105284649447/6 96 -211600449597266513097597728109824233673043954389060234150638733420050668349987259/4501770 98 67908260672905495624051117546403605607342195728504487509073961249992947058239/6 100 -94598037819122125295227433069493721872702841533066936133385696204311395415197247711/33330 102 3204019410860907078243020782116241775491817197152717450679002501086861530836678158791/4326 104 -319533631363830011287103352796174274671189606078272738327103470162849568365549721224053/1590 106 36373903172617414408151820151593427169231298640581690038930816378281879873386202346572901/642 108 -3469342247847828789552088659323852541399766785760491146870005891371501266319724897592306597338057/209191710 110 7645992940484742892248134246724347500528752413412307906683593870759797606269585779977930217515/1518 112 -2650879602155099713352597214685162014443151499192509896451788427680966756514875515366781203552600109/1671270 114 21737832319369163333310761086652991475721156679090831360806110114933605484234593650904188618562649/42 116 -309553916571842976912513458033841416869004128064329844245504045721008957524571968271388199595754752259/1770 118 366963119969713111534947151585585006684606361080699204301059440676414485045806461889371776354517095799/6 120 -51507486535079109061843996857849983274095170353262675213092869167199297474922985358811329367077682677803282070131/2328255930 122 49633666079262581912532637475990757438722790311060139770309311793150683214100431329033113678098037968564431/6 124 -95876775334247128750774903107542444620578830013297336819553512729358593354435944413631943610268472689094609001/30 126 5556330281949274850616324408918951380525567307126747246796782304333594286400508981287241419934529638692081513802696639/4357878 128 -267754707742548082886954405585282394779291459592551740629978686063357792734863530145362663093519862048495908453718017/510 130 1928215175136130915645299522271596435307611010164728458783733020528548622403504078595174411693893882739334735142562418015/8646 132 -410951945846993378209020486523571938123258077870477502433469747962650070754704863812646392801863686694106805747335370312946831/4206930 134 264590171870717725633635737248879015151254525593168688411918554840667765591690540727987316391252434348664694639349484190167/6 136 -84290226343367405131287578060366193649336612397547435767189206912230442242628212786558235455817749737691517685781164837036649737/4110 138 2694866548990880936043851683724113040849078494664282483862150893060478501559546243423633375693325757795709438325907154973590288136429/274386 140 -3289490986435898803930699548851884006880537476931130981307467085162504802973618096693859598125274741604181467826651144393874696601946049/679470 142 14731853280888589565870080442453214239804217023990642676194878997407546061581643106569966189211748270209483494554402556608073385149191/6 144 -3050244698373607565035155836901726357405007104256566761884191852434851033744761276392695669329626855965183503295793517411526056244431024612640493/2381714790 146 4120570026280114871526113315907864026165545608808541153973817680034790262683524284855810008621905238290240143481403022987037271683989824863/6 148 -1691737145614018979865561095112166189607682852147301400816480675916957871178648433284821493606361235973346584667336181793937950344828557898347149/4470 150 463365579389162741443284425811806264982233725425295799852299807325379315501572305760030594769688296308375193913787703707693010224101613904227979066275/2162622 152 -3737018141155108502105892888491282165837489531488932951768507127182409731328472084456653639812530140212355374618917309552824925858430886313795805601/30 154 10259718682038021051027794238379184461025738652460569233992776489750881337506863808448685054322627708245455888249006715516690124228801409697850408284121/138 156 -81718086083262628510756459753673452313595710396116467582152090596092548699138346942995509488284650803976836337164670494733866559829768848363506624334818961419869/1794590070 158 171672676901153210072183083506103395137513922274029564150500135265308148197358551999205867870374013289728260984269623579880772408522396975250682773558018919/6 160 -4240860794203310376065563492361156949989398087086373214710625778458441940477839981850928830420029285687066701804645453159767402961229305942765784122421197736180867/230010 162 1584451495144416428390934243279426140836596476080786316960222380784239380974799880364363647978168634590418215854419793716549388865905348534375629928732008786233507729/130074 164 -20538064609143216265571979586692646837805331023148645068133372383930344948316600591203926388540940814833173322793804325084945094828524860626092013547281335356200073083/2490 166 5734032969370860921631095311392645731505222358555208498573088911303001784652122964703205752709194193095246308611264121678834250704468082648313788124754168671815815821441/1002 168 -13844828515176396081238346585063517228531109156984345249260453934317772754836791258987516540324983611569758649525983347408589045734176589270143058509026392246407576578281097477/3404310 170 195334207626637530414976779238462234481410337350988427215139995707346979124686918267688171536352650572535330369818176979951931477427594872783018749894699157917782460035894085/66 172 -11443702211333328447187179942991846613008046506032421731755258148665287832264931024781365962633301701773088470841621804328201008020129996955549467573217659587609679405537739509973/5190 174 4166161554662042831884959593250717297395614318182561412048180684077407803317591270831194619293832107482426945655143357909807251852859279483176373435697607639883085093246499347128331/2478 176 -1369347910486705707645621362512824332220360774476594348356938715366608044588614657557436131706543948464159947970464346070253278291989696390096800799614617317655510118710460076077638883999/1043970 178 1124251816617941290026484851206299982774720467712867275292043701618829826708395745459654170718363182143418314514085426692857018428614935412736063946853033094328968069656979232446257101741/1074 180 -6173136454016248924640522272263470960199559328290655337530202055853397791747341312347030141906500993752700612233695954532816018207721731818225290076670213481102834647254685911917265818955932383093313/7225713885390 182 4277269279349192541137304400628629348327468135828402291661683018622451659989595510712915810436238721139546963558655260384328988773219688091443529626531335687951612545946030357929306651006711/6 184 -857321333523056180131194437347933216431403305730705359015465649285681432317514010686029079324479659634642384809061711319481020030715989009140595170556956196762318625529645723516532076273012244047/1410 186 22258646098436968050639602221816385181596567918515338169946670500599612225742487595012775838387331550474751212260636163500086787417640903770807353228157478339547041472679880890292167353534100797481/42 188 -14158277750623758793309386870401397333112823632717478051426522029712001260747920789473711562165031101665618225654329210473605281619696918061316240634857984019071572591940586875558943580878119388321001/30 190 5411555842544259796131885546196787277987837486638756184149141588783989774511509608733429067517383750706299486822702171672522203106730993581242777825864203487238429479957280273093904025319950569633979493395/12606 192 -346465752997582699690191405750952366871923192340955593486485715370392154894102000406980162521728492501917598012711402163530166516991115122131398542029056286959857727373568402417020319761912636411646719477318166587/868841610 194 2269186825161532962833665086968359967389321429297588337232986752409765414223476696863199759981611817660735753831323900456495253961837175924312108872915089534970310604331636484174526399721365966337809334021247/6 196 -62753135110461193672553106699893713603153054153311895305590639107017824640241378480484625554578576142115835788960865534532214560982925549798683762705231316611716668749347221458005671217067357943416524984438771831113/171390 198 88527914861348004968400581010530565220544526400339548429439843908721196349579494069282285662653465989920237253162555666526385826449862863083834096823053048072002986184254693991336699593468906111158296442729034119206322233/244713882 200 -498384049428333414764928632140399662108495887457206674968055822617263669621523687568865802302210999132601412697613279391058654527145340515840099290478026350382802884371712359337984274122861159800280019110197888555893671151/1366530
-- Gerard Schildberger (talk) 09:04, 11 March 2014 (UTC)
Algorithm?
How about adding info on how to generate the numbers to the task description? --Paddy3118 (talk) 13:47, 11 March 2014 (UTC)
- I didn't want to suggest or demand any particular method on how to generate Bernoulli numbers. People are free to use any method they want. The link to the Wolfram MathWorld (TM) has some examples, the REXX example uses formula 33. -- Gerard Schildberger (talk) 16:11, 11 March 2014 (UTC)
- Yes, it sure looks simple. The programming solutions that used it also looks simple and clean. -- Gerard Schildberger (talk) 01:56, 12 March 2014 (UTC)
The double sum formula used by the REXX example is #33 from the entry Bernoulli number on The Eric Weisstein's World of Mathematics (TM).
- where is a binomial coefficient.
- where is a binomial coefficient.
-- Gerard Schildberger (talk) 01:00, 18 March 2014 (UTC)
use of memoization
In the REXX programming solution, I added memoization for the COMB and PERM functions (combinations and permutations).
For computing the Bernoulli numbers up to (and including): 50 values, memoization was 17% faster. 100 " " " 38% " 200 " " " 42% "
B(1)?
I propose to leave the choice of B(1)=1/2 or B(1)=-1/2 open to solution authors. It's essentially arbitrary anyway, and the solutions already provided are split. I see no reason to prescribe one over the other. Thebigh (talk) 13:39, 27 October 2021 (UTC)