Volumes of polytopes in hypercubes

[N. J. A. Sloane]

Roland said:

It would be interesting to know them:
The first value $n! Vol(P(0))=n! Vol(P(n-1))$ is always one.
The triangle of these Volumes starts thus

 1:               1
 2:            1     1
 3:         1     4     1
 4:      1    11    11     1
 5:   1     x     y     x     1

where 2+2x+y=120 and y/120 is the volume of the convex hull
P(2)=P(2)_5 of the 20 integral points with coordinates in {0,1}^5 and 
coordinate sum either 2 or 3. (Remark that this polytope is symmetric
with respect to central symmetry in its barycenter 1/5(1,1,1,1,1),
this holds of course for all "middle" polytopes P(n)_{2n+1}
in odd dimension 2n+1).

Me:  Looks like the Eulerian numbers, A008292 !

NJAS