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I'm not too sure what the exact term for this type of tetration is, but you'll probably recognize it as the newton series
${\ ^x} b = \sum^\infty _{k=0} {x\choose{k}}\Delta^k \text{sexp}_b(0)$
Where $\Delta f(x)$ is the difference operator $f(x+1) - f(x)$.
The above series has two main problems:
1. It has horrifyingly bad convergence
2. It only converges for bases $1

The first has two solutions:
1.Expanding around -1 instead of 0 actually seems to help convergence. Not a lot, but it does help.
2.The series seems to converge faster at larger arguments. This can be exploited by taking the logarithm of the series and increasing the argument appropriately. This helps tremendously, with 40 logs and 190 terms, $\ ^{\frac{1}{2}}\sqrt{2}$ has 40 correct digits. Without the logs, it only has 4.

This ends up allowing this tetration be be calculated fast and accurately enough that we can help solve the second problem by taylor expanding the in the base.
Thus, here are the first 20 coefficients of $\ ^{\frac{1}{2}}x$ around 1.3:
Code:
1.18938676815057446038958978533952183643013189670322056055481057298164131225527684427735915948183638577039636148478045857061959630059209396667384426975, 0.502010375341689917463980699942849515217180567032456403096253323808817519839299401993748785203409898436669803385695469896066815361054606935920440249897, -0.262759788721756864581601122842893801277751036360441951655201206112118171206055203331662497461495616704883313354811962146537537522652596017799716984049, 0.252541433354585431957607810065749107577232333317254003936920345075890384451085222527914626348907743064183299869580062147394545557290468315440354334807, -0.352612282449298556037058962871272764222633755255262271381414574681831425906537015507092391324710037580722776608642134517129513748910717693940021788039, 0.621677298164652736531936223803703122798845331895259122967269609718567408818646902493674955196358849620675027493712391738200958924868991814377511409157, -1.26050780940949293309473496751457508803155869212975193602873050623150572857720148361865858042292939837099015175537045195113300841486392202214221157598, 2.78946894926307250419609012944237141327971047365764644180937634359092733505462942006856627879977802915804677312596759441403910348973540678219087075504, -6.55130450767494919665255117058858039633497329452433984571224787280239805541843479702057849808492419468918495110076687736921823268369546258458551623048, 16.0710090228845933132534615450640023013886687687167747617039630933751129706165865500814356888785226546490852687225827248617581496737696817168093153429, -40.7704259182153375530604086443593919521475614769493253756483954258517708616052988082048357384027386210545375335546113749939214696714432741264720158991, 106.244321126292059196752083390024857789027574924140570504522491688951114502955395593102728514269806301547513915489115405515891366398589937001672403523, -283.023065848868812346563707317507012808599394093222261325251615876830618173347110349520513434919295375400097755630023208473741240599411576425055827909, 767.933763786027965831407740576122497561217686342251260748284176663708140871513390685625726433475251079261484458238695383917961576775641177247147520503, -2116.44561270241051080236792320537583316910647423316698320246454576890631421345287600136864420938113116159861645314023708929380344723655318159752190274, 5911.8594138127759329749422943752047603337761813433546786311935528993537980987722892605576055010848848272468910526372583051327151512324971334609620443, -16707.3334717021267293040481463596170312951888040432337228406433675057696360597132222789111854366949803489111005938596369105185867941219689060361131505, 47700.5349235131417564363370889439820833084022424615928142803634329053272283017103738924412877180486952747228851252753475003864826221543486125144067987, -137421.741181992684270720845646461088437555676807641552339277946116606194777054566765162517003931685067266949282565592941858857284964758394951437621541, 399099.764178539676938227138702779209771226135938677676356183560670501048089604499650618860208276875975877505538229622254784064064072823768230700329241, -1167441.59720175702159986672888446578396885416629335195276020166949794005477053704724756023751465315420620439408155825183831357730080864875865877178256
I estimate that those are accurate to around 30 coefficients
The series seems to have a radius of convergence of around .4
Hi Fivexthethird -

perhaps this might be interesting for you: I've made a discussion of the Newtonian method in comparision with the diagonalization-approach in 2008; perhaps today I could write this better (but unfortunately usually don't have time for this); see here: http://go.helms-net.de/math/tetdocs/Bino...zation.htm

I came to tetration via consideration of infinite matrices and powers of them, so possibly the whole article looks strange to you but I hope it is self-expalining enough. Don't hesitate to ask if something might look explainable by not too much effort...

Gottfried