Matrixexponential and Normaldistribution/Gauss' function (II)

[Paul D. Hanna]

Gottfried (and Seqfans),
> With another important known function the matrix-exponential
> can be associated to provide an extremely simple structure.
     Yes, the diagonals of a triangle are directly related to the 
e.g.f. of the columns of the matrix exponential of that triangle. 
 
If we construct a lower triangular matrix T in the following way: 
    T(k,k)=0 (zeros in main diagonal);
    T(n+k,k) = d(k)*C(n+k,k) for k>0  (k-th secondary diagonal)
                      for any arbitrary sequence of coefficients d(k), 
Then exp(T) = E  is given by: 
     E(n,k) = e(n-k)*C(n,k) 
where {e(n)| n>=0} has e.g.f.: 
     Sum_{n>=0} e(n)/n!*x^n = Sum_{n>=1} d(n)*x^n/n! 
         = exp( d(1)*x + d(2)*x^2/2! + d(3)*x^3/3! +...)
Thus, each k-th diagonal of T has its contribution simply as d(k)*x^k/k! 
in the e.g.f. of column 0. 
 
I am sure there is a simple formula for the e.g.f. of T and E 
when they have the above form (using binomial coefficients). 
 
There are various examples of this in the OEIS, as you have observed. 
 
Paul