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A004247
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Multiplication table read by antidiagonals: T(i,j) = i*j (i>=0, j>=0). Alternatively, multiplication triangle read by rows: P(i,j) = j*(i-j) (i>=0, 0<=j<=i).
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27
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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
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OFFSET
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0,8
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COMMENTS
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Table of x*y, 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, Oct 21 2003
From Dennis P. Walsh, Nov 10 2009: (Start)
Triangle P(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.]
P(n,k) is also the number of ways to form an opposite-sex dance couple from k women and (n-k) men. (End)
P(n,k) is the number of negative products of two numbers from a set of n real numbers, k of which are negative. - Logan Pipes, Jul 08 2021
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LINKS
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T. D. Noe, Rows n = 0..50 of triangle, flattened
Dennis Walsh, Variance bounds on binary data sets
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FORMULA
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a(n) = A002262(n) * A025581(n). - Antti Karttunen
From Ridouane Oudra, Dec 14 2019: (Start)
a(n) = A004197(n)*A003984(n).
a(n) = (3/4 + n)*t^2 - (1/4)*t^4 - (1/2)*t - n^2 - n, where t = floor(sqrt(2*n+1)+1/2). (End)
P(n,k) = (P(n-1,k-1) + P(n-1,k) + n) / 2. - Robert FERREOL, Jan 16 2020
P(n,floor(n/2)) = A002620(n). - Logan Pipes, Jul 08 2021
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EXAMPLE
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As the triangle P, sequence begins:
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;
...
From Dennis P. Walsh, Nov 10 2009: (Start)
P(5,2)=T(2,3)=6 since the variance of the data set <0,0,1,1,1> equals 6/25.
P(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
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seq(seq(k*(n-k), k=0..n), n=0..13); # Dennis P. Walsh, Nov 10 2009
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MATHEMATICA
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Table[(x - y) y, {x, 0, 13}, {y, 0, x}] // Flatten (* Robert G. Wilson v, Oct 06 2007 *)
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PROG
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(PARI) T(i, j)=i*j \\ Charles R Greathouse IV, Jun 23 2017
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CROSSREFS
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See A003991 for another version with many more comments.
Cf. A002262, A025581, A003056, A004197, A003984, A048720, A325820, A000292 (row sums of triangle).
Sequence in context: A059692 A353109 A336225 * A271916 A327031 A014473
Adjacent sequences: A004244 A004245 A004246 * A004248 A004249 A004250
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KEYWORD
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tabl,nonn,easy,nice
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AUTHOR
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David W. Wilson
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EXTENSIONS
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Edited by N. J. A. Sloane, Sep 30 2007
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STATUS
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approved
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