A346906 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, where one of the vertices is the origin; 0 <= k <= n.

1, 1, 1, 1, 3, 1, 1, 7, 3, 1, 1, 15, 9, 4, 1, 1, 31, 25, 10, 5, 1, 1, 63, 70, 35, 15, 6, 1, 1, 127, 196, 140, 35, 21, 7, 1, 1, 255, 553, 476, 175, 56, 28, 8, 1, 1, 511, 1569, 1624, 1071, 126, 84, 36, 9, 1, 1, 1023, 4476, 6070, 4935, 1197, 210, 120, 45, 10, 1

Offset: 0

Author

Peter Kagey, Aug 06 2021

Keywords

• nonn
• tabl

Examples

			Triangle begins:
n\k | 0     1     2     3     4     5    6    7   8   9
----+--------------------------------------------------
  0 | 1;
  1 | 1,    1;
  2 | 1,    3,    1;
  3 | 1,    7,    3,    1;
  4 | 1,   15,    9,    4,    1;
  5 | 1,   31,   25,   10,    5,    1;
  6 | 1,   63,   70,   35,   15,    6,   1;
  7 | 1,  127,  196,  140,   35,   21,   7,   1;
  8 | 1,  255,  553,  476,  175,   56,  28,   8,  1;
  9 | 1,  511, 1569, 1624, 1071,  126,  84,  36,  9,  1
One of the T(7,3) = 140 ways of choosing a 3-cube from the vertices of a 7-cube where one of the vertices is the origin is the cube with the following eight points:
(0,0,0,0,0,0,0);
(1,1,0,0,0,0,0);
(0,0,1,0,0,1,0);
(0,0,0,0,1,0,1);
(1,1,1,0,0,1,0);
(1,1,0,0,1,0,1);
(0,0,1,0,1,1,1); and
(1,1,1,0,1,1,1).
		

Crossrefs

Columns: A000012 (k=0), A000225 (k=1), A097861 (k=2), A344559 (k=3).

Cf. A346905.

Programs

Mathematica

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

Formula

T(n,k) = A346905(n,k)/2^(n-k).