

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
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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, (n1)*2, (n2)*3, ..., 2*(n1), 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 kdimensional subspaces of a (fixed) ndimensional 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 (nk) 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 oppositesex dance couple from k women and (nk) 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(nk) 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(n1,k1) + T(n1,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*(nk), 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



