Does anyone have taylor series approximations for tetration and slog base e^(1/e)?

[sheldonison]

This is more what I was looking for, thanks. And another question, do you have a similar taylor series for \( \text{slog}_\eta(z) \)? That would be the inverse of the lower super function.
......
Woah! have you ever thought to consider that since \( \text{cheta}(0) = 2*e = e + e \), \( \text{cheta}(1) = e^2 = e*e \), and \( \text{cheta}(2) = \text{sexp}_e(2) = e^e \) that the cheta function maps the growth of \( e\, \{x\}\, e \), where \( \{x\} \) is a hyper operator of x order (0 is addition, 1 is multiplication etc etc); or mathematically speaking, another conjecture:
\( \text{cheta}(x)\, =\, e\, \{x\}\, e\, =\, e\,\{x+1\}\,2 \), which should at least be true over domain [0, 2]. To see if its universally true would be very difficult, though.

It is an interesting sequence. Here is the series, centered at z=1, \( \text{slog}_\eta(z) = \sum_{n=0}^{\infty} a_n (z-1)^n \)

Code:

a0=   0
a1=   1.6364055628757310098612069643305
a2=   1.0153219515675015927934054348231
a3=   0.60179106451341218323473861841285
a4=   0.35339094138233716197130631267800
a5=   0.20678022805222642138569218148044
a6=   0.12077316515278617589978715489636
a7=   0.070466597739279935817347004921549
a8=   0.041088245566444697493192432669165
a9=   0.023947858691724371628412673016534
a10=  0.013953603992552690728627252149490
a11=  0.0081285329698961693398261557991908
a12=  0.0047344290157003070913140831631485
a13=  0.0027572046775250114750438564059555
a14=  0.0016055653508202650722548361996376
a15=  0.00093487396331912588479860322878510
a16=  0.00054431539381581329680504421391946
a17=  0.00031690239123736519247306265808875
a18=  0.00018449370377075905076738832103314
a19=  0.00010740430864046301276907787649912
a20=  0.000062524232227043156013995698181196
a21=  0.000036396822638059218270554447632005
a22=  0.000021186957652934110380738251130135
a23=  0.000012332895471487133422245016435244
a24=  0.0000071788340691456493380397479368256
a25=  0.0000041786510411693011327284124119868
a26=  0.0000024322734042405353969581364435823
a27=  0.0000014157397670132180924517455634643
a28=  0.00000082404283673548797945624589827040
a29=  0.00000047963619402863627513438889010921
a30=  0.00000027917101829902645542548058586310
a31=  0.00000016248949999299272203056483378299
a32=  0.000000094575181457179592966349170760430
a33=  0.000000055046061997819873417493220126620
a34=  0.000000032038543589826489735755462890716
a35=  0.000000018647341967554525035509582154381
a36=  0.000000010853228813698946831281796207325
a37=  0.0000000063168274147666049763200649142710
a38=  0.0000000036765226692784746720709040758475
a39=  0.0000000021398030836117500316624962487179
a40=  0.0000000012453998812027015780706541464576
a41=  0.0000000007248404052267904098654234587018
a42=  0.0000000004218661070727302103010309894460
a43=  0.0000000002455306059973809636590738454709
a44=  1.4290106986271220339032492794899 E-10
a45=  8.3169529782673189838917026421766 E-11
a46=  4.8405198310832728584491149379096 E-11
a47=  2.8172073806752434116210055497734 E-11
a48=  1.6396258116384352659166197756211 E-11
a49=  9.5426686705080144354640068061238 E-12
a50=  5.5538502152136607519127775538049 E-12
a51=  3.2323452797056846816888732034200 E-12
a52=  1.8812245466465630762639581584087 E-12
a53=  1.0948708034538274092484245688059 E-12
a54=  6.3721280701164076567624351207206 E-13
a55=  3.7085619145590230253906096801015 E-13
a56=  2.1583706178992941955046700446313 E-13
a57=  1.2561629614172847097262503849877 E-13
a58=  7.3108092357697884172557590560006 E-14
a59=  4.2548519124264617256971973371109 E-14
a60=  2.4762985306030106868904605580111 E-14
a61=  1.4411896206100053414406968180449 E-14
a62=  8.3876219415829221560909589871987 E-15
a63=  4.8815324822646088477541099811656 E-15
a64=  2.8410125546371166957177683205131 E-15
a65=  1.6534450727661439551155006062390 E-15
a66=  9.6229018805487868709553017018049 E-16
a67=  5.6004383866369015723533210751934 E-16
a68=  3.2594001769485484340769051661933 E-16
a69=  1.8969376400216642523206535563610 E-16
a70=  1.1039976379052598207387127395496 E-16
a71=  6.4251454988114292160593966066487 E-17
a72=  3.7393621961593764263505452461775 E-17
a73=  2.1762653817172712533134127189169 E-17
a74=  1.2665604966549427015232899034031 E-17
a75=  7.3712271908237475668169947042325 E-18