A346905 Triangle read by rows: T(n,k) is the number of ways of choosing a k-dimensional cube from the vertices of an n-dimensional hypercube; 0 <= k <= n.

1, 2, 1, 4, 6, 1, 8, 28, 6, 1, 16, 120, 36, 8, 1, 32, 496, 200, 40, 10, 1, 64, 2016, 1120, 280, 60, 12, 1, 128, 8128, 6272, 2240, 280, 84, 14, 1, 256, 32640, 35392, 15232, 2800, 448, 112, 16, 1, 512, 130816, 200832, 103936, 34272, 2016, 672, 144, 18, 1

Offset: 0

Author

Peter Kagey, Aug 06 2021

Keywords

• nonn
• tabl

Examples

			Table begins:
n\k |   0       1       2       3      4     5    6    7   8  9
----+----------------------------------------------------------
  0 |   1;
  1 |   2,      1;
  2 |   4,      6,      1;
  3 |   8,     28,      6,      1;
  4 |  16,    120,     36,      8,     1;
  5 |  32,    496,    200,     40,    10,    1;
  6 |  64,   2016,   1120,    280,    60,   12,   1;
  7 | 128,   8128,   6272,   2240,   280,   84,  14,   1;
  8 | 256,  32640,  35392,  15232,  2800,  448, 112,  16,  1;
  9 | 512, 130816, 200832, 103936, 34272, 2016, 672, 144, 18, 1
One of the T(7,3) = 2240 ways of choosing a 3-cube from the vertices of a 7-cube is the cube with the following eight points:
(0,0,0,1,1,1,0);
(1,1,0,1,1,1,0);
(0,0,1,1,1,0,0);
(0,0,0,1,0,1,1);
(1,1,1,1,1,0,0);
(1,1,0,1,0,1,1);
(0,0,1,1,0,0,1); and
(1,1,1,1,0,0,1).
		

Crossrefs

Columns: A000079 (k=0), A006516 (k=1), A345340 (k=2).

Cf. A346906.

Programs

Mathematica

T[n_, 0] := 2^n
T[n_, k_] := 2^(n - k)*Sum[n!/(k!*(i!)^k*(n - i*k)!), {i, 1, n/k}]

Formula

T(n,0) = 2^n.

T(n,k) = 2^(n-k) * Sum_{i=1..floor(n/k)} n!/(k!*(i!)^k*(n-i*k)!).

T(n,k) = 2^(n-k) * A346906(n,k).