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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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..48.

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}]][[1 ;; 47]]  (* Jean-Fran├žois 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

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Last modified August 10 02:02 EDT 2020. Contains 336365 sequences. (Running on oeis4.)