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

Sun Feb 28 11:38:38 CET 2016

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