Matrixexponential and Bernoulli Numbers (I)

[Gottfried Helms]

Hi,
 a time ago I posted already my amazing discovery

          exp("counting") = "binomial-Matrix"

 As ascii:
 "Counting" :
     	| 0         0         0         0         0         0         0         0         0        |
     	| 1         0         0         0         0         0         0         0         0        |
     	| 0         2         0         0         0         0         0         0         0        |
     	| 0         0         3         0         0         0         0         0         0        |
     	| 0         0         0         4         0         0         0         0         0        |
     	| 0         0         0         0         5         0         0         0         0        |
     	| 0         0         0         0         0         6         0         0         0        |
     	| 0         0         0         0         0         0         7         0         0        |
     	| 0         0         0         0         0         0         0         8         0        |

  "binomial"  = exp("counting")
     	| 1         0         0         0         0         0         0         0         0        |
     	| 1         1         0         0         0         0         0         0         0        |
     	| 1         2         1         0         0         0         0         0         0        |
     	| 1         3         3         1         0         0         0         0         0        |
     	| 1         4         6         4         1         0         0         0         0        |
     	| 1         5         10        10        5         1         0         0         0        |
     	| 1         6         15        20        15        6         1         0         0        |
     	| 1         7         21        35        35        21        7         1         0        |
     	| 1         8         28        56        70        56        28        8         1        |

===== Bernoulli-Zahlen ==========================================================================

 Of even more interest may the observation be, that this
 matrix is a simple basis to construct the Bernoulli-numbers.

 I extract the diagonal and get
 "Binom_Reduz"
     	| 1         0         0         0         0         0         0         0        |
     	| 1         2         0         0         0         0         0         0        |
     	| 1         3         3         0         0         0         0         0        |
     	| 1         4         6         4         0         0         0         0        |
     	| 1         5         10        10        5         0         0         0        |
     	| 1         6         15        20        15        6         0         0        |
     	| 1         7         21        35        35        21        7         0        |
     	| 1         8         28        56        70        56        28        8        |

If I compute from this the inverse, I get in the first column
the bernoulli-nubers, and additionally in the following
columns an interesting partitioning of that numbers,
which may be helpful to detect more properties of that
bernoulli-numbers:

          Bernoulli  |
          nubers     |                 rowsum + Bernoullinubers = 0 for rowindexes>1
        -------------+--------------------------------------------------------------------|
     	|   1.0000   | 0.0000    0.0000    0.0000   -0.0000   -0.0000   -0.0000    0.0000 |
     	|  -0.5000   | 0.5000   -0.0000   -0.0000    0.0000    0.0000   -0.0000    0.0000 |
     	|   0.1667   |-0.5000    0.3333   -0.0000   -0.0000   -0.0000    0.0000   -0.0000 |
     	|   0.0000   | 0.2500   -0.5000    0.2500    0.0000    0.0000   -0.0000    0.0000 |
     	|  -0.0333   |-0.0000    0.3333   -0.5000    0.2000   -0.0000    0.0000    0.0000 |
     	|   0.0000   |-0.0833   -0.0000    0.4167   -0.5000    0.1667   -0.0000   -0.0000 |
     	|   0.0238   |-0.0000   -0.1667    0.0000    0.5000   -0.5000    0.1429   -0.0000 |
     	|   0.0000   | 0.0833    0.0000   -0.2917    0.0000    0.5833   -0.5000    0.1250 |

I didn't find a good definition for the structure of these coefficients
of partitioning; it seems, that along the subdiagonals they form poly-
nomial sequences of a degree according to their distance to the
main diagonal (only the subdiagonals with odd index:
 (1/2,1/3,1/4,1/5...; (-1/2,-1/2,...) , ) but I didn't manage to find the
polynomial coefficients so far.
Perhaps someone may find them more easily?

Regards -

Gottfried