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%I A004247 #69 Aug 25 2024 18:48:22
%S A004247 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,
%T A004247 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,
%U A004247 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
%N 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).
%C A004247 Table of x*y, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...
%C A004247 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.
%C A004247 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
%C A004247 From _Dennis P. Walsh_, Nov 10 2009: (Start)
%C A004247 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.]
%C A004247 P(n,k) is also the number of ways to form an opposite-sex dance couple from k women and (n-k) men. (End)
%C A004247 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
%H A004247 T. D. Noe, <a href="/A004247/b004247.txt">Rows n = 0..50 of triangle, flattened</a>
%H A004247 Dennis Walsh, <a href="https://web.archive.org/web/20150911024909/http://capone.mtsu.edu/dwalsh/VBOUND2.pdf">Variance bounds on binary data sets</a>
%F A004247 a(n) = A002262(n) * A025581(n). - _Antti Karttunen_
%F A004247 From _Ridouane Oudra_, Dec 14 2019: (Start)
%F A004247 a(n) = A004197(n)*A003984(n).
%F A004247 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)
%F A004247 P(n,k) = (P(n-1,k-1) + P(n-1,k) + n) / 2. - _Robert FERREOL_, Jan 16 2020
%F A004247 P(n,floor(n/2)) = A002620(n). - _Logan Pipes_, Jul 08 2021
%F A004247 From _Stefano Spezia_, Aug 19 2024: (Start)
%F A004247 G.f. as array: x*y/((1 - x)^2*(1 - y)^2).
%F A004247 E.g.f. as array: exp(x+y)*x*y. (End)
%e A004247 As the triangle P, sequence begins:
%e A004247   0;
%e A004247   0,0;
%e A004247   0,1,0;
%e A004247   0,2,2,0;
%e A004247   0,3,4,3,0;
%e A004247   0,4,6,6,4,0,;
%e A004247   0,5,8,9,8,5,0;
%e A004247   ...
%e A004247 From _Dennis P. Walsh_, Nov 10 2009: (Start)
%e A004247 P(5,2)=T(2,3)=6 since the variance of the data set <0,0,1,1,1> equals 6/25.
%e A004247 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)
%p A004247 seq(seq(k*(n-k),k=0..n),n=0..13); # _Dennis P. Walsh_, Nov 10 2009
%t A004247 Table[(x - y) y, {x, 0, 13}, {y, 0, x}] // Flatten (* _Robert G. Wilson v_, Oct 06 2007 *)
%o A004247 (PARI) T(i,j)=i*j \\ _Charles R Greathouse IV_, Jun 23 2017
%Y A004247 See A003991 for another version with many more comments.
%Y A004247 Cf. A002262, A025581, A003056, A004197, A003984, A048720, A325820, A000292 (row sums of triangle), A002620.
%K A004247 tabl,nonn,easy,nice
%O A004247 0,8
%A A004247 _David W. Wilson_
%E A004247 Edited by _N. J. A. Sloane_, Sep 30 2007

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