# [seqfan] Re: Request for some help for understanding an empirically found triangle

Max Alekseyev maxale at gmail.com
Wed Mar 2 15:34:59 CET 2016

```Hi Thomas,
It is not quite clear what you are asking. To guess formula from the
given list of numbers? It would be much helpful if you explain how
exactly you generated those numbers.
Regards,
Max

On Sun, Feb 28, 2016 at 5:38 AM, Thomas Baruchel <baruchel at gmx.com> wrote:
> Dear fellow contributors,
>
> I have been working for some time on a formula I would be verry happy to
> complete. This formula is related to généralized continued fractions as
> are all others I previously found (see
> https://oeis.org/wiki/User:Thomas_Baruchel ).
>
> With the help of well-known algorithms (LLL, PSLQ, Padé, etc.), I detect
> sequences of integer numbers related to this formula-to-come, and
> understanding
> the logic behind these sequences is each time a step forward for my purpose.
>
> Unfortunately, one "parameter" is related to a sequence of rational
> functions
> I could fully "detect" with some feeling about its general shape but not
> fully understand.
>
> I know there are some wizzards here that may quickly understand the logic
> or at least notice some things.
>
> Maybe this mailing list is not the most relevant place for asking this; in
> which
> case I apologize. If someone can help me for that without taking too much of
> his/her time, I would be very happy to be able to go further in working of
> the whole formula. Thank you by advance, regards, tb.
>
> My rational functions are (2D display for the first lines):
>
> n° 1 :
>
>       4
> - ---------
>   3 (x + 1)
>
>
> n° 2 :
>          1                 5
> -------------------- - --------- 24 (x + 1) (3 x + 1)   3 (x + 1)
>
>
> n° 3 :
>                  6                           3               2
> - ------------------------------- + -------------------- - -----
>   125 (x + 1) (4 x + 1) (4 x + 2)   25 (x + 1) (4 x + 1)   x + 1
>
>
> n° 4 :
>                    13                                     101
> ----------------------------------------- - -------------------------------
> 135 (x + 1) (5 x + 1) (5 x + 2) (5 x + 3)   540 (x + 1) (5 x + 1) (5 x + 2)
>
>                                                        7                 7
>                                             + -------------------- -
> ---------
>                                               30 (x + 1) (5 x + 1)   3 (x +
> 1)
>
> etc.
>
> I display it that way because it was very easy to use LLL in order to find
> the
> coefficients in the numerators. The most convenient way for working on that
> triangle is probably rather to write it; I show it below for the initial 28
> lines.
> I could easely find some formulas for each column but no general formula.
>
> (coefficients and a formula below for building back the rational functions)
>
> m: [
>     [-8],
>     [-20,2],
>     [-40,12,-24],
>     [-70,42,-202,624],
>     [-112,112,-944,6800,-28160],
>     [-168,252,-3240,39990,-378180,1956240],
>     [-240,504,-9120,168780,-2691500,31299660,-193818240],
>     [-330,924,-22308,573210,-13533950,262134768,-3604679456,25969798400],
>
> [-440,1584,-49104,1665048,-54028800,1533955752,-34784795304,551021454648,-4524877873152],
>
> [-572,2574,-99528,4294290,-182338520,7056003570,-232622918920,6027044680440,-107934603537600,994719833856000],
>
> [-728,4004,-188760,10081500,-540836660,27196564920,-1213669081240,45402131767300,-1320731548020500,26362209822109700,-269367401834854400],
>
> [-910,6006,-338910,21926190,-1447260750,91405905570,-5269521952170,265270592109600,-11076506267112000,357029036918928000,-7854973969921056000,88120488036962304000],
>
> [-1120,8736,-581152,44753280,-3559649600,275205604224,-19825712025728,1282796713117920,-71715923149544960,3301368476366175072,-116699890447511246304,2804668390029759121440,-34267109445760293273600],
>
> [-1360,12376,-958256,86572772,-8159187400,756765661176,-66440724681072,5348371530601974,-382626987630417516,23479203132873399912,-1180109553700743730064,45368332310341259611392,-1182254848944902971766528,15625389962188145791748096],
>
> [-1632,17136,-1527552,159942120,-17614027080,1928217475800,-202304078444160,19770089799495420,-1752820980572484900,137106641163083684400,-9150469205270049657000,498255795785126793714300,-20689280727893354516039100,580954533672149606803333500,-8258153843323806482092032000],
>
> [-1938,23256,-2364360,283936380,-36111651940,4603291360920,-568072401325800,66113606126531250,-7091616708136156350,685093642937381277600,-58084005963489507601600,4185232111643968748563200,-245300606116021459523520000,10937978741665549577203200000,-329201103515195643674342400000,5008018989272579747902955520000],
