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A071223
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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.
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4
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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
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OFFSET
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2,1
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COMMENTS
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This can also be regarded as the lower triangular part of an infinite square array - see Example section and A198889.
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REFERENCES
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T. M. Cover, The number of linearly inducible orderings of points in d-space, SIAM J. Applied Math., 15 (1967), 434-439.
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LINKS
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Table of n, a(n) for n=2..48.
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FORMULA
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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.
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EXAMPLE
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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 ...
....
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MAPLE
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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);
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MATHEMATICA
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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]] (* From Jean-François Alcover, Apr 27 2011 *)
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CROSSREFS
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Cf. A087644. Second and third columns = A002378, A087645.
Sequence in context: A110765 A176991 A091818 * A055934 A096217 A098555
Adjacent sequences: A071220 A071221 A071222 * A071224 A071225 A071226
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KEYWORD
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nonn,tabl,easy,nice
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AUTHOR
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N. J. A. Sloane, Oct 26 2003
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EXTENSIONS
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More terms from Emeric Deutsch, May 24 2004
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STATUS
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approved
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