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%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
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