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A004247
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Multiplication table read by antidiagonals: T(i,j) = ij (i>=0, j>=0).
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14
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0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 12, 15, 16, 15, 12, 7, 0, 0, 8, 14, 18, 20, 20, 18, 14, 8, 0, 0, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 0, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0, 0, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| Table of xy, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...
Or, triangle read by rows, in which row n gives the numbers 0, n*1, (n-1)*2, (n-2)*3, ..., 2*(n-1), 1*n, 0.
Letting T(n,k) be the (k+1)st entry in the (n+1)st row (same numbering used for Pascal's triangle), T(n,k) is the dimension of the space of all k-dimensional subspaces of a (fixed) n-dimensional real vector space. - Paul Boddington (psb(AT)maths.warwick.ac.uk), Oct 21 2003
Contribution from Dennis P. Walsh (dwalsh(AT)mtsu.edu), Nov 10 2009: (Start)
Triangle T(n,k), 0<=k<=n, equals n^2 x the variance of a binary data set with k zeros and (n-k) ones. [For the case when n=0, let the variance of the empty set be defined as 0.]
T(n,k) is also the number of ways to form an opposite-sex dance couple from k women and (n-k) men. (End)
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LINKS
| T. D. Noe, Rows n=0..50 of triangle, flattened
Dennis Walsh, Variance bounds on binary data sets, [From Dennis P. Walsh (dwalsh(AT)mtsu.edu), Nov 10 2009]
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FORMULA
| a(n) = (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n) * (n-((trinv(n)*(trinv(n)-1))/2)); # A002262[ n ]*A025581[ n ] - Antti Karttunen
T(n,k)=k(n-k) for 0<=k<=n. [From Dennis P. Walsh (dwalsh(AT)mtsu.edu), Nov 10 2009]
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EXAMPLE
| 0; 0,0; 0,1,0; 0,2,2,0; 0,3,4,3,0; 0,4,6,6,4,0,; 0,5,8,9,8,5,0; ...
Contribution from Dennis P. Walsh (dwalsh(AT)mtsu.edu), Nov 10 2009: (Start)
For example, T(5,2)=6 since the variance of the data set <0,0,1,1,1> equals 6/25.
For example, T(5,2)=6 since, with 2 women, say Alice and Betty, and with 3 men, say Charles, Dennis, and Ed, the dance couple is one of the following: {Alice, Charles}, {Alice, Dennis}, {Alice, Ed}, {Betty, Charles}, {Betty, Dennis} and {Betty, Ed}. (End)
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MAPLE
| seq(seq(k*(n-k), k=0..n), n=0..13); [From Dennis P. Walsh (dwalsh(AT)mtsu.edu), Nov 10 2009]
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MATHEMATICA
| Table[(x - y) y, {x, 0, 13}, {y, 0, x}] // Flatten (* Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 06 2007 *)
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CROSSREFS
| See A003991 for another version with many more comments.
Cf. A048720, A003056.
Sequence in context: A048720 A067138 A059692 * A014473 A101164 A062275
Adjacent sequences: A004244 A004245 A004246 * A004248 A004249 A004250
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KEYWORD
| tabl,nonn,easy,nice
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AUTHOR
| David W. Wilson (davidwwilson(AT)comcast.net)
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Sep 30 2007
Corrected url address for link to "Variance bounds on binary data sets" Dennis Walsh (dwalsh(AT)mtsu.edu), Nov 11 2009
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