A071223 Triangle T(n,k) (n >= 2, 1 <= k <= n) read by rows: number of linearly inducible orderings of n points in k-dimensional Euclidean space.

2, 2, 6, 2, 12, 24, 2, 20, 72, 120, 2, 30, 172, 480, 720, 2, 42, 352, 1512, 3600, 5040, 2, 56, 646, 3976, 14184, 30240, 40320, 2, 72, 1094, 9144, 45992, 143712, 282240, 362880, 2, 90, 1742, 18990, 128288, 557640, 1575648, 2903040, 3628800, 2, 110, 2642, 36410, 318188, 1840520, 7152048, 18659520, 32659200, 39916800

Offset

2,1

Comments

This can also be regarded as the lower triangular part of an infinite square array - see Example section and A198889.

Second and third columns are A002378 and A087645.

Links

Table of n, a(n) for n=2..56.

T. M. Cover, The number of linearly inducible orderings of points in d-space, SIAM J. Applied Math., 15 (1967), 434-439.

Formula

T(n, 1) = 2 for n >= 2, T(2, k) = 2 for k >= 1, T(n+1, k) = T(n, k) + n*T(n, k-1). Also T(n, k) = n! for k >= n-1.

Example

Triangle begins:
  2
  2  6
  2 12  24
  2 20  72 120
  2 30 172 480 720
  ...
This triangle is the lower triangular part of a square array which begins
  2   2   2   2   2 ...
  2   6   6   6   6 ...
  2  12  24  24  24 ...
  2  20  72 120 120 ...
  2  30 172 480 720 ...
  ...

Maple

T:=proc(n, k) if k>=n then 0 elif k=1 and n>=2 then 2 elif n=2 and k>=1 then 2 elif k=n-1 then n! else T(n-1, k)+(n-1)*T(n-1, k-1) fi end:seq(seq(T(n, k), k=1..n-1), n=2..12);

Mathematica

T[n_ /; n >= 2, 1] = 2; T[2, k_ /; k >= 1] = 2;
T[n_, k_] := T[n, k] = T[n-1, k] + (n-1)*T[n-1, k-1];
T[n_, k_] /; k >= n-1 = n!;
Flatten[Table[T[n, k], {n, 2, 11}, {k, 1, n-1}]]  (* Jean-François Alcover, Apr 27 2011 *)

Crossrefs

Cf. A087644, A002378, A087645.

Sequence in context: A286374 A284012 A278243 * A249631 A055934 A096217

Adjacent sequences: A071220 A071221 A071222 * A071224 A071225 A071226

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane, Oct 26 2003

Extensions

More terms from Emeric Deutsch, May 24 2004

Status

approved