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
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).
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).