Volumes of polytopes in hypercubes

[Roland Bacher]

On Fri, Aug 18, 2006 at 09:13:00AM -0400, N. J. A. Sloane wrote:
> 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

The evidence is not overhelming. I would like to have
at least three more rows (n=5,6,7) before betting any 
valuable body-parts.    Roland