Generating Function For Eulerian Numbers

The first few rows of Eulerian Numbers are

                         1
                       1   1
              m      1   4   1      n
                   1  11  11   1
                 1  26  66  26   1
               1  57 302 302  57   1
                       etc

These number have a great many interesting properties, and they can
be generated by the simple recurrence E[m,n] = nE[m-1,n] + mE[m,n-1].
However, I never noticed before that they also have the nice generating
function
                           1      / xe^(-yt) - ye^(-xt) \
          f(x,y,t)   =   ----- ln( --------------------- )
                          xyt     \       x - y         /

The coefficient of  x^(m-1) y^(n-1) (-t)^(m+n-1) / (m+n)!  is
the Eulerian number E[m,n], so we have

                 1         x + y           x^2 + 4xy + y^2
 f(x,y,t)  =  - --- t  +  ------- t^2  -  ----------------- t^3  + ...
                 2           6                   24

Is there a simple procedure for deducing the generating function (if
one exists) for any given array of numbers?