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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.

%I A346906 #15 Apr 18 2022 12:30:08
%S A346906 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 A346906 127,196,140,35,21,7,1,1,255,553,476,175,56,28,8,1,1,511,1569,1624,
%U A346906 1071,126,84,36,9,1,1,1023,4476,6070,4935,1197,210,120,45,10,1
%N 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.
%F A346906 T(n,k) = A346905(n,k)/2^(n-k).
%e A346906 Triangle begins:
%e A346906 n\k | 0     1     2     3     4     5    6    7   8   9
%e A346906 ----+--------------------------------------------------
%e A346906   0 | 1;
%e A346906   1 | 1,    1;
%e A346906   2 | 1,    3,    1;
%e A346906   3 | 1,    7,    3,    1;
%e A346906   4 | 1,   15,    9,    4,    1;
%e A346906   5 | 1,   31,   25,   10,    5,    1;
%e A346906   6 | 1,   63,   70,   35,   15,    6,   1;
%e A346906   7 | 1,  127,  196,  140,   35,   21,   7,   1;
%e A346906   8 | 1,  255,  553,  476,  175,   56,  28,   8,  1;
%e A346906   9 | 1,  511, 1569, 1624, 1071,  126,  84,  36,  9,  1
%e A346906 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 A346906 (0,0,0,0,0,0,0);
%e A346906 (1,1,0,0,0,0,0);
%e A346906 (0,0,1,0,0,1,0);
%e A346906 (0,0,0,0,1,0,1);
%e A346906 (1,1,1,0,0,1,0);
%e A346906 (1,1,0,0,1,0,1);
%e A346906 (0,0,1,0,1,1,1); and
%e A346906 (1,1,1,0,1,1,1).
%t A346906 T[n_, 0] := 1
%t A346906 T[n_, k_] := Sum[n!/(k!*(i!)^k*(n - i*k)!), {i, 1, n/k}]
%Y A346906 Columns: A000012 (k=0), A000225 (k=1), A097861 (k=2), A344559 (k=3).
%Y A346906 Cf. A346905.
%K A346906 nonn,tabl
%O A346906 0,5
%A A346906 _Peter Kagey_, Aug 06 2021