Triangle

[Gottfried Helms]

Dear seqfans,

a short review and resume.

Assume a matrix-function rowshift(M) which
computes M1 = rowshift(M) in the following way:
M = [a,b,c,...]
    [k,l,m,...]
    [r,s,t,...]
    [...      ]

M1 = [a,b,c, ...   ]
     [0,k,l,m, ... ]
     [0,0,r,s,t,...]
     [ ...         ]

Assume the lower-trianguler matrix of Stirling-numbers 2'nd kind
S = [1  0  0  0 ...]
    [1  1  0  0 ...]
    [1  3  1  0 ...]
    [1  7  6  1 ...]
    [ ... ]
then with
H0 = [1]
     [1]
     [1]
     [1]
    ...
we have the iterative Mathar-products
H1 = S * rowshift(H0)  \\ = S
H2 = S * rowshift(H1)
H3 = S * rowshift(H2)
...
H1 =
  1   .   .   .  .
  1   1   .   .  .
  1   3   1   .  .
  1   7   6   1  .
  1  15  25  10  1
H2=
  1   .   .   .   .   .   .   .  .
  1   1   1   .   .   .   .   .  .
  1   3   4   3   1   .   .   .  .
  1   7  13  19  13   6   1   .  .
  1  15  40  85  96  75  35  10  1
H3=
  1   .   .    .    .    .    .    .    .   .   .   .  .
  1   1   1    1    .    .    .    .    .   .   .   .  .
  1   3   4    6    4    3    1    .    .   .   .   .  .
  1   7  13   26   31   31   25   13    6   1   .   .  .
  1  15  40  100  171  220  255  215  156  85  35  10  1

(construction based on a interpretation of
the Maple-implementation of R.Mathar)

----------------------------------

In my basic problem-description I also had the vector D,
which I also should index now.

It contains the coefficients of the polynomials in u:

 Dm = polcoeffs(prod(k=1,m-1,u^k-1))
Say
 D3 = columnvector([1 -1 -1 1])

Then define the matrix MD3 as the concatenation of shifted D3
 MD3 =
   1
  -1  1
  -1 -1  1
   1 -1 -1
      1 -1  ...
         1
      ...
up to the required dimension for the following matrix-multiplication
to obtain the k'th coefficient in the original powerseries
  Ut(x,h) = a_1(t,h)/1!*x + a_2(t,h)/2!*x^2 + ...

Then the coefficients for the bivariate polynomial in u=log(t) and
v=u^h of the k'th coefficient a_k(t,h) occur in the vector Vk

 Vk = MDk * transpose(Hj[k,])

where [k,] denotes the k'th row and the index j at Hj indicates the
 1/2*k*(k^2-4k+5)  'th iterate/row-shift

------------------------------------

This is an eminent simple recursive scheme to obtain the coefficients
for the integer (and fractional!) iteration of x -> t^x - 1.

However it is again enormous consumtive: the required iterations of
the Mathar-products is cubic with the index!

So for index k=20 I need already 8000 iterations with always growing
matrices... surely some shortcuts may be implemented, but this may
well be the same amount of time and memory as I would need for
the symbolic eigensystem-analysis. Hmmm....

However - the fact, that this simple scheme is able to *mimic* the
symbolic eigendecomposition of a matrix-operator is a very astonishing
aspect.

Gottfried Helms