<pre>
Take any vertex of the hypercube, and ask how many k-V's it
participates in.  To make a k-V it needs to combine with k adjacent and
orthogonal vertices, and there are (nCk) distinct ways of doing this
(that is, choose k directions out of n possible ones).  Then multiply
by 2^n, the total number of vertices.  But this involves multiple
counting, since each k-V is shared by 2^k vertices.  So divide by 2^k,
and this yields the answer: (nCk)*2^{n-k}.


For example, 12d hypercube:

  0-v:   4,096
  1-v:  24,576
  2-v:  67,584
  3-v: 112,640
  4-v: 126,720
  5-v: 101,376
  6-v:  59,136
  7-v:  25,344
  8-v:   7,920
  9-v:   1,760
 10-v:     264
 11-v:      24
 12-v:       1
</pre>
