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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 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-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

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