Tetra-series

[Gottfried]

If I Euler-sum the list of coefficients, which I get with the formula at the superroot-msg for heights h=0..63 , then the resulting powerseries seems to begin with

Code:

0 + 1/2x  -1/2 x^2 + 1/4 x^3  -1/6 x^4 + 5/12 x^5 + ??? + ??? , ... ]

where for the question-marks I needed higher Euler-orders.

Because this is tidy to avoid the Euler-summation at all we can do a trick:
Let S(x,h) be the formal powerseries for the height h for

S(x,h) = (1+x)^(1+x)^...^(1+x) - 1

and S(x) the series for the limit when h-> infinity, then by definition,

AS(x) = S(x,0) - S(x,1) + S(x,2) - ... + // Euler-sum

Since the coefficients of any height converge to that of the S(x)-series I compute the difference D(x,h) = S(x,h) - S(x) and rewrite

AS(x) = D(x,0) - D(x,1) + D(x,2) ... + aeta(0)*S(x)

where aeta(0) is the alternating zeta-series zeta(0) meaning

aeta(0) = 1-1+1-1+1-... = 1/2

Because the coefficients with index k<h in D(x,h) vanish, I get exact rational values in the coefficients of the formal powerseries in AS(x)

Code:

0 * x^0
                     1/2 * x^1
                    -1/2 * x^2
                     1/4 * x^3
                    -1/6 * x^4
                    5/12 * x^5
                  -23/80 * x^6
                  97/720 * x^7
              -1801/3360 * x^8
                619/5040 * x^9
            -4279/15120 * x^10
          106549/151200 * x^11
        2586973/5702400 * x^12
        2111317/1425600 * x^13
   777782953/1037836800 * x^14
  3321778277/4358914560 * x^15
...

In float-numerical display this is

Code:

0 * x^0
    0.500000000000 * x^1
   -0.500000000000 * x^2
    0.250000000000 * x^3
   -0.166666666667 * x^4
    0.416666666667 * x^5
   -0.287500000000 * x^6
    0.134722222222 * x^7
   -0.536011904762 * x^8
    0.122817460317 * x^9
  -0.283002645503 * x^10
   0.704689153439 * x^11
   0.453663895903 * x^12
    1.48100238496 * x^13
   0.749427032266 * x^14
   0.762065470951 * x^15
   -2.02559608779 * x^16
   -4.93868722102 * x^17
   -11.5286692883 * x^18
   -17.6563985780 * x^19
   -24.5338937285 * x^20
   -22.4594923016 * x^21
   -4.19284436502 * x^22
    53.8185412606 * x^23
    176.092085183 * x^24
    405.014519784 * x^25
    772.287054778 * x^26
    1291.34671701 * x^27
    1872.07516409 * x^28
    2213.27210256 * x^29
    1537.71737942 * x^30
   -1795.52581418 * x^31
...

Because the constant term for S(x,h)+1 = 1 its sum is the alternating sum 1-1+1-1... and we should set the constant term in AS(x) to 1/2.

Check, for base x=1/2 I get AS(x) = 0.938253002500