Unification of Eulerians and Binomials

The most obvious recurrence relation satisfied by the binomial 
coefficients is the familiar

            C(m,n)  =  C(m-1,n)  +  C(m-1,n-1)

Somewhat less familiar is the recurrence

            C(m,n)  =  (n+1) C(m+1,n)  -  (m+2) C(m,n-1)

The nice thing about this formula is that it generates the Eulerian 
Numbers as well as the binomial coefficients, so that both sets of 
numbers can be regarded as simply different regions of a single array.  
Here is a brief table to illustrate:

                  The Binomial & Eulerian Numbers

                                       n
      -1      0       1      2      3      4       5       6       7
 m
-8     0      1     120    4293   88234 1310354
-7     0      1      57    1191   15619  156190 1310354
-6     0      1      26     302    2416   15619   88234  455192
-5     0      1      11      66     302    1191    4293   14608   47840
-4     0      1       4      11      26      57     120     247     502
-3     1      1       1       1       1       1       1       1       1
-2     0      0       0       0       0       0       0       0       0
-1     1      0       0       0       0       0       0       0       0
 0     0      1       0       0       0       0       0       0       0
 1     0      1       1       0       0       0       0       0       0
 2     0      1       2       1       0       0       0       0       0
 3     0      1       3       3       1       0       0       0       0
 4     0      1       4       6       4       1       0       0       0
 5     0      1       5      10      10       5       1       0       0
 6     0      1       6      15      20      15       6       1       0
 7     0      1       7      21      35      35      21       7       1