[seqfan] Looking for a simpler pattern for that matrix

[Gottfried Helms]

Hi -

 I'm experimenting with some generating functions
 and by analysis of the resulting approximative values
 I get the following matrix of coefficients. I'm unable
 to find a simpler pattern (for instance recursive);
 possibly there are the Bell-numbers involved because
 the matrix describes compositions of exponentials.

 Here is the matrix
 A =
     2        .          .          .           .          .           .         .         .      .      .   .
    -5        3          .          .           .          .           .         .         .      .      .   .
    10      -22          4          .           .          .           .         .         .      .      .   .
   -17      108        -57          5           .          .           .         .         .      .      .   .
    26     -432        504       -116           6          .           .         .         .      .      .   .
   -37     1520      -3510       1600        -205          7           .         .         .      .      .   .
    50    -4896      21060     -16960        4050       -330           8         .         .      .      .   .
   -65    14784    -113967     152320      -60375       8820        -497         9         .      .      .   .
    82   -42496     571536   -1218560      752500    -175392       17248      -712        10      .      .   .
  -101   117504   -2703132    8945664    -8268750    2884896     -440412     31104      -981     11      .   .
   122  -314880   12203460  -61440000    82687500  -41477184     9219840   -990720     52650  -1310     12   .
  -145   822272  -53045685  400097280  -768281250  538876800  -167498562  25681920  -2044845  84700  -1705  13

I do not yet know, whether it is meaningful to prefix it with an additional
column 1,0,0,0,... and also whether the row- and column-indices
(let's use "r" and "c" for that) better begin at zero or at 1.

It seems, that along the columns, consecutive powers of the column-nr are
involved, so 1^r in column 1 , 2^r in column 2, 3^r in column 3 and so on.
For column 1 WolframAlpha gives abs(a_{r,1}) = r^2+1, but for the following
columns I've not yet an exponential or polynomial formula.
What I finally want is a formula which allows to compute the columns to
arbitrary index so that I can approximate the column-sums
(divergent (but Euler-summable) for columns of A,
 possibly convergent for columns of F^-1 * A * F )

Thank you in advance for any helpful idea...

Gottfried

P.s.

A bit more context, possibly helpful to suggest an idea:

I get it as precept of coefficients for composition with the columnvector

        E = e * [1 , e, e^2, e^3, ...]         where   e=exp(1)

and the diagonalvector

        F = [0!, 1!, 2!, 3! , ...]


by
        C = (F^-1 * A * F) * E