OFFSET
0,8
COMMENTS
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
LINKS
T. D. Noe, Rows n = 0..50 of triangle, flattened
Dennis Walsh, Variance bounds on binary data sets
FORMULA
From Ridouane Oudra, Dec 14 2019: (Start)
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
From Stefano Spezia, Aug 19 2024: (Start)
G.f. as array: x*y/((1 - x)^2*(1 - y)^2).
E.g.f. as array: exp(x+y)*x*y. (End)
EXAMPLE
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)
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
KEYWORD
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, Sep 30 2007
STATUS
approved