Tetration series for integer exponent. Can you find the pattern?

[marraco]

Those series are for the base not the exponent! That's why they needed to be generated with tetcomplex rather than kneser; because getting such amazingly good analytic convergence for such a series requires complex base tetration, with results for n bases, equally spaced around a circle centered on base=2. fatou.gp (with small code updates) could also generate such results. To evaluate those two referenced Taylor series, substitute x=(b-2), to evaluate the Taylor series.

...I'm royally confused now.

the coefficients in the last code tag of this post are the Taylor series coefficients of the function \( \\[15pt]

{f_2(x)\,=\,^{-0.5}(x+2)} \) ?

Then it should be
\( \\[15pt]

{f_{2\,(0)}\,=\, ^{-0.5}(2)=0.544764121459557...} \)
\( \\[15pt]

{f_{2\,(1)}\,=\, ^{-0.5}(3)=0.486648923935738...} \)
\( \\[15pt]

{f_{2\,(2)}\,=\, ^{-0.5}(4)=0.457213238216139...} \)

but this code gives

Code:

\p 100
/*coefficients of Taylor around 2 of ^{-0.5}x*/

a=[0.5447641214595567339801218858257244685854,-0.09026490293475114180982800726025252487179,0.05334642698935378617403396491528890594804,-0.03638190492562309183765608353362070821840,0.02665589484943122254265742189263438424835,-0.02047608577133435850738520805893632252252,0.01628939391559684527389871185757624228228,-0.01331802035638468229849633176805710250959,0.01113080347039454404398917618932270486539,-0.009471945601741301301799666960159500493414,0.008181870472918983418952481797363865140773,-0.007156971109633091475785436209879906176635,0.006327698270413005651257016844418549882893,-0.005646005057506155565996841622134687059648,0.005077852297548008377590502397935756807242,-0.004598579564709383679003395147264261288216,0.004189960720720897899813797031566114828076,-0.003838281581489355718364575037825080832826,0.003533055202798846826754080155559369484869,-0.003266144873505376098605478730250127897586,0.003031153706186763516973507099991793317656,-0.002822992160610217843850530938287773588695,0.002637566583362648226391359841543536218954,-0.002471551499838161939220088380143251371568,0.002322220812129558330686955182616565785365,-0.002187321052201244713039707812416923885954,0.002064975080105974232286030081964966694861,-0.001953608108640999822645409586956590179027,0.001851890298539576237491912671987439471748,-0.001758691790434811807511081574014212915353,0.001673047168806168273884584564550319035826,-0.001594127148975512837191984637091772279620,0.001521215846034519175841930450931000214014,-0.001453692394304569109742037483974061901089,0.001391015984731342378663288972983370893753,-0.001332713607717317327426366538502721958881,0.001278369952559800466677607874323180950794,-0.001227619037446880124253911053592339307366,0.001180137236855264191223682764928143626889,-0.001135637444029102801867724419804092337925,0.001093864160641107689052187995385697434389,-0.001054589347847405322195327651270883669674,0.001017608905751088252280461602251417505390,-0.0009827396740070429313553656082778187553405,0.0009498168665858747234148639733725614346048,-0.0009186918698079533934930814875314534697736,0.0008892303455966866573662840017051026789547,-0.0008613105921955727892242590292733293823957,0.0008348221228916550711398042195621478636400,-0.0008096644300085595816921514966352297536765,0.0007857459069005338594524734325749675459648,-0.0007629829051481069438849922812669102687654,0.0007412989078245228555308404776565708380003,-0.0007206238027261275601524070871984237792707,0.0007008932419630012770449395923516096144246,-0.0006820480763866325438818551382445843370545,0.0006640338550679383137667663988829019940730,-0.0006468003814946490331294868382872885969872,0.0006303013193831178111678644623963748370560,-0.0006144938420376038744680011450272121474617,0.0005993383200741946673108758684076009577129,-0.0005847980430851115584178527733490186488280,0.0005708389714760195065962019338923580583096,-0.0005574295152845303003325133006371836953863,0.0005445403373002463827629604348689289744143,-0.0005321441782716469402918017155794455019026,0.0005202157024181592772390603384936993523783,-0.0005087313618820156701812678182987572351923,0.0004976692791697413648392248040346423784153,-0.0004870091470646722871348673605764087543724,0.0004767321459596961918548061350965022366495,-0.0004668208790873150944172435667760750155470,0.0004572593267416065803546985077871002284578,-0.0004480328213309398865880156803821028719749,0.0004391280460191844112860195774985849996584,-0.0004305330608687577109307488887504560310821,0.0004222373618726151852083171686555860996115,-0.0004142319801615488633254732943499557223787,0.0004065096311401751693759190311085411363661,-0.0003990649265288728503867805273227150792807,0.0003918946665218884447060007230972331199284,-0.0003849982348510417760694970639142865634935,0.0003783781269217428875539830317674205408416,-0.0003720406509700054967447945428476416910439,0.0003659968551918890293717641839010994444982,-0.0003602637511201750326035146298122448667960,0.0003548659266520138706765657962715630833711,-0.0003498376730740476659615080146624803134097,0.0003452257919088022517001222379805220095853,-0.0003410933031098378510324434621473247433045,0.0003375243510812609314744505648626776451409,-0.0003346307060211903759381304723940593677783,0.0003325603945037557882292885310966766766416,-0.0003315091777419398624384857779237117250571,0.0003317358460153907833479388101334190239424,-0.0003335826371421790840947235619676003126446,0.0003375025483293688889372157683210232309158,-0.0003440959391986930633575807516000693209207,0.0003541596811015852018702302245010936694298,-0.0003687532792553722994412436565871821184048,0.0003892879974033369355506985563538084697749,-0.0004176472122482091920832394022681079546693,0.0004563492419325118494433069912923916819317,-0.0005087680413650084606119114211931555096770,0.0005794328703416784956152178590887291116773,-0.0006744359202315312364732927380874961661109,0.0008019877695049482375578463387186199880001,-0.0009731755956181394060092634776613016580966,0.00120299993096644682942613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/*Taylor series around 2 of ^{-0.5}x*/
f2(x)=sum(n=0,150,a[n+1]/n!*x^n)

\( \\[15pt]

{f_{2\,(0)}\,=\, 0.5447641214595567...} \)
\( \\[15pt]

{f_{2\,(1)}\,=\, 0.4760690431769200...} \)
\( \\[15pt]

{f_{2\,(2)}\,=\, 0.4358972773362133...} \)


... and why textcomplex.gp "generates tetration for arbitrary bases" instead of "arbitrary exponents"? Sounds like the same as knesser and fatou (the base is constant, and the exponent is the variable)