>     [-2280, 31008, -3565920, 486748080, -70778344000, 10387428137520,
> -1488100140390640, 203083381548077800, -25863156057665295600,
> 3013571052233142231600, -314587396914476897195600,
> 28707485207091210763643400, -2219712979095004695654352000,
> 139280212909636939833667975000, -6636253210525016113608507935000,
> 213097407927242925450657097805000, -3454359206881951401298519654400000],
>     [-2660, 40698, -5255856, 809056860, -133342927200, 22312129321188,
> -3669893137886976, 579814786246525830, -86348985070472364240,
> 11912853180399067114218, -1495810636771876664855616,
> 167605806446114219615512080, -16367656617234522958801322880,
> 1351158196468373411398992679968, -90343714080278210268675472690176,
> 4580006947756999980100721131530240, -156282112623791407868541623144448000,
> 2689249605403395016547015123312640000],
>     [-3080, 52668, -7589208, 1308328296, -242549291800, 45885645345792,
> -8583630937867664, 1553160300088906908, -267115992196423617852,
> 42987425869754619349668, -6375040043352284259869056,
> 857073372770354425645988028, -102516882515527648341577456052,
> 10661831289669551213862356652936, -935559064657406376990369478217592,
> 66392317932445459737857422287168372, -3567745082228659013145697404664186980,
> 128910715024676759785064905466962718772,
> -2346816894979813166169796326498705604608],
>     [-3542, 67298, -10758066, 2064221940, -427579512340, 90781666964580,
> -19156534075381220, 3933373867476907490, -773001449364837212650,
> 143337480955932385601050, -24740379167353794462482250,
> 3919619596881620663698352400, -561261767174736596981182684800,
> 71335052610001221107048830848000, -7868282527494059201650532567232000,
> 731130238635173198053153940388864000,
> -54874347623661338563124599643795456000,
> 3115425378725886177371208787427500032000,
> -118823016992948260463398185243235328000000,
> 2281667516033630958272834061325819904000000],
>     [-4048, 85008, -14997840, 3185310480, -732817955520, 173481958077120,
> -40999952833277760, 9476856735932804400, -2109071440724185828800,
> 445975726476730199677200, -88512496883173204230589200,
> 16287555278375689955554908000, -2742762926014265565760465116000,
> 416464503552989975128829015232000, -56022670706540265707212091974752000,
> 6530157038781041616772104173250906000,
> -640432883214307884870393296312193408000,
> 50678900925449456482492390925833162170000,
> -3030888088511779915449136680430563767970000,
> 121680563022029894895609769265080802797710000,
> -2457881871574620592645101983798188100812800000],
>     [-4600, 106260, -20594200, 4817335050, -1224368027500, 321314420301000,
> -84511921344096400, 21837002485484744250, -5460560343151268855500,
> 1305129863913508752619500, -294830327010826293740053000,
> 62266523988286549276643448150, -12155836300655550173555485100500,
> 2166552581042493037698711451044000, -347528815311777085029924893736724000,
> 49310886906769450018463698053718956000,
> -6055094240863690023007706913404687496000,
> 624931908053334691607080995207899763360000,
> -51995246297017813506321403783934541271680000,
> 3267058354801244958789636823999905461614080000,
> -137714123890197288080314052933290069397790720000,
> 2919082688078782495755638635126117397822177280000],
>     [-5200, 131560, -27890720, 7153246100, -1998828604500, 578492714221380,
> -168381395188292160, 48336833335182707700, -13488618129875116559100,
> 3616198939768951478832120, -921761072167762889584562040,
> 221195726134157418021734756400, -49475332499912242476184032601200,
> 10205195620733840007949834678752240, -1918172758706697630190525216812587520,
> 323985355991420046527323571346102988380,
> -48344306336853827546608471043597263180580,
> 6236410605945196309646575950593070996242112,
> -675574363650914146210117776862820284274422024,
> 58952707582889583114849809235709300486236865900,
> -3882542732229399143933348322987701786016750303500,
> 171441026104474223322555734221353597751602880467500,
> -3804948188770142619704669041367337098336518799360000],
>     [-5850, 161460, -37297260, 10445304870, -3194948279250,
> 1014976460259720, -325284627793625640, 103172770888592820030,
> -31935449045956426109370, 9539216762922608385767220,
> -2723057324780894154211144140, 736160112029528961953639831370,
> -186796816819349796321290668269630, 44075061969736070597572230035573520,
> -9572302178958218895730236218841324000,
> 1891505089131626077283496419557169149440,
> -335445207934760792427195571281784372615680,
> 52500216486691388374580499101901862088712192,
> -7097170407703652020998162722033686958068359168,
> 805065643420847023775013665323598164107193712640,
> -73517194182833158880223919841841477149672840232960,
