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

[marraco]

According of OEIS, the \( b_j \) sequence for \( \\[15pt]

{^z(x+1)} \) seems to be number of "forests" with n nodes and height at most z-1 (OEIS A210725)

Following this, I had been trying to figure what a forest of fractional height is.

\( \\[25pt]

{^{1+\frac{1}{n}}(x+1)} \) should correspond to forests with height =\( \\[15pt]

{\frac{1}{n}} \).

I tried various definitions for counting forests of half height, but none seems to work. Them I attempted to reverse engineer it by getting the\( \\[15pt]

{b_j} \) coefficients first, and then trying to figure how to get them.

My plan was to use Sheldonison's code to calculate the values of \( \\[25pt]

{^{1+\frac{1}{2}}(x+1)} \) at various points, and numerically obtaining the coefficients of his Taylor series by applying finite differences.



The problem is that for \( \\[25pt]

{^{1+\frac{1}{2}}(x+1)} \) we need his Taylor series developed around x=1, and both knesser.gp and fatou.gp do not converge here.

So, I planned to get the Taylor series around r, and with it, calculate back the Taylor around 1.

I tried an r=1.01. calculated n (41) points of \( \\[15pt]

{^{1.5}(r+n.\Delta x)} \) using knesser.gp, and from those points estimated the derivatives, and the bj values.


This is the Pari/GP code I used.

Code:

\r kneser.gp
\p 400
z=1.5 /*fractional exponent*/
r=1.01; /*place where derivatives will be calculated. Initially, coefficientes of Taylor developped around x=r*/
dx=1e-15;
num=20; /*number of derivatives to calculate*/
T=vector(1+2*num,i,0); /*vector where tetrations are stored*/

/*^z(x+r) at various points separated by dx*/
{for(n=1,1+2*num,
  init(r+(-1-num+n)*dx);
  T[n]=real(sexp(z));
);}


/*finding the coefficients for taylor series around x=r*/
/*using finite differences of order 'num'*/
m=matrix(1+2*num,1+2*num,r,c,((-(1+num)+r)*dx)^(c-1));
Tcoeff_at_r=m^-1*T~


/*indentification of the converged coefficients*/
/*calculating ^{1.5}1=1+tolerance*/
tolerance=abs(2*r);
maxConvergedCoeff=0;
{T1=0;
for(n=1,num,
    T1=T1+Tcoeff_at_r[n]*(-(r-1))^(n-1);
    if(abs(T1)>tolerance,break,maxConvergedCoeff=n);
)}

/*finding the coefficients for taylor series around x=1*/
f(x)=sum(n=1,maxConvergedCoeff,Tcoeff_at_r[n]*(x-(r-1))^(n-1));
Tcoeff_at_1=vector(maxConvergedCoeff,n,polcoeff(f(x),n-1));
cj=vector(maxConvergedCoeff-1,n,Tcoeff_at_1[n+1]*(n)!)
bj=vector(length(cj),n,sum(j=1,n,stirling(n,j,2)*cj[j]))

Unfortunately, even using a large precision (it took more than one day running on an i7 920 processor), I only got 4 derivatives. The other have stability problems, and the ones I got are not very precise.

Anyways, these are the closest numbers I got for bj of \( \\[15pt]

{^{1.5}(x+1)} \):

Code:

%38 = [1.000152704474621237210741951681248733595287259524430467296, 2.861320494145353285475346429548911736678675196129994723805, -13.6475978846314412502384523327046630578, 756.92186972158813546692266102007258863]

Those should be "number of forests with height 1/2" for 1,2,3 and 4 nodes.
The first number is the more accurate, but not for much. It should be 1, and the rest loses many digits of accuracy each one.
So, for 2 nodes, there should be 2.86 forests of 1/2 height. But that number is surely a rough approximation. It may well be 2.5 instead of 2.86, and I find dubious that 3 nodes have a negative number of forests.

I have no idea on how to improve precision. Increasing the decimals in Pari/GP does nothing.
Maybe using a more accurate numerical difference equation? I used finite difference of order 40.