A346905 -id:A346905 - cp's OEIS frontend

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

Peter Kagey, Aug 06 2021

			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).

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

Cf. A346905.

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

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

A346911 Triangle read by rows: T(n,k) is the number of k-dimensional simplices with vertices from the n-dimensional cross polytope; 0 <= k < n.

Original entry on oeis.org

2, 4, 6, 6, 15, 8, 8, 28, 32, 16, 10, 45, 80, 80, 32, 12, 66, 160, 240, 192, 64, 14, 91, 280, 560, 672, 448, 128, 16, 120, 448, 1120, 1792, 1792, 1024, 256, 18, 153, 672, 2016, 4032, 5376, 4608, 2304, 512

Offset: 1

Peter Kagey, Aug 06 2021

			Table begins:
n\k |  0    1    2     3     4     5     6     7    8
----+-------------------------------------------------
  1 |  2
  2 |  4,   6
  3 |  6,  15,   8
  4 |  8,  28,  32,   16
  5 | 10,  45,  80,   80,   32
  6 | 12,  66, 160,  240,  192,   64
  7 | 14,  91, 280,  560,  672,  448,  128
  8 | 16, 120, 448, 1120, 1792, 1792, 1024,  256
  9 | 18, 153, 672, 2016, 4032, 5376, 4608, 2304, 512
Three of the T(3,1) = 15 1-simplices (line segments) in the 3-dimensional cross-polytope have vertices {(1,0,0), (-1,0,0)}, {(1,0,0), (0,1,0)}, and {(0,1,0), (0,0,-1)}.
One of the T(5,3) = 80 of the 3-simplices (tetrahedra) in the 5-dimensional cross-polytope has vertices {(1,0,0,0,0), (0,0,1,0,0), (0,0,0,-1,0), (0,0,0,0,1)}.

Cf. A013609, A346905.

T(n,0) = 2*n;
T(n,1) = 2*n^2-n;
T(n,k) = A013609(n,k+1) when k > 1.

cp's OEIS Frontend

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.

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