0,8

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)

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]

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]

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)

seq(seq(k*(n-k), k=0..n), n=0..13); [From Dennis P. Walsh (dwalsh(AT)mtsu.edu), Nov 10 2009]

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

See A003991 for another version with many more comments.

Cf. A048720, A003056.

Sequence in context: A048720 A067138 A059692 this_sequence A014473 A101164 A062275

Adjacent sequences: A004244 A004245 A004246 this_sequence A004248 A004249 A004250

tabl,nonn,easy,nice,new

David W. Wilson (davidwwilson(AT)comcast.net)

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