> 5063910198045757853788580192610834249566744036769792,
> -233753499049244474320664989973779541903817843793723392,
> 5421019686584291940290446961296577366550977042379177984],
>     [-6552, 196560, -49299120, 15019547400, -5008903972800,
> 1739242914960600, -610937879343586200, 213044845306016606400,
> -72754198545840383754000, 24070115922338692488744000,
> -7644318575328042577024440000, 2310967474707680781916704026400,
> -659643825387695670545486563900800, 176310960419571195542552313305220000,
> -43737562948690701530900945112487860000,
> 9971569996183917023985409450427829525000,
> -2065816103048706564571284238051613500970000,
> 383694616243443437568263013812039800791990000,
> -62837694530317608096687694020291847950887850000,
> 8882027287256940966231272654559742520461647675000,
> -1052798325643449998635040396582752781540903534400000,
> 100403132568689053077039174326824794886423800369525000,
> -7219010863790387928075409606725932651241502240697925000,
> 347694912384706890967941398378351924881505314746114375000,
> -8410211480678023320925498721110295938761779872530432000000],
>     [-7308, 237510, -64467000, 21292941150, -7714097921400,
> 2916404269698150, -1118190692314588200, 426802089352299911250,
> -160027260019840672173000, 58331250782529140212904250,
> -20490396263678036008373775000, 6882240483264899117171621464650,
> -2193778617416596625925140519889000, 658701474719992054755835904812589250,
> -184849219993358885493062726683645011000,
> 48071268295949005824672264507034120713000,
> -11474627010559184001964832102040305176064000,
> 2486300080568850616517044780683050329279500000,
> -482557829071841856097474441995986751588159600000,
> 82519812703910109373696671538744758363463431600000,
> -12171485983756715937697088833853438913213231139200000,
> 1504622344753359530620585743898465267509637578086400000,
> -149578514098722961283327295745791298648255342896128000000,
> 11206095897313941255176473673738843538864273909178368000000,
> -562168844769302825642413825011269789161124218168180736000000,
> 14158685308482113187947385422334786985368814916547903488000000],
>     [-8120, 285012, -83467800, 29793593700, -11686535347500,
> 4793533613422200, -1998556890132738600, 831586576427972253000,
> -340848228875785588089000, 136238023124447243505363000,
> -52660395629469710899852464000, 19538961040560845384445253280400,
> -6910955344258841823610759577910000, 2314356603026298502745751283117512000,
> -728675543038928155386332670093531348000,
> 214094057758291962937552370682805572874500,
> -58219061862163874792551203092332714335562500,
> 14516218849791911496733341909570194788693657500,
> -3282604618552285952731079609822870039677157280000,
> 664411186688887595279287675566612107186757131137500,
> -118409734996254908439089485033112187166889903398912500,
> 18191629115197225627312502465880976532595946618885575000,
> -2341224306602995570500391595079223927042601832325548375000,
> 242207299534450314667366061440535601337557549950222099062500,
> -18876082685899484114785412705333370447849929199143588260312500,
> 984735360414688845599685871865811518568346909249116274589062500,
> -25783538810629517021266249543138106125941453578155830804480000000],
>     [-8990, 339822, -107076294, 41184403650, -17437036139250,
> 7734593443379670, -3494544596518289790, 1579284525442183941150,
> -704824443778733315833950, 307603458576783505012150890,
> -130224815626952571544054365330, 53104886131252741135008778869750,
> -20725045171508268666798696739515750, 7692149381874530736969377052576265170,
> -2697951867406866051352410906078675593130,
> 888312186434005905720786992903999283113280,
> -272592942573130969721591283451860861263314880,
> 77338601225244978709744671633189447473744656512,
> -20101127086406857123404360233486723068548779401984,
> 4734712312004040060142996428231341452704698192056320,
> -997560027458845642233086777856612791698591648386293760,
> 184958250344358621036304852979454062719166092717119602688,
> -29548178532481410695731393820614988207097594428678936395776,
> 3952663598502220791832289869795663784663381872742938194739200,
> -424872593639734370119419756220910340050772936517217787445248000,
> 34392492457805374774120867589709950335110603672365392763289600000,
> -1863044156298137915554810903420018148506651555374947876496998400000,
> 50638772787167674049719466457371229923270837855299620820746240000000]
> ]\$
>
> ny: length(m)\$
>
> m2: makelist(
>       sum( m[k-2][i] / k^i * x /
>            ( (x+1)*product( (k-1)*x+j-1, j, 1, i ) ),
>            i, 1, k-2), k, 3, ny+2 )\$
>
> (Maxima syntax, easy to understand however).
>
>
> --
> Thomas Baruchel
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
```