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A004247 Multiplication table read by antidiagonals: T(i,j) = ij (i>=0, j>=0). 21
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; text; internal format)
OFFSET

0,8

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, Oct 21 2003

From Dennis P. Walsh, 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)

LINKS

T. D. Noe, Rows n=0..50 of triangle, flattened

Dennis Walsh, Variance bounds on binary data sets

FORMULA

a(n) = A002262(n) * A025581(n). - Antti Karttunen

T(n,k) = k(n-k) for 0<=k<=n. - Dennis P. Walsh, Nov 10 2009

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)

T(n,k) = (T(n-1,k-1) + T(n-1,k) + n) / 2. - Robert FERREOL, Jan 16 2020

EXAMPLE

As a triangle, 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)

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)

MAPLE

seq(seq(k*(n-k), k=0..n), n=0..13); # Dennis P. Walsh, Nov 10 2009

MATHEMATICA

Table[(x - y) y, {x, 0, 13}, {y, 0, x}] // Flatten (* Robert G. Wilson v, Oct 06 2007 *)

PROG

(PARI) T(i, j)=i*j \\ Charles R Greathouse IV, Jun 23 2017

CROSSREFS

See A003991 for another version with many more comments.

Cf. A002262, A025581, A003056, A004197, A003984, A048720, A325820.

Sequence in context: A067138 A059692 A336225 * A271916 A327031 A014473

Adjacent sequences:  A004244 A004245 A004246 * A004248 A004249 A004250

KEYWORD

tabl,nonn,easy,nice

AUTHOR

David W. Wilson

EXTENSIONS

Edited by N. J. A. Sloane, Sep 30 2007

STATUS

approved

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Last modified October 24 10:53 EDT 2020. Contains 337975 sequences. (Running on oeis4.)