A346906 - OEIS

%I #15 Apr 18 2022 12:30:08

%S 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,

%T 127,196,140,35,21,7,1,1,255,553,476,175,56,28,8,1,1,511,1569,1624,

%U 1071,126,84,36,9,1,1,1023,4476,6070,4935,1197,210,120,45,10,1

%N 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.

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

%e Triangle begins:

%e n\k | 0     1     2     3     4     5    6    7   8   9

%e ----+--------------------------------------------------

%e   0 | 1;

%e   1 | 1,    1;

%e   2 | 1,    3,    1;

%e   3 | 1,    7,    3,    1;

%e   4 | 1,   15,    9,    4,    1;

%e   5 | 1,   31,   25,   10,    5,    1;

%e   6 | 1,   63,   70,   35,   15,    6,   1;

%e   7 | 1,  127,  196,  140,   35,   21,   7,   1;

%e   8 | 1,  255,  553,  476,  175,   56,  28,   8,  1;

%e   9 | 1,  511, 1569, 1624, 1071,  126,  84,  36,  9,  1

%e 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:

%e (0,0,0,0,0,0,0);

%e (1,1,0,0,0,0,0);

%e (0,0,1,0,0,1,0);

%e (0,0,0,0,1,0,1);

%e (1,1,1,0,0,1,0);

%e (1,1,0,0,1,0,1);

%e (0,0,1,0,1,1,1); and

%e (1,1,1,0,1,1,1).

%t T[n_, 0] := 1

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

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

%Y Cf. A346905.

%K nonn,tabl

%O 0,5

%A _Peter Kagey_, Aug 06 2021