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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.
2
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
0,2
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).
EXAMPLE
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).
MATHEMATICA
T[n_, 0] := 2^n
T[n_, k_] := 2^(n - k)*Sum[n!/(k!*(i!)^k*(n - i*k)!), {i, 1, n/k}]
CROSSREFS
Columns: A000079 (k=0), A006516 (k=1), A345340 (k=2).
Cf. A346906.
Sequence in context: A114192 A114656 A294440 * A075497 A158983 A261642
KEYWORD
nonn,tabl
AUTHOR
Peter Kagey, Aug 06 2021
STATUS
approved