A004247 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).

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

Offset: 0

Author

David W. Wilson

Keywords

• tabl
• nonn
• easy
• nice

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

Examples

			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)
		

Links

• T. D. Noe, Rows n = 0..50 of triangle, flattened
• Dennis Walsh, Variance bounds on binary data sets

Crossrefs

See A003991 for another version with many more comments.

Cf. A002262, A025581, A003056, A004197, A003984, A048720, A325820, A000292 (row sums of triangle), A002620.

Programs

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 *)

PARI

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

Formula

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

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)

Extensions